research article symbolic solution to complete ordinary...

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 518194 15 pageshttpdxdoiorg1011552013518194

Research ArticleSymbolic Solution to Complete Ordinary Differential Equationswith Constant Coefficients

Juan F Navarro and Antonio Peacuterez-Carrioacute

Department of Mathematics University of Alicante Carretera San Vicente del Raspeig sn 03690 San Vicente del RaspeigAlicante Spain

Correspondence should be addressed to Juan F Navarro jfnavarrouaes

Received 7 May 2013 Accepted 17 July 2013

Academic Editor Debasish Roy

Copyright copy 2013 J F Navarro and A Perez-Carrio This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The aim of this paper is to introduce a symbolic technique for the computation of the solution to a complete ordinary differentialequation with constant coefficients The symbolic solution is computed via the variation of parameters method and thusconstructed over the exponential matrix of the linear system associated with the homogeneous equation This matrix is alsosymbolically determined The accuracy of the symbolic solution is tested by comparing it with the exact solution of a test problem

1 Introduction

Perturbation theories for differential equations containinga small parameter 120598 are quite old The small perturbationtheory originated by Sir Isaac Newton has been highlydeveloped by many others and an extension of this theoryto the asymptotic expansion consisting of a power seriesexpansion in the small parameter was devised by Poincare[1] The main point is that for the most of the differentialequations it is not possible to obtain an exact solutionIn cases where equations contain a small parameter wecan consider it as a perturbation parameter to obtain anasymptotic expansion of the solution In practice the workinvolved in the application of this approach to compute thesolution to a differential equation cannot be performed byhand and algebraic processors result in being a very usefultool

As explained inHenrard [2] the first symbolic processorswere developed toworkwith Poisson series that ismultivari-ate Fourier series whose coefficients are multivariate Laurentseries

sum

1198941119894119899

sum

1198951119895119898

1198621198951119895119898

1198941119894119899

1199091198941

1sdot sdot sdot 119909

119894119899

119899

cossin(119895

11206011+ sdot sdot sdot + 119895

119898120601119898) (1)

where 1198621198951 1198951198981198941119894119899

isin R 1198941 119894

119899 119895

1 119895

119898isin Z and 119909

1 119909

119899

and 1206011 120601

119898are called polynomial and angular variables

respectively These processors were applied to problemsin nonlinear mechanics or nonlinear differential equationproblems in the field of celestial mechanics One of the firstapplications of these processorwas concernedwith the theoryof the Moon Delaunay invented his perturbation methodto treat it and spent 20 years doing algebraic manipulationsby hand to apply it to the problem Deprit et al [3 4]prolongated the solution of Delaunayrsquos work with the helpof a special purpose symbolic processor and Henrard [5]pushed to order 25 This solution was improved by iterationby Chapront-Touze [6] and planetary perturbations werealso introduced by Chapront-Touze [7] At present themost complete solution Ephemeride Lunaire Parisien (ELP)contains more than 50 000 periodic terms But the motion ofthe Moon is not the only application of algebraic processorsThere are many problems where the facilities provided byPoisson series processors can lead rather quickly to veryaccurate results As examples we would like to mentionplanetary theories the theory of the rotation of the Earth(see eg Navarro and Ferrandiz [8]) and artificial satellitetheories

In order to achieve better accuracies in the applications ofanalytical theories high orders of the approximate solutionmust be computed making necessary a continuous main-tenance and revision of the existing symbolic manipulationsystems as well as the development of new packages adapted

2 Journal of Applied Mathematics

to the peculiarities of the problem to be treated RecentlyNavarro [9 10] developed a symbolic processor to deal withthe solution to second order differential equations of the form

+ 1198861 + 119886

0119909 = 119906 (119905) + 120598119891 (119909 ) (2)

with initial conditions

119909 (0) = 1199090 (0) = 0

(3)

where 1198860 119886

1 119909

0 and

0isin R 119906(119905) is a quasipolynomial that

is an object of the form

119906 (119905) = sum

]ge0

119905119899]119890120572]119905(120582] cos (120596]119905) + 120583] sin (120596]119905)) (4)

where 119899] isin N 120572] 120596] 120582] and 120583] isin R and 119891(119909 ) admits theexpansion

119891 (119909 ) =

119872

sum

120581=0

sum

0le]le120581

119891]120581minus]119909]120581minus] 119891]120581minus] isin R (5)

To face this problem the algebraic processor handles objectscalled quasipolynomials and it has resulted in being a usefultool in the computation of the solution to (2) by the appli-cation of the asymptotic expansion method A modificationof this processor has been employed to compute periodicsolutions in equations of type (2) via the Poincare-Lindstedtmethod in Navarro [9 10] To that end the idea is to expandboth the solution and the modified frequency with respectto the small parameter allowing to kill secular terms whichappear in the recursive scheme The elimination of secularterms is performed through a manipulation system whichworks with modified quasipolynomials that is quasipolyno-mials containing undetermined constants

119906 (119905) = sum

]ge0

1205911205901

1times sdot sdot sdot times 120591

120590119876

119876119905119899]119890120572]119905

times (120582] cos (120596]119905) + 120583] sin (120596]119905)) (6)

where 119899] isin N 120572] 120596] 120582] and 120583] isin R 120590] isin Z and 1205911 120591119876are real constants with unknown value

One year later Navarro [11] presented a symbolic com-putation package based on the object-oriented philosophyfor manipulating matrices whose elements lie on the set ofquasipolynomials of type (4) The kernel of the symbolicprocessor was developed in C++ defining a class for thisnew object as well as a set of functions that operate onthe data structure addition substraction differentiation andintegration with respect to 119905 substitution of an undeterminedcoefficient by a series and many others The goal of thisprocessor is to provide a tool to solve a perturbed 119899-orderdifferential equation of the class

119909(119899)+ 119886

119899minus1119909(119899minus1)+ sdot sdot sdot + 119886

0119909 = 119906 (119905) + 120598119891 (119909 119909

(119899minus1))

(7)


119909 (0) = 11990910 (0) = 11990920

119909(119899minus1)(0) = 1199091198990

(8)

where 120598 is a small real parameter 1198860 119886

1 119886

119899minus1isin R 119906(119905) is

a quasipolynomial and 119891 is such that

119891 (119909 119909(119899minus1)) = sum

0le]1]119899le119872

119891]1]119899

119909]1sdot sdot sdot (119909

(119899minus1))

]119899

(9)

with119872 isin N ]1 ]

119899isin N and 119891]

1]119899

isin RAs a first step Navarro and Perez [12] have developed

a symbolic technique for the computation of the principalmatrix of the linear system associated with a homogeneousordinary differential equation with constant coefficients ofthe form

119909(119899)+ 119886


1 + 119886

0119909 = 0 (10)

where 1198860 119886

1 119886

119899minus1isin R with initial conditions

119909 (0) = 11990910 (0) = 11990920

119909(119899minus1)(0) = 1199091198990

(11)

being 11990910 119909

1198990isin R This method provides a final

analytical solution which can be completely computed in asymbolic way

The second step in the construction of the solution to (7)is to obtain a symbolic procedure for computing the solutionto the nonhomogeneus equation

119909(119899)+ 119886


0119909 = 119906 (119905) (12)

where as above 119906(119905) is a quasipolynomialThis new symbolic package can be useful for research and

educational purposes in the field of differential equations ordynamical systems Nowadays there aremany open problemswhich requires massive symbolic computation as to cite oneexample the analytical theory of the resonant motion ofMercury We would like to stress that the aim of this work isto develop a symbolic tool not a numericmethod as numericsolutions cannot be used in perturbation methods for differ-ential equations However we have performed comparisonsbetween the symbolic solution with a numeric one just toshow the efficiency of the technique

In next section we summarize the scheme proposedin [12] to calculate the solution to (10) which is neededto express the solution to the complete problem (12) Thetechnique to construct this solution will be described inSection 3

2 Solution to the Homogeneous Problem

As mentioned Navarro and Perez [12] have proposed asymbolic method to compute the solution to (10)

119909(119899)+ 119886


1 + 119886

0119909 = 0 (13)

where 1198860 119886

1 119886


119909 (0) = 11990910 (0) = 11990920

119909(119899minus1)(0) = 1199091198990

(14)

being 11990910 119909



Journal of Applied Mathematics 3

21 Description of the Method With the aid of the substitu-tions

1199091= 119909 119909

2= 119909

119899= 119909

(119899minus1) (15)

(10) is transformed into the system of differential equationsgiven by

(119905) = 119860119883 (119905) 119883 (0) = 1198830 (16)

where 119860 is the companion matrix

119860 = (

0 1 0 sdot sdot sdot 0


d

minus1198860minus119886

1minus119886

2sdot sdot sdot minus119886

119899minus1

)

119883 (119905) = (

1199091 (119905)

1199092 (119905)

119909119899 (119905)

) 1198830= (

11990910

11990920

1199091198990

)

(17)

To compute the exponential of 119860 the matrix is splitted into119861 + 119862 where

119861 = (



d


)

119862 = (



d


1minus119886


119899minus1

)

(18)

Then we use the approximation

119890119860asymp (119890

119861119898119890119862119898)

119898

(19)

with119898 isin NThis approach to obtain 119890119860 is of potential interestwhen the exponentials of matrices 119861 and 119862 can be efficientlycomputed and requires 119886

119899minus1to be nonequal to zero In our

case

119890119861119898=(

1198920 (119898) 1198921 (

119898) 1198922 (119898) sdot sdot sdot 119892119899minus1 (

119898)

0 1198920 (119898) 1198921 (

119898) sdot sdot sdot 119892119899minus2 (119898)

0 0 1198920 (119898) sdot sdot sdot 119892119899minus3 (

119898)

d

0 0 0 sdot sdot sdot 1198920 (119898)

)

119890119862119898= 119868 minus

1

119886119899minus1

(119890minus119886119899minus1

119898minus 1)119862

=(




d

120596012059611205962sdot sdot sdot 120596

119899minus1

)

(20)

being for any 119899 isin N

119892119899 (119898) =

1

119899119898119899 (21)

and for any 119894 = 0 119899 minus 2

120596119894=

119886119894

119886119899minus1

(119890minus119886119899minus1

119898minus 1) 120596

119899minus1= 119890


119898 (22)

As it is established in Moler and van Loan [13] for a generalsplitting 119860 = 119861 + 119862119898 can be determine from

100381710038171003817100381710038171003817

119890119860minus (119890

119861119898119890119862119898)

119898100381710038171003817100381710038171003817

le

1

2119898

[119861 119862] 119890119861+119862

(23)

Thus with 119860 being the companion matrix we get that

100381710038171003817100381710038171003817

119890119860minus (119890

119861119898119890119862119898)

119898100381710038171003817100381710038171003817

le

1

119898

1198901+119886119886 (24)

where 119886 = (1198860 119886

1 119886

119899minus1) Here

119909 = 1199091=10038161003816100381610038161199091

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816119909119899

1003816100381610038161003816 (25)

for any 119909 = (1199091 119909

119899) isin R119899 and 119860 = max

1199091=1119860119909

1=

max](|1198861]| + sdot sdot sdot + |119886119899]|) for any matrix

119860 = (

1198861111988612sdot sdot sdot 119886

1119899

1198862111988622sdot sdot sdot 119886

2119899

d

11988611989911198861198992sdot sdot sdot 119886

119899119899

) (26)

22 Adaptation to a Symbolic Formalism Navarro and Perez[12 14] have proposed an adaptation of themethod describedabove in order to compute the matrix 119890119860119905 instead of 119890119860 If wedo so we obtain the principal matrix of (16) whose elementslie on the set of quasipolynomials and the symbolic processorresults in being suitable to work with those matrices Theapproach is to compute

119890119860119905≃ (119890

119861119905119898119890119862119905119898)

119898

(27)

taking into account that

119890119861119905119898= (

1198920 (119905 119898) 1198921 (

119905 119898) sdot sdot sdot 119892119899minus1 (119905 119898)

0 1198920 (119905 119898) sdot sdot sdot 119892119899minus2 (

119905 119898)

d

0 0 sdot sdot sdot 119892

0 (119905 119898)

)

119890119862119905119898=(




d

1205960 (119905) 1205961 (


119905)

)

(28)


119892119899 (119905 119898) =

119905119899

119899119898119899 (29)


and for any 119894 = 0 119899 minus 2

120596119894 (119905) =

119886119894

119886119899minus1

(119890minus119886119899minus1

119905119898minus 1) 120596

119899minus1 (119905) = 119890


119905119898

(30)

Thus 119890119860119905 is amatrix of quasipolynomials that can be com-pletely computed through the symbolic processor developedby Navarro [11] The procedure for the computation of theexponential matrix is substantially simplified by using thefollowing equation which avoids the symbolic multiplicationof 119890119861119905119898 and 119890119862119905119898

119890119861119905119898119890119862119905119898= 119890

119861119905119898+ 119867 (119905119898) (31)

where

119867(119905119898) =

1

119886119899minus1

119891 (119905 119898)(

119892119899minus1 (119905 119898)

1198920 (119905 119898)

) (11988601198861sdot sdot sdot 119886

119899minus1)

119891 (119905 119898) = 119890minus119886119899minus1

119905119898minus 1

(32)

This relation can be obtained directly from themultiplicationof matrices 119890119861119905119898 and 119890119862119905119898 as given in (28)

23 Symbolic Expansion of the Exponential Matrix As it hasbeen stated in Section 22 the exponential matrix 119890119860119905 can besymbolically calculated through (27) and (31)

119890119860119905≃ (119890

119861119905119898119890119862119905119898)

119898

= (119890119861119905119898+ 119867 (119905119898))

119898

(33)

This expression can be calculated symbolically Let us express119867(119905119898) as

119867(119905119898) = 120582 (119905 119898) 119869 (119905 119898) (34)

with

120582 (119905 119898) =

1

119886119899minus1

119891 (119905 119898)

119869 (119905 119898) = (

119892119899minus1 (119905 119898)

1198920 (119905 119898)

) (11988601198861sdot sdot sdot 119886

119899minus1)

(35)

Thus

(119890119861119905119898+ 119867 (119905119898))

119898

=

119898

sum

119896=0

120582119896(119905 119898) (119890

119861119905119898 119869 (119905 119898))

119898

119896

(36)

where (119860 119861)119898119896is the so-called noncommutative parenthesis

defined as

(119860 119861)119898

119896=

(119898

119896)

sum

119894=1

120590(119898119896)

prod

119895=1

(119860(1minus(minus1)

119895)2119861(1+(minus1)

119895)2)

119888119894119895

(37)

where 119860 and 119861 are two noncommutative 119899 times 119899 matrices119898 119896 119888

119894119895isin Z+cup 0

120590 (119898 119896) =

2119896 + 1 if 2119896 le 1198982 (119898 minus 119896 + 1) if 2119896 gt 119898

(38)

and 119888119894119895is determined by the following properties

(1) sum120590(119898119896)

119895=1119888119894119895= 119898 for all 119894 = 1 2 (119898119896 )

(2) sum[120590(119898119896)2]

119895=1119888119894(2119895)= 119898 minus 119896 for all 119894 = 1 2 (119898119896 ) with

[120590(119898 119896)2] being the entire part of 120590(119898 119896)2(3) 119888

1198942= 0 for all 119894 = 1 2 (119898119896 )

(4) if 119888119894119895= 0 and 119888

119894(119895+2)= 0 then 119888

119894(119895+1)= 0 for all 119894 =

1 2 (119898

119896 ) and 119895 = 1 2 120590(119898 119896) minus 2

In the following we summarize some expressions whichsimplify the way in which the matrix (119890119861119905119898 + 119867(119905119898))119898 issymbolically computed First let us introduce the followingmatrices

119878119899minus1 (119905 119898) = (1198860


119899minus1) 119865 (119905 119898)

119865 (119905 119898) =(

(

(

119905119899minus1

(119899 minus 1) 119905119899minus2119898

(119899 minus 1) (119899 minus 2) 119905119899minus3119898

2

(119899 minus 1)119905

0119898

119899minus1

)

)

)

(39)

Lemma 1 For any 119896 isin Z such that 119896 gt 1

(119890119861119905119898)

119896

=(

1198660119896 (119905 119898) 1198661119896 (



119905 119898)

d


0119896 (119905 119898)

)

(40)

with

119866]119896 (119905 119898) = 119896]119892] (119905 119898) (41)

for each ] = 0 119899 minus 1 where 119892](119905 119898) is given by (29)


(119867 (119905 119898))119901= (

119891 (119905 119898)

119886119899minus1 (119899 minus 1)119898

119899minus1)

119901

(119878119899minus1 (119905 119898))

119901minus1

times 119865 (119905 119898) (11988601198861sdot sdot sdot 119886

119899minus1)

(42)

where 119878119899minus1(119905 119898) and 119865(119905 119898) are given by (39)

Lemma 3 For any 119901 119896 isin Z such that 119901 119896 ge 1

(119890119861119905119898)

119896

(119867 (119905 119898))119901= (

119891 (119905 119898)

119886119899minus1 (119899 minus 1)119898

119899minus1)

119901

(119878119899minus1 (119905))

119901minus1

times Ω119865 (119905 119898) (11988601198861sdot sdot sdot 119886

119899minus1)

(43)


where 119878119899minus1(119905 119898) and 119865(119905 119898) are given by (39) and

Ω =(

(119896 + 1)119899minus1

0 sdot sdot sdot 0

0 (119896 + 1)119899minus2sdot sdot sdot 0

d


) (44)


119867119901(119905 119898) (119890

119861119905119898)

119896

= (

119891 (119905 119898)

119886119899minus1 (119899 minus 1)119898

119899minus1)

119901

times (119878119899minus1 (119905))

119901minus1119865 (119905 119898)119860 (119905 119898)

(45)


119860 (119905119898) = (1198600 (119905 119898) 1198601 (

119905 119898) sdot sdot sdot 119860119899minus1 (119905 119898))

119860120583 (119905 119898) =

120583

sum

]=0

119886]119896120583minus]119892120583minus]

(46)

for any 120583 = 0 119899 minus 1 and 119892] is given by (29)

For the sake of simplicity we have omitted the depen-dence on119898 and 119905 of 119892]

3 Solution to the Nonhomogeneous Problem

The general solution to a nonhomogeneous linear differentialequation of order 119899 can be expressed as the sum of the generalsolution to the corresponding homogeneous linear differen-tial equation and any solution to the complete equation Thesymbolic manipulation system calculates the solution to anonperturbed differential equation with initial conditions ofthe form (12)

119909(119899)+ 119886


1 + 119886

0119909 = 119906 (119905) (47)


119909 (0) = 11990910 119909

(119899minus1)(0) = 1199091198990

(48)

where 1198860 119886

1 119886

119899minus1isin R 119909

10 119909

1198990isin R and

119906 (119905) = sum

]ge0

119905119899]119890120572]119905(120582] cos (120596]119905) + 120583] sin (120596]119905)) (49)

being 119899] isin N and 120572] 120596] 120582] 120583] isin R With the aid of thesubstitutions

1199091= 119909 119909

2= 119909

119899= 119909

(119899minus1) (50)

(12) is reduced to the system of differential equations given by

(119905) = 119860119883 (119905) + 119861 (119905) 119883 (0) = 1198830 (51)

where

119860 = (



d


1minus119886


119899minus1

)

119861 (119905) = (

0

0

119906 (119905)

) 119883 (119905) = (

1199091 (119905)

1199092 (119905)

119909119899 (119905)

)

1198830= (

11990910

11990920

1199091198990

)

(52)

The computation of the solution to the constant coeffi-cients linear part requires the calculation of the exponentialof the matrix 119860 Φ(119905) = 119890119860119905

119883 (119905) = Φ (119905)1198830+ Φ (119905) int

119905

0

exp (119860120591) 119861 (120591) 119889120591 (53)

31 Symbolic Expansion of the Solution In the following wegive a formula for the symbolic expansion of the solution tothe complete ordinary differential equation To that end letus express the noncommutative parenthesis (119890119861119905119898 119869(119905 119898))

119898

119896

as

(119890119861119905119898 119869(119905 119898))

119898

119896= (

119901(119898119896)

11990311

(119905) sdot sdot sdot 119901(119898119896)

1199031119899

(119905)

d

119901(119898119896)

1199031198991

(119905) sdot sdot sdot 119901(119898119896)

119903119899119899

(119905)

)

= (119901(119898119896)

119903119894119895

(119905))

119899

119894119895=1

= 119875 (119898 119896) (119905)

(54)

where each 119901(119898119896)

1199031119899

(119905) is a polynomial of degree 119903119894119895le 119898(119899 minus 1)

in the indeterminate 119905 with coefficients from R and119898 = 0Let us also express

120582119896(119905 119898) = (

1

119886119899minus1

(119890119886119899minus1

119905119898minus 1))

119896

=

119896

sum

]=0

120572]119890120573]119905 (55)

being

120572] = (1

119886119899minus1

)

119896

(

119896

]) (minus1)

]

120573] = minus119886119899minus1

119898

(119896 minus ])

(56)


Thus taking into account (33) the exponential matrix of 119860can be arranged as

119890119860119905≃

119898

sum

119896=0

(

120601119896 (119905) 119901

(119898119896)

11990311

(119905) sdot sdot sdot 120601119896 (119905) 119901

(119898119896)

1199031119899

(119905)

d

120601119896 (119905) 119901

(119898119896)

1199031198991

(119905) sdot sdot sdot 120601119896 (119905) 119901

(119898119896)

119903119899119899

(119905)

)

=(

(

119898

sum

119896=0

120601119896 (119905) 119901

(119898119896)

11990311

(119905) sdot sdot sdot

119898

sum

119896=0

120601119896 (119905) 119901

(119898119896)

1199031119899

(119905)

d

119898

sum

119896=0

120601119896 (119905) 119901

(119898119896)

1199031198991


119898

sum

119896=0

120601119896 (119905) 119901

(119898119896)

119903119899119899

(119905)

)

)

(57)

where

120601119896 (119905) =

119896

sum

]=0

120572]119890120573]119905 (58)

In order to develop a symbolic expression of the solutionto the complete differential equation let us express 119906(119905) as

119906 (119905) = 119890120575119905(119876

119886 (119905) cos (120596119905) + 119879119887 (119905) sin (120596119905)) (59)

with 119886 119887 isin Z cup 0 120575 120596 isin R and 119876119886(119905) and 119879

119887(119905) being

real polynomials of degrees 119886 and 119887 respectively in theindeterminate 119905

The product 119890minus119860120591119861(120591) can be arranged as follows

119890minus119860120591119861 (120591)

=

(

(

(

(

(

(

(

(

int

119905

0

(

119898

sum

119896=0

(

119896

sum

]=0

120572]119890120573]120591)119901

(119898119896)

1199031119899

(minus120591)) 119906 (120591) 119889120591

int

119905

0

(

119898

sum

119896=0

(

119896

sum

]=0

120572]119890120573]120591)119901

(119898119896)

1199032119899

(minus120591)) 119906 (120591) 119889120591

int

119905

0

(

119898

sum

119896=0

(

119896

sum

]=0

120572]119890120573]120591)119901

(119898119896)

119903119899119899

(minus120591)) 119906 (120591) 119889120591

)

)

)

)

)

)

)

)

(60)

Here the integrand of each element of the above matrix canbe written in the form of a quasipolynomial Hence we arriveto the formulae

int

119905

0

(

119898

sum

119896=0

(

119896

sum

]=0

120572]119890120573]120591)119901

(119898119896)

119903119895119899

(minus120591)) 119906 (120591) 119889120591

= sum

120572119895119899

(120578120572119895119899

119862120572119895119899120573119895119899120574119895119899(119905) + ]120572

119895119899

119878120572119895119899120573119895119899120574119895119899(119905))

(61)

where 120572119895119899isin Z+cup 0 120573

119895119899 120574

119895119899isin R 119895 = 1 119899 and

119862119898120573120574 (119905) = int 119905

119898119890120573119905 cos (120574119905) 119889119905

119878119898120573120574 (119905) = int 119905

119898119890120573119905 sin (120574119905) 119889119905

(62)

These functions are computed recursively through

1198620120573120574 (119905) =

120574

1205732+ 120574

2119890120573119905 sin (120574119905) +

120573

1205732+ 120574

2119890120573119905 cos (120574119905)

1198780120573120574 (119905) =

120573

1205732+ 120574

2119890120573119905 sin (120574119905) minus

120574

1205732+ 120574

2119890120573119905 cos (120574119905)

(63)

and for any119898 ge 1

119862119898120573120574 (119905) = 119905

1198981198620120573120574 (119905) minus 119898

120574

1205732+ 120574

2119878119898minus1120573120574

minus 119898

120573

1205732+ 120574

2119862119898minus1120573120574 (

119905)

119878119898120573120574 (119905) = 119905

1198981198780120573120574 (119905) minus 119898

120573

1205732+ 120574

2119878119898minus1120573120574

+ 119898

120574

1205732+ 120574

2119862119898minus1120573120574 (

119905)

(64)

Thus (53) can be expressed via a quasipolynomial andthus obtained through the designed symbolic system

4 Symbolic and Numeric Results

41 On the Form of the Exponential Matrix The aim of thissection is to illustrate the form that the approximation of theexponential matrix adopts For that purpose let us considerthe differential equation given by

119909(4)+ 4 + 3 + 2 + 119909 = 0 (65)


119909 (0) = 1 (0) = 0 (0) = 0 (0) = 0

(66)

This equation is transformed into the system of differentialequations given by

(119905) = 119860119883 (119905) (67)


119860 = (

0 1 0 0

0 0 1 0

0 0 0 1

minus1 minus2 minus3 minus4

) (68)

According to the technique described in Section 2 we splitmatrix 119860 into 119861 + 119862 where

119861 = (

0 1 0 0

0 0 1 0

0 0 0 1

0 0 0 0

) 119862 = (

0 0 0 0

0 0 0 0

0 0 0 0


)

(69)


Equation (36)

(119890119861119905119898+ 119867 (119905119898))

119898

=

119898

sum

119896=0

120582119896(119905 119898) (119890

119861119905119898 119869(119905 119898))

119898

119896

(70)

allows us to compute a symbolic approximation to thesolution to the differential equation In the problem we arediscussing 119890119861119905119898119867(119905119898) 120582(119905 119898) and 119869(119905 119898) can be writtenas

119890119861119905119898=

(

(

(

(

(

(

(

1

119905

119898

1199052

(2119898)

1199053

(6119898)

0 1

119905

119898

1199052

(2119898)

0 0 1

119905

119898

0 0 0 1

)

)

)

)

)

)

)

119867(119905119898) = 120582 (119905 119898) 119869 (119905 119898)

(71)

with

120582 (119905 119898) =

1

4

(eminus4119905119898 minus 1)

119869 (119905 119898) =

(

(

(

(

(

(

1199053

(61198983)

21199053

(61198983)

31199053

(61198983)

41199053

(61198983)

1199052

(21198982)

21199052

(21198982)

31199052

(21198982)

41199052

(21198982)

119905

119898

2119905

119898

3119905

119898

4119905

119898

1 2 3 4

)

)

)

)

)

)

(72)

Taking for instance119898 = 3 we get that

119890119860119905≃

3

sum

119896=0

120582119896(119905 3) (119890

1198611199053 119869 (119905 3))

3

119896

= A3+ 120582 (A

2B +BA

2+ABA)

+ 1205822(AB

2+B

2A +BAB) + 120582

3B

3

(73)

For the sake of simplicity we have introduced the followingnotation

A = 1198901198611199053 B = 119869 (119905 3) (74)

0 10987654321

05

minus05

0

x1(t

)

t

Figure 1 Comparison of the solution computed through thesymbolic method presented in this paper and the numeric solutionto the problem calculated by a Runge-Kutta fourth order methodwith a step of ℎ = 0001

Matrices A3 A2B BA2 ABA AB2 B2A BABandB3 are computed with the help of the designed symbolicprocessor with the result

A3=

(

(

(

(

1 119905

1199052

2

1199053

6

0 1 119905

1199052

2

0 0 1 119905

0 0 0 1

)

)

)

)

A2B =(

(

(

(

1199053

6

1199053

3

1199053

2

21199053

3

1199052

2

1199052 3119905

2

2

21199052

119905 2119905 3119905 4119905

1 2 3 4

)

)

)

)

AB2

=(

(AB2)11(AB2

)12(AB2

)13(AB2

)14

(AB2)21(AB2

)22(AB2

)23(AB2

)24

(AB2)31(AB2

)32(AB2

)33(AB2

)34

(AB2)41(AB2

)42(AB2

)43(AB2

)44

)

(75)

where

(AB2)11=

2

6561

1199056+

4

729

1199055+

4

81

1199054+

16

81

1199053

(AB2)12=

4

6561

1199056+

8

729

1199055+

8

81

1199054+

32

81

1199053


Exponential matrix

Dep variable Indep variable

Coeff and func

ODE

Initial conditions

ODE order

Initial cond

2

0 1

1 1 1

x t

u(t)

sin(t)

t = 0

a1 a0a2

x10 x20

Figure 2 The first window shown by the symbolic processor allows us to introduce the parameters which define the differential equation tobe solved including the initial conditions

(AB2)13=

2

2187

1199056+

4

243

1199055+

4

27

1199054+

16

27

1199053

(AB2)14=

8

6561

1199056+

16

729

1199055+

16

81

1199054+

64

81

1199053

(AB2)21=

1

729

1199055+

2

81

1199054+

2

9

1199053+

8

9

1199052

(AB2)22=

2

729

1199055+

4

81

1199054+

4

9

1199053+

16

9

1199052

(AB2)23=

1

243

1199055+

2

27

1199054+

2

3

1199053+

8

3

1199052

(AB2)24=

4

729

1199055+

8

81

1199054+

8

9

1199053+

32

9

1199052

(AB2)31=

1

243

1199054+

2

27

1199053+

2

3

1199052+

8

3

119905

(AB2)32=

2

243

1199054+

4

27

1199053+

4

3

1199052+

16

3

119905

(AB2)33=

1

81

1199054+

2

9

1199053+ 2119905

2+ 8119905

(AB2)34=

4

243

1199054+

8

27

1199053+

8

3

1199052+

32

3

119905

(AB2)41=

1

162

1199053+

1

9

1199052+ 119905 + 4

(AB2)42=

1

81

1199053+

2

9

1199052+ 2119905 + 8

(AB2)43=

1

54

1199053+

1

3

1199052+ 3119905 + 12

(AB2)44=

2

81

1199053+

4

9

1199052+ 4119905 + 16

ABA

= (

(ABA)11 (ABA)12 (ABA)13 (ABA)14(ABA)21 (ABA)22 (ABA)23 (ABA)24(ABA)31 (ABA)32 (ABA)33 (ABA)34(ABA)41 (ABA)42 (ABA)43 (ABA)44

)

(76)

where

(ABA)11 =4

81

1199053

(ABA)12 =4

243

1199054+

8

81

1199053

(ABA)13 =2

729

1199055+

8

243

1199054+

4

27

1199053

(ABA)14 =2

6561

1199056+

4

729

1199055+

4

81

1199054+

16

81

1199053

(ABA)21 =2

9

1199052

(ABA)22 =2

27

1199053+

4

9

1199052

(ABA)23 =1

81

1199054+

4

27

1199053+

2

3

1199052

(ABA)24 =1

729

1199055+

2

81

1199054+

2

9

1199053+

8

9

1199052

(ABA)31 =2

3

119905

(ABA)32 =2

9

1199052+

4

3

119905


(ABA)33 =1

27

1199053+

4

9

1199052+ 2119905

(ABA)34 =1

243

1199054+

2

27

1199053+

2

3

1199052+

8

3

119905

(ABA)41 = 1

(ABA)42 =1

3

119905 + 2

(ABA)43 =1

18

1199052+

2

3

119905 + 3

(ABA)44 =1

162

1199053+

1

9

1199052+ 119905 + 4

B3=(

(

(B3)11(B3)12(B3)13(B3)14

(B3)21(B3)22(B3)23(B3)24

(B3)31(B3)32(B3)33(B3)34

(B3)41(B3)42(B3)43(B3)44

)

)

(77)

where

(B3)11=

1

4251528

1199059+

1

118098

1199058+

1

6561

1199057+

11

6561

1199056

+

17

1458

1199055+

4

81

1199054+

8

81

1199053

(B3)12=

1

2125764

1199059+

1

59049

1199058+

2

6561

1199057+

22

6561

1199056

+

17

729

1199055+

8

81

1199054+

16

81

1199053

(B3)13=

1

1417176

1199059+

1

39366

1199058+

1

2187

1199057+

11

2187

1199056

+

17

486

1199055+

4

27

1199054+

8

27

1199053

(B3)14=

1

1062882

1199059+

2

59049

1199058+

4

6561

1199057+

44

6561

1199056

+

34

729

1199055+

16

81

1199054+

32

81

1199053

(B3)21=

1

472392

1199058+

1

13122

1199057+

1

729

1199056+

11

729

1199055

+

17

162

1199054+

4

9

1199053+

8

9

1199052

(B3)22=

1

236196

1199058+

1

6561

1199057+

2

729

1199056+

22

729

1199055

+

17

81

1199054+

8

9

1199053+

16

9

1199052

(B3)23=

1

157464

1199058+

1

4374

1199057+

1

243

1199056+

11

243

1199055

+

17

54

1199054+

4

3

1199053+

8

3

1199052

(B3)24=

1

118098

1199058+

2

6561

1199057+

4

729

1199056+

44

729

1199055

+

34

81

1199054+

16

9

1199053+

32

9

1199052

(B3)31=

1

78732

1199057+

1

2187

1199056+

2

243

1199055+

22

243

1199054

+

17

27

1199053+

8

3

1199052+

16

3

119905

(B3)32=

1

39366

1199057+

2

2187

1199056+

4

243

1199055+

44

243

1199054

+

34

27

1199053+

16

3

1199052+

32

3

119905

(B3)33=

1

26244

1199057+

1

729

1199056+

2

81

1199055+

22

81

1199054

+

17

9

1199053+ 8119905

2+ 16119905

(B3)34=

1

19683

1199057+

4

2187

1199056+

8

243

1199055+

88

243

1199054

+

68

27

1199053+

32

3

1199052+

64

3

119905

(B3)41=

1

26244

1199056+

1

729

1199055+

2

81

1199054+

22

81

1199053

+

17

9

1199052+ 8119905 + 16

(B3)42=

1

13122

1199056+

2

729

1199055+

4

81

1199054+

44

81

1199053

+

34

9

1199052+ 16119905 + 32

(B3)43=

1

8748

1199056+

1

243

1199055+

2

27

1199054+

22

27

1199053

+

17

3

1199052+ 24119905 + 48

(B3)44=

1

6561

1199056+

4

729

1199055+

8

81

1199054+

88

81

1199053

+

68

9

1199052+ 32119905 + 64

B2A

=(

(B2A)11(B2A)

12(B2A)

13(B2A)

14

(B2A)21(B2A)

22(B2A)

23(B2A)

24

(B2A)31(B2A)

32(B2A)

33(B2A)

34

(B2A)41(B2A)

42(B2A)

43(B2A)

44

)

(78)


where

(B2A)

11=

1

26244

1199056+

1

1458

1199055+

1

162

1199054+

2

81

1199053

(B2A)

12=

1

78732

1199057+

2

6561

1199056+

5

1458

1199055+

5

243

1199054+

4

81

1199053

(B2A)

13=

1

472392

1199058+

5

78732

1199057+

2

2187

1199056+

11

1458

1199055

+

17

486

1199054+

2

27

1199053

(B2A)

14=

1

4251528

1199059+

1

118098

1199058+

1

6561

1199057+

11

6561

1199056

+

17

1458

1199055+

4

81

1199054+

8

81

1199053

(B2A)

21=

1

2916

1199055+

1

162

1199054+

1

18

1199053+

2

9

1199052

(B2A)

22=

1

8748

1199056+

2

729

1199055+

5

162

1199054+

5

27

1199053+

4

9

1199052

(B2A)

23=

1

52488

1199057+

5

8748

1199056+

2

243

1199055+

11

162

1199054

+

17

54

1199053+

2

3

1199052

(B2A)

24=

1

472392

1199058+

1

13122

1199057+

1

729

1199056+

11

729

1199055

+

17

162

1199054+

4

9

1199053+

8

9

1199052

(B2A)

31=

1

486

1199054+

1

27

1199053+

1

3

1199052+

4

3

119905

(B2A)

32=

1

1458

1199055+

4

243

1199054+

5

27

1199053+

10

9

1199052+

8

3

119905

(B2A)

33=

1

8748

1199056+

5

1458

1199055+

4

81

1199054+

11

27

1199053+

17

9

1199052+ 4119905

(B2A)

34=

1

78732

1199057+

1

2187

1199056+

2

243

1199055+

22

243

1199054

+

17

27

1199053+

8

3

1199052+

16

3

119905

(B2A)

41=

1

162

1199053+

1

9

1199052+ 119905 + 4

(B2A)

42=

1

486

1199054+

4

81

1199053+

5

9

1199052+

10

3

119905 + 8

(B2A)

43=

1

2916

1199055+

5

486

1199054+

4

27

1199053+

11

9

1199052+

17

3

119905 + 12

(B2A)

44=

1

26244

1199056+

1

729

1199055+

2

81

1199054+

22

81

1199053+

17

9

1199052

+ 8119905 + 16

BA2

=(

(

(BA2)11(BA2

)12(BA2

)13(BA2

)14

(BA2)21(BA2

)22(BA2

)23(BA2

)24

(BA2)31(BA2

)32(BA2

)33(BA2

)34

(BA2)41(BA2

)42(BA2

)43(BA2

)44

)

)

(79)

where

(BA2)11=

1

162

1199053

(BA2)12=

1

243

1199054+

1

81

1199053

(BA2)13=

1

729

1199055+

2

243

1199054+

1

54

1199053

(BA2)14=

2

6561

1199056+

2

729

1199055+

1

81

1199054+

2

81

1199053

(BA2)21=

1

18

1199052

(BA2)22=

1

27

1199053+

1

9

1199052

(BA2)23=

1

81

1199054+

2

27

1199053+

1

6

1199052

(BA2)24=

2

729

1199055+

2

81

1199054+

1

9

1199053+

2

9

1199052

(BA2)31=

1

3

119905

(BA2)32=

2

9

1199052+

2

3

119905

(BA2)33=

2

27

1199053+

4

9

1199052+ 119905

(BA2)34=

4

243

1199054+

4

27

1199053+

2

3

1199052+

4

3

119905

(BA2)41= 1

(BA2)42=

2

3

119905 + 2

(BA2)43=

2

9

1199052+

4

3

119905 + 3

(BA2)44=

4

81

1199053+

4

9

1199052+ 2119905 + 4


BAB

= (

(BAB)11 (BAB)12 (BAB)13 (BAB)14(BAB)21 (BAB)22 (BAB)23 (BAB)24(BAB)31 (BAB)32 (BAB)33 (BAB)34(BAB)41 (BAB)42 (BAB)43 (BAB)44

)

(80)

where

(BAB)11 =2

6561

1199056+

2

729

1199055+

1

81

1199054+

2

81

1199053

(BAB)11 =4

6561

1199056+

4

729

1199055+

2

81

1199054+

4

81

1199053

(BAB)11 =2

2187

1199056+

2

243

1199055+

1

27

1199054+

2

27

1199053

(BAB)11 =8

6561

1199056+

8

729

1199055+

4

81

1199054+

8

81

1199053

(BAB)11 =2

729

1199055+

2

81

1199054+

1

9

1199053+

2

9

1199052

(BAB)11 =4

729

1199055+

4

81

1199054+

2

9

1199053+

4

9

1199052

(BAB)11 =2

243

1199055+

2

27

1199054+

1

3

1199053+

2

3

1199052

(BAB)11 =8

729

1199055+

8

81

1199054+

4

9

1199053+

8

9

1199052

(BAB)11 =4

243

1199054+

4

27

1199053+

2

3

1199052+

4

3

119905

(BAB)11 =8

243

1199054+

8

27

1199053+

4

3

1199052+

8

3

119905

(BAB)11 =4

81

1199054+

4

9

1199053+ 2119905

2+ 4119905

(BAB)11 =16

243

1199054+

16

27

1199053+

8

3

1199052+

16

3

119905

(BAB)11 =4

81

1199053+

4

9

1199052+ 2119905 + 4

(BAB)11 =8

81

1199053+

8

9

1199052+ 4119905 + 8

(BAB)11 =4

27

1199053+

4

3

1199052+ 6119905 + 12

(BAB)11 =16

81

1199053+

16

9

1199052+ 8119905 + 16

(81)

So the noncommutative parentheses are given by

(AB)3

0= A

3 (AB)

3

3=B

3

(AB)3

1= (A

2B +BA

2+ABA)

= (

12057211120572121205721312057214

12057221120572221205722312057224

12057231120572321205723312057234

12057241120572421205724312057244

)

(82)

where

12057211=

2

9

1199053

12057212=

4

9

1199053+

5

243

1199054

12057213=

2

3

1199053+

1

243

1199055+

10

243

1199054

12057214=

8

9

1199053+

4

6561

1199056+

2

243

1199055+

5

81

1199054

12057221=

7

9

1199052

12057222=

14

9

1199052+

1

9

1199053

12057223=

7

3

1199052+

2

81

1199054+

2

9

1199053

12057224=

28

9

1199052+

1

243

1199055+

4

81

1199054+

1

3

1199053

12057231= 2119905

12057232= 4119905 +

4

9

1199052

12057233= 6119905 +

1

9

1199053+

8

9

1199052

12057234= 8119905 +

5

243

1199054+

2

9

1199053+

4

3

1199052

12057241= 3

12057242= 6 + 119905

12057243= 9 +

5

18

1199052+ 2119905

12057244= 12 +

1

18

1199053+

5

9

1199052+ 3119905

(AB)3

2= (AB

2+B

2A +BAB)

= (

12057311120573121205731312057314

12057321120573221205732312057324

12057331120573321205733312057334

12057341120573421205734312057344

)

(83)

being

12057311=

17

26244

1199056+

13

1458

1199055+

11

162

1199054+

20

81

1199053

12057312=

10

6561

1199056+

29

1458

1199055+

35

243

1199054+

40

81

1199053+

1

78732

1199057

12057313=

2

729

1199056+

47

1458

1199055+

107

486

1199054+

20

27

1199053

+

1

472392

1199058+

5

78732

1199057


12057314=

1

243

1199056+

65

1458

1199055+

8

27

1199054+

80

81

1199053+

1

4251528

1199059

+

1

118098

1199058+

1

6561

1199057

12057321=

13

2916

1199055+

1

18

1199054+

7

18

1199053+

4

3

1199052

12057322=

8

729

1199055+

7

54

1199054+

23

27

1199053+

8

3

1199052+

1

8748

1199056

12057323=

5

243

1199055+

35

162

1199054+

71

54

1199053+ 4119905

2+

1

52488

1199057+

5

8748

1199056

12057324=

23

729

1199055+

49

162

1199054+

16

9

1199053+

16

3

1199052+

1

472392

1199058

+

1

13122

1199057+

1

729

1199056

12057331=

11

486

1199054+

7

27

1199053+

5

3

1199052+

16

3

119905

12057332=

14

243

1199054+

17

27

1199053+

34

9

1199052+

32

3

119905 +

1

1458

1199055

12057333=

1

9

1199054+

29

27

1199053+

53

9

1199052+ 16119905 +

1

8748

1199056+

5

1458

1199055

12057334=

14

81

1199054+

41

27

1199053+ 8119905

2+

64

3

119905 +

1

78732

1199057

+

1

2187

1199056+

2

243

1199055

12057341=

5

81

1199053+

2

3

1199052+ 4119905 + 12

12057342=

13

81

1199053+

5

3

1199052+

28

3

119905 + 24 +

1

486

1199054

12057343=

17

54

1199053+

26

9

1199052+

44

3

119905 + 36 +

1

2916

1199055+

5

486

1199054

12057344=

40

81

1199053+

37

9

1199052+ 20119905 + 48 +

1

26244

1199056

+

1

729

1199055+

2

81

1199054

(84)

In Figure 1 we show a comparison of the solution com-puted through the symbolic method presented in this paperand the numeric solution to the problem calculated by aRunge-Kutta fourth order method with a step of ℎ = 0001Let us stress that the difference between both solutions at anytime is smaller than 10minus7

42 Description of the Program In order to describe the alge-braic processor let us introduce the following test problem

+ + 119909 = sin 119905 (85)

Matrix A

0 1

minus1 minus1

Figure 3 Matrix 119860

Matrix B

0 1

0 0



119909 (0) = 0 (0) = 1 (86)

This equation describes the (small) angular position inradians 119909(119905) of a forced damped pendulum with a periodicdriving force In this equation (119905) represents the inertia(119905) represents the friction and sin(119905) represents a sinusoidaldriving torque applied at the pivot of the pendulumThe exactsolution to this problem is given by

119909 (119905) = minus cos 119905 + 119890minus1199052 cos(radic3

2

119905) + radic3119890minus1199052 sin(

radic3

2

119905)

(87)

The program proceeds as followsThe first window (Figure 2)allows us introducing the definition of variables the orderof the ODE and its coefficients and the function of thenonhomogeneous term 119906(119905)

The next window visualizes the expression of the com-panion matrix related to the ODE (Figure 3)

Then matrices 119861 exp(119861119905119898) 119862 and exp(119862119905119898) whichare used in the calculation of the exponential matrix are


exp (Bm)

0

((tm)0)(10) ((tm)1)(11)

((tm)0)(10)

Figure 5 Matrix exp(119861119905119898)

Matrix C

0 0

minus1 minus1


computed by using a value of119898 satisfying (23)Thesematricesare presented in Figures 4 5 6 and 7

In Figures 8 and 9 we evaluate the exponential matrix 119890119860119905in 119905 = 1 and for119898 = 1000 The algebraic processor enables usto change these two parameters We should emphasize thatthe symbolic solution to the problem is dependent on thevalue of parameter 119898 We have computed this solution fordifferent values of this parameter (119898 = 10 100 1000 and10000) In Figures 10 and 11 we compare the real solutionwithsymbolic solutions 119909(119905) for these values of119898 We observe thatthe symbolic solution gets closer to the real solution as thevalue of119898 is increased

Table 1 shows the numerical evolution of the symbolicapproximation 119909

119878(119905) computed for 119898 = 1000 These values

have been obtained by substituting variable 119905 in the symbolicsolution by values 01 02 1

In Figure 12 we compare the symbolic approximationcomputed via the symbolic processor with the exact solution(87) We also compare these two functions with a numericsolution computed through a Runge-Kutta 4th order methodwith step ℎ = 01 We see that the solution symbolicallycomputed fits better to the exact solution to than thedescribed numerical solution In Table 2 we show the error

exp (Cm)

01

exp(minustm) exp(minustm)


Evaluation on a value of t

Automatic procedure

Variation of the matrix norm

Precision

Digits

t

Value of m

18

13

1

1000

Aproximation of eAt for t

Figure 8 Parameters related to the accuracy of the solution to thedifferential equation

exp (A)

06596974858612 05335045275926

minus05335071951242 01261956258061


on the solution computed using both methods We observethat error obtained in the symbolic approximation is quitesmaller that the numeric oneThis fact can be better observedin Figure 13 where we show the difference in absolute valuebetween the exact solution and the symbolic and numericsolutions Nevertheless it is not the goal of this paper todevelop a numeric tool The symbolic technique we havedeveloped provides an analytical solution which can be used


0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15

Figure 10 Comparison of real solution (black) and symbolicsolutions for119898 = 10 (light blue line) and119898 = 100 (yellow line)

minus15

minus1

minus05

0

05

1

15

0 2 4 6 8 10

Figure 11 Comparison of real solution (black) and symbolicsolutions for119898 = 1000 (dark blue line) and119898 = 10000 (green line)

0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15

Figure 12 Exact solution (black) symbolic solution (blue) andnumeric solution (green)

Table 1 Numerical evolution of the symbolic approximation 119909119878(119905)

for119898 = 1000

Step 119905 119909(119905)

1 01 009516653368102 02 018133052249203 03 025948012999554 04 033058564791315 05 039559200286866 06 045541222101157 07 051092181897118 08 056295408367889 09 0612296198681210 10 06596861707192

Table 2 Numerical error in the symbolic (119909119878(119905)) and numeric

(119909119877(119905)) approximations with respect to the exact solution 119909(119905)

119905 |119909119878(119905) minus 119909(119905)| |119909

119877(119905) minus 119909(119905)|

01 00000047853239 0000000167203104 00000712453497 0000035875126707 00002074165163 0000354460586810 00003710415717 0001430787361713 00004547161744 0003825311783616 00003284030320 0008043689956219 00000622406880 0014419715717422 00006216534547 0023032955185125 00011277539714 0033667545654128 00013572764337 0045813471294231 00012347982225 0058707643120834 00008953272672 0071407807759437 00006002231245 0082889689449740 00005492334199 0092155687292443 00007154938930 0098343146001446 00008300921248 0100820433067849 00005487250652 0099260855115152 00002963137375 0093686827986155 00015298326576 0084479859363158 00026838757810 0072355393614361 00032571351779 0058304626062764 00030289419613 0043509187995267 00022189180928 0029236498885170 00013691340100 0016725709670673 00010043530807 0007075055562276 00012821439144 0001140917647379 00018689726696 0000541733952482 00021436005756 0002189733032385 00016148602301 0009108229122588 00002869277587 0019614707845191 00012796703448 0032789720352394 00022771057968 0047474001777797 00021518593816 00623710756305100 00009803998394 00761625070468


0 2 4 6 8 100

002

004

006

008

01

012

Figure 13 Error in the symbolic (black) and numeric (blue)approximations

as a kernel to apply perturbation methods to compute thesolution to a perturbed differential equation depending on asmall parameter

5 Conclusions

Wehave developed a symbolic processor as well as a symbolictechnique in order to deal with the solution to a linearordinary differential equation with constant coefficients oforder 119899The solution to this equation is computed completelyin a symbolic way that is the solution is expressed as afunction of 119905 This function can be of great interest to facea perturbed differential equation by means of a perturbationmethod The aim of this work is not to develop a numericmethod However we have compared the symbolic solutionobtained via our symbolic processor with a numeric solutioncomputed with a Runge-Kutta method in order to show howour solution is closer to the exact solution

References

[1] H Poincare Les Methodes Nouvelles de la Mecanique Celeste IGauthiers-Villars Paris France 1892

[2] J Henrard ldquoA survey of Poisson series processorsrdquo CelestialMechanics vol 45 no 1ndash3 pp 245ndash253 1989

[3] A Deprit J Henrard and A Rom ldquoLa theorie de la lune deDelaunay et son prolongementrdquo Comptes Rendus de lrsquoAcademiedes Sciences vol 271 pp 519ndash520 1970

[4] A Deprit J Henrard and A Rom ldquoAnalytical LunarEphemeris Delaunayrsquos theoryrdquo The Astronomical Journalvol 76 pp 269ndash272 1971

[5] J Henrard ldquoA new solution to the Main Problem of LunarTheoryrdquo Celestial Mechanics vol 19 no 4 pp 337ndash355 1979

[6] M Chapront-Touze ldquoLa solution ELP du probleme central dela Lunerdquo Astronomy amp Astrophysics vol 83 pp 86ndash94 1980

[7] M Chapront-Touze ldquoProgress in the analytical theories for theorbital motion of the Moonrdquo Celestial Mechanics vol 26 no 1pp 53ndash62 1982

[8] J F Navarro and J M Ferrandiz ldquoA new symbolic processorfor the Earth rotation theoryrdquo Celestial Mechanics amp DynamicalAstronomy vol 82 no 3 pp 243ndash263 2002

[9] J F Navarro ldquoOn the implementation of the Poincare-Lindstedttechniquerdquo Applied Mathematics and Computation vol 195 no1 pp 183ndash189 2008

[10] J F Navarro ldquoComputation of periodic solutions in perturbedsecond-order ODEsrdquo Applied Mathematics and Computationvol 202 no 1 pp 171ndash177 2008

[11] J F Navarro ldquoOn the symbolic computation of the solution to adifferential equationrdquo Advances and Applications in Mathemat-ical Sciences vol 1 no 1 pp 1ndash21 2009

[12] J F Navarro and A Perez ldquoPrincipal matrix of a linear systemsymbolically computedrdquo in Proceedings of the InternationalConference on Numerical Analysis and Applied Mathematics(ICNAAM rsquo08) pp 400ndash402 September 2008

[13] C Moler and C van Loan ldquoNineteen dubious ways to computethe exponential of a matrix twenty-five years laterrdquo SIAMReview vol 45 no 1 pp 3ndash49 2003

[14] J F Navarro and A Perez ldquoSymbolic computation of thesolution to an homogeneous ODE with constant coefficientsrdquoin Proceedings of the International Conference on NumericalAnalysis and AppliedMathematics (ICNAAM rsquo09) pp 400ndash4022009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


to the peculiarities of the problem to be treated RecentlyNavarro [9 10] developed a symbolic processor to deal withthe solution to second order differential equations of the form

+ 1198861 + 119886

0119909 = 119906 (119905) + 120598119891 (119909 ) (2)


119909 (0) = 1199090 (0) = 0

(3)

where 1198860 119886

1 119909

0 and

0isin R 119906(119905) is a quasipolynomial that

is an object of the form

119906 (119905) = sum

]ge0

119905119899]119890120572]119905(120582] cos (120596]119905) + 120583] sin (120596]119905)) (4)

where 119899] isin N 120572] 120596] 120582] and 120583] isin R and 119891(119909 ) admits theexpansion

119891 (119909 ) =

119872

sum

120581=0

sum

0le]le120581

119891]120581minus]119909]120581minus] 119891]120581minus] isin R (5)

To face this problem the algebraic processor handles objectscalled quasipolynomials and it has resulted in being a usefultool in the computation of the solution to (2) by the appli-cation of the asymptotic expansion method A modificationof this processor has been employed to compute periodicsolutions in equations of type (2) via the Poincare-Lindstedtmethod in Navarro [9 10] To that end the idea is to expandboth the solution and the modified frequency with respectto the small parameter allowing to kill secular terms whichappear in the recursive scheme The elimination of secularterms is performed through a manipulation system whichworks with modified quasipolynomials that is quasipolyno-mials containing undetermined constants

119906 (119905) = sum

]ge0

1205911205901

1times sdot sdot sdot times 120591

120590119876

119876119905119899]119890120572]119905

times (120582] cos (120596]119905) + 120583] sin (120596]119905)) (6)

where 119899] isin N 120572] 120596] 120582] and 120583] isin R 120590] isin Z and 1205911 120591119876are real constants with unknown value

One year later Navarro [11] presented a symbolic com-putation package based on the object-oriented philosophyfor manipulating matrices whose elements lie on the set ofquasipolynomials of type (4) The kernel of the symbolicprocessor was developed in C++ defining a class for thisnew object as well as a set of functions that operate onthe data structure addition substraction differentiation andintegration with respect to 119905 substitution of an undeterminedcoefficient by a series and many others The goal of thisprocessor is to provide a tool to solve a perturbed 119899-orderdifferential equation of the class

119909(119899)+ 119886


0119909 = 119906 (119905) + 120598119891 (119909 119909

(119899minus1))

(7)


119909 (0) = 11990910 (0) = 11990920

119909(119899minus1)(0) = 1199091198990

(8)

where 120598 is a small real parameter 1198860 119886

1 119886

119899minus1isin R 119906(119905) is

a quasipolynomial and 119891 is such that

119891 (119909 119909(119899minus1)) = sum

0le]1]119899le119872

119891]1]119899

119909]1sdot sdot sdot (119909

(119899minus1))

]119899

(9)

with119872 isin N ]1 ]

119899isin N and 119891]

1]119899

isin RAs a first step Navarro and Perez [12] have developed

a symbolic technique for the computation of the principalmatrix of the linear system associated with a homogeneousordinary differential equation with constant coefficients ofthe form

119909(119899)+ 119886


1 + 119886

0119909 = 0 (10)

where 1198860 119886

1 119886


119909 (0) = 11990910 (0) = 11990920

119909(119899minus1)(0) = 1199091198990

(11)

being 11990910 119909



The second step in the construction of the solution to (7)is to obtain a symbolic procedure for computing the solutionto the nonhomogeneus equation

119909(119899)+ 119886


0119909 = 119906 (119905) (12)

where as above 119906(119905) is a quasipolynomialThis new symbolic package can be useful for research and

educational purposes in the field of differential equations ordynamical systems Nowadays there aremany open problemswhich requires massive symbolic computation as to cite oneexample the analytical theory of the resonant motion ofMercury We would like to stress that the aim of this work isto develop a symbolic tool not a numericmethod as numericsolutions cannot be used in perturbation methods for differ-ential equations However we have performed comparisonsbetween the symbolic solution with a numeric one just toshow the efficiency of the technique

In next section we summarize the scheme proposedin [12] to calculate the solution to (10) which is neededto express the solution to the complete problem (12) Thetechnique to construct this solution will be described inSection 3

2 Solution to the Homogeneous Problem

As mentioned Navarro and Perez [12] have proposed asymbolic method to compute the solution to (10)

119909(119899)+ 119886


1 + 119886

0119909 = 0 (13)

where 1198860 119886

1 119886


119909 (0) = 11990910 (0) = 11990920

119909(119899minus1)(0) = 1199091198990

(14)

being 11990910 119909





1199091= 119909 119909

2= 119909

119899= 119909

(119899minus1) (15)


(119905) = 119860119883 (119905) 119883 (0) = 1198830 (16)


119860 = (



d


1minus119886


119899minus1

)

119883 (119905) = (

1199091 (119905)

1199092 (119905)

119909119899 (119905)

) 1198830= (

11990910

11990920

1199091198990

)

(17)


119861 = (



d


)

119862 = (



d


1minus119886


119899minus1

)

(18)


119890119860asymp (119890

119861119898119890119862119898)

119898

(19)



case

119890119861119898=(

1198920 (119898) 1198921 (


119898)

0 1198920 (119898) 1198921 (



119898)

d

0 0 0 sdot sdot sdot 1198920 (119898)

)

119890119862119898= 119868 minus

1

119886119899minus1

(119890minus119886119899minus1

119898minus 1)119862

=(




d

120596012059611205962sdot sdot sdot 120596

119899minus1

)

(20)


119892119899 (119898) =

1

119899119898119899 (21)

and for any 119894 = 0 119899 minus 2

120596119894=

119886119894

119886119899minus1

(119890minus119886119899minus1

119898minus 1) 120596

119899minus1= 119890


119898 (22)


100381710038171003817100381710038171003817

119890119860minus (119890

119861119898119890119862119898)

119898100381710038171003817100381710038171003817

le

1

2119898

[119861 119862] 119890119861+119862

(23)


100381710038171003817100381710038171003817

119890119860minus (119890

119861119898119890119862119898)

119898100381710038171003817100381710038171003817

le

1

119898

1198901+119886119886 (24)

where 119886 = (1198860 119886

1 119886

119899minus1) Here

119909 = 1199091=10038161003816100381610038161199091

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816119909119899

1003816100381610038161003816 (25)

for any 119909 = (1199091 119909

119899) isin R119899 and 119860 = max

1199091=1119860119909

1=


119860 = (

1198861111988612sdot sdot sdot 119886

1119899

1198862111988622sdot sdot sdot 119886

2119899

d

11988611989911198861198992sdot sdot sdot 119886

119899119899

) (26)


119890119860119905≃ (119890

119861119905119898119890119862119905119898)

119898

(27)


119890119861119905119898= (

1198920 (119905 119898) 1198921 (



119905 119898)

d


0 (119905 119898)

)

119890119862119905119898=(




d

1205960 (119905) 1205961 (


119905)

)

(28)


119892119899 (119905 119898) =

119905119899

119899119898119899 (29)


and for any 119894 = 0 119899 minus 2

120596119894 (119905) =

119886119894

119886119899minus1

(119890minus119886119899minus1

119905119898minus 1) 120596

119899minus1 (119905) = 119890


119905119898

(30)


119890119861119905119898119890119862119905119898= 119890

119861119905119898+ 119867 (119905119898) (31)

where

119867(119905119898) =

1

119886119899minus1

119891 (119905 119898)(

119892119899minus1 (119905 119898)

1198920 (119905 119898)

) (11988601198861sdot sdot sdot 119886

119899minus1)

119891 (119905 119898) = 119890minus119886119899minus1

119905119898minus 1

(32)



119890119860119905≃ (119890

119861119905119898119890119862119905119898)

119898

= (119890119861119905119898+ 119867 (119905119898))

119898

(33)


119867(119905119898) = 120582 (119905 119898) 119869 (119905 119898) (34)

with

120582 (119905 119898) =

1

119886119899minus1

119891 (119905 119898)

119869 (119905 119898) = (

119892119899minus1 (119905 119898)

1198920 (119905 119898)

) (11988601198861sdot sdot sdot 119886

119899minus1)

(35)

Thus

(119890119861119905119898+ 119867 (119905119898))

119898

=

119898

sum

119896=0

120582119896(119905 119898) (119890

119861119905119898 119869 (119905 119898))

119898

119896

(36)


defined as

(119860 119861)119898

119896=

(119898

119896)

sum

119894=1

120590(119898119896)

prod

119895=1


119895)2119861(1+(minus1)

119895)2)

119888119894119895

(37)


119894119895isin Z+cup 0

120590 (119898 119896) =

2119896 + 1 if 2119896 le 1198982 (119898 minus 119896 + 1) if 2119896 gt 119898

(38)


(1) sum120590(119898119896)

119895=1119888119894119895= 119898 for all 119894 = 1 2 (119898119896 )

(2) sum[120590(119898119896)2]



1198942= 0 for all 119894 = 1 2 (119898119896 )

(4) if 119888119894119895= 0 and 119888

119894(119895+2)= 0 then 119888

119894(119895+1)= 0 for all 119894 =

1 2 (119898

119896 ) and 119895 = 1 2 120590(119898 119896) minus 2


119878119899minus1 (119905 119898) = (1198860


119899minus1) 119865 (119905 119898)

119865 (119905 119898) =(

(

(

119905119899minus1

(119899 minus 1) 119905119899minus2119898


2

(119899 minus 1)119905

0119898

119899minus1

)

)

)

(39)


(119890119861119905119898)

119896

=(

1198660119896 (119905 119898) 1198661119896 (



119905 119898)

d


0119896 (119905 119898)

)

(40)

with

119866]119896 (119905 119898) = 119896]119892] (119905 119898) (41)



(119867 (119905 119898))119901= (

119891 (119905 119898)

119886119899minus1 (119899 minus 1)119898

119899minus1)

119901

(119878119899minus1 (119905 119898))

119901minus1


119899minus1)

(42)



(119890119861119905119898)

119896

(119867 (119905 119898))119901= (

119891 (119905 119898)

119886119899minus1 (119899 minus 1)119898

119899minus1)

119901

(119878119899minus1 (119905))

119901minus1


119899minus1)

(43)



Ω =(

(119896 + 1)119899minus1

0 sdot sdot sdot 0


d


) (44)


119867119901(119905 119898) (119890

119861119905119898)

119896

= (

119891 (119905 119898)

119886119899minus1 (119899 minus 1)119898

119899minus1)

119901

times (119878119899minus1 (119905))

119901minus1119865 (119905 119898)119860 (119905 119898)

(45)


119860 (119905119898) = (1198600 (119905 119898) 1198601 (


119860120583 (119905 119898) =

120583

sum

]=0

119886]119896120583minus]119892120583minus]

(46)





119909(119899)+ 119886


1 + 119886

0119909 = 119906 (119905) (47)


119909 (0) = 11990910 119909

(119899minus1)(0) = 1199091198990

(48)

where 1198860 119886

1 119886


10 119909

1198990isin R and

119906 (119905) = sum

]ge0

119905119899]119890120572]119905(120582] cos (120596]119905) + 120583] sin (120596]119905)) (49)


1199091= 119909 119909

2= 119909

119899= 119909

(119899minus1) (50)


(119905) = 119860119883 (119905) + 119861 (119905) 119883 (0) = 1198830 (51)

where

119860 = (



d


1minus119886


119899minus1

)

119861 (119905) = (

0

0

119906 (119905)

) 119883 (119905) = (

1199091 (119905)

1199092 (119905)

119909119899 (119905)

)

1198830= (

11990910

11990920

1199091198990

)

(52)


119883 (119905) = Φ (119905)1198830+ Φ (119905) int

119905

0

exp (119860120591) 119861 (120591) 119889120591 (53)


119898

119896

as

(119890119861119905119898 119869(119905 119898))

119898

119896= (

119901(119898119896)

11990311

(119905) sdot sdot sdot 119901(119898119896)

1199031119899

(119905)

d

119901(119898119896)

1199031198991

(119905) sdot sdot sdot 119901(119898119896)

119903119899119899

(119905)

)

= (119901(119898119896)

119903119894119895

(119905))

119899

119894119895=1

= 119875 (119898 119896) (119905)

(54)

where each 119901(119898119896)

1199031119899



120582119896(119905 119898) = (

1

119886119899minus1

(119890119886119899minus1

119905119898minus 1))

119896

=

119896

sum

]=0

120572]119890120573]119905 (55)

being

120572] = (1

119886119899minus1

)

119896

(

119896

]) (minus1)

]

120573] = minus119886119899minus1

119898

(119896 minus ])

(56)



119890119860119905≃

119898

sum

119896=0

(

120601119896 (119905) 119901

(119898119896)

11990311

(119905) sdot sdot sdot 120601119896 (119905) 119901

(119898119896)

1199031119899

(119905)

d

120601119896 (119905) 119901

(119898119896)

1199031198991

(119905) sdot sdot sdot 120601119896 (119905) 119901

(119898119896)

119903119899119899

(119905)

)

=(

(

119898

sum

119896=0

120601119896 (119905) 119901

(119898119896)

11990311


119898

sum

119896=0

120601119896 (119905) 119901

(119898119896)

1199031119899

(119905)

d

119898

sum

119896=0

120601119896 (119905) 119901

(119898119896)

1199031198991


119898

sum

119896=0

120601119896 (119905) 119901

(119898119896)

119903119899119899

(119905)

)

)

(57)

where

120601119896 (119905) =

119896

sum

]=0

120572]119890120573]119905 (58)


119906 (119905) = 119890120575119905(119876

119886 (119905) cos (120596119905) + 119879119887 (119905) sin (120596119905)) (59)


119887(119905) being



119890minus119860120591119861 (120591)

=

(

(

(

(

(

(

(

(

int

119905

0

(

119898

sum

119896=0

(

119896

sum

]=0

120572]119890120573]120591)119901

(119898119896)

1199031119899

(minus120591)) 119906 (120591) 119889120591

int

119905

0

(

119898

sum

119896=0

(

119896

sum

]=0

120572]119890120573]120591)119901

(119898119896)

1199032119899

(minus120591)) 119906 (120591) 119889120591

int

119905

0

(

119898

sum

119896=0

(

119896

sum

]=0

120572]119890120573]120591)119901

(119898119896)

119903119899119899

(minus120591)) 119906 (120591) 119889120591

)

)

)

)

)

)

)

)

(60)


int

119905

0

(

119898

sum

119896=0

(

119896

sum

]=0

120572]119890120573]120591)119901

(119898119896)

119903119895119899

(minus120591)) 119906 (120591) 119889120591

= sum

120572119895119899

(120578120572119895119899

119862120572119895119899120573119895119899120574119895119899(119905) + ]120572

119895119899

119878120572119895119899120573119895119899120574119895119899(119905))

(61)

where 120572119895119899isin Z+cup 0 120573

119895119899 120574

119895119899isin R 119895 = 1 119899 and

119862119898120573120574 (119905) = int 119905

119898119890120573119905 cos (120574119905) 119889119905

119878119898120573120574 (119905) = int 119905

119898119890120573119905 sin (120574119905) 119889119905

(62)


1198620120573120574 (119905) =

120574

1205732+ 120574

2119890120573119905 sin (120574119905) +

120573

1205732+ 120574

2119890120573119905 cos (120574119905)

1198780120573120574 (119905) =

120573

1205732+ 120574

2119890120573119905 sin (120574119905) minus

120574

1205732+ 120574

2119890120573119905 cos (120574119905)

(63)


119862119898120573120574 (119905) = 119905

1198981198620120573120574 (119905) minus 119898

120574

1205732+ 120574

2119878119898minus1120573120574

minus 119898

120573

1205732+ 120574

2119862119898minus1120573120574 (

119905)

119878119898120573120574 (119905) = 119905

1198981198780120573120574 (119905) minus 119898

120573

1205732+ 120574

2119878119898minus1120573120574

+ 119898

120574

1205732+ 120574

2119862119898minus1120573120574 (

119905)

(64)




119909(4)+ 4 + 3 + 2 + 119909 = 0 (65)


119909 (0) = 1 (0) = 0 (0) = 0 (0) = 0

(66)


(119905) = 119860119883 (119905) (67)


119860 = (

0 1 0 0

0 0 1 0

0 0 0 1


) (68)


119861 = (

0 1 0 0

0 0 1 0

0 0 0 1

0 0 0 0

) 119862 = (

0 0 0 0

0 0 0 0

0 0 0 0


)

(69)


Equation (36)

(119890119861119905119898+ 119867 (119905119898))

119898

=

119898

sum

119896=0

120582119896(119905 119898) (119890

119861119905119898 119869(119905 119898))

119898

119896

(70)


119890119861119905119898=

(

(

(

(

(

(

(

1

119905

119898

1199052

(2119898)

1199053

(6119898)

0 1

119905

119898

1199052

(2119898)

0 0 1

119905

119898

0 0 0 1

)

)

)

)

)

)

)

119867(119905119898) = 120582 (119905 119898) 119869 (119905 119898)

(71)

with

120582 (119905 119898) =

1

4

(eminus4119905119898 minus 1)

119869 (119905 119898) =

(

(

(

(

(

(

1199053

(61198983)

21199053

(61198983)

31199053

(61198983)

41199053

(61198983)

1199052

(21198982)

21199052

(21198982)

31199052

(21198982)

41199052

(21198982)

119905

119898

2119905

119898

3119905

119898

4119905

119898

1 2 3 4

)

)

)

)

)

)

(72)


119890119860119905≃

3

sum

119896=0

120582119896(119905 3) (119890

1198611199053 119869 (119905 3))

3

119896

= A3+ 120582 (A

2B +BA

2+ABA)

+ 1205822(AB

2+B

2A +BAB) + 120582

3B

3

(73)


A = 1198901198611199053 B = 119869 (119905 3) (74)

0 10987654321

05

minus05

0

x1(t

)

t



A3=

(

(

(

(

1 119905

1199052

2

1199053

6

0 1 119905

1199052

2

0 0 1 119905

0 0 0 1

)

)

)

)

A2B =(

(

(

(

1199053

6

1199053

3

1199053

2

21199053

3

1199052

2

1199052 3119905

2

2

21199052

119905 2119905 3119905 4119905

1 2 3 4

)

)

)

)

AB2

=(

(AB2)11(AB2

)12(AB2

)13(AB2

)14

(AB2)21(AB2

)22(AB2

)23(AB2

)24

(AB2)31(AB2

)32(AB2

)33(AB2

)34

(AB2)41(AB2

)42(AB2

)43(AB2

)44

)

(75)

where

(AB2)11=

2

6561

1199056+

4

729

1199055+

4

81

1199054+

16

81

1199053

(AB2)12=

4

6561

1199056+

8

729

1199055+

8

81

1199054+

32

81

1199053


Exponential matrix


Coeff and func

ODE

Initial conditions

ODE order

Initial cond

2

0 1

1 1 1

x t

u(t)

sin(t)

t = 0

a1 a0a2

x10 x20


(AB2)13=

2

2187

1199056+

4

243

1199055+

4

27

1199054+

16

27

1199053

(AB2)14=

8

6561

1199056+

16

729

1199055+

16

81

1199054+

64

81

1199053

(AB2)21=

1

729

1199055+

2

81

1199054+

2

9

1199053+

8

9

1199052

(AB2)22=

2

729

1199055+

4

81

1199054+

4

9

1199053+

16

9

1199052

(AB2)23=

1

243

1199055+

2

27

1199054+

2

3

1199053+

8

3

1199052

(AB2)24=

4

729

1199055+

8

81

1199054+

8

9

1199053+

32

9

1199052

(AB2)31=

1

243

1199054+

2

27

1199053+

2

3

1199052+

8

3

119905

(AB2)32=

2

243

1199054+

4

27

1199053+

4

3

1199052+

16

3

119905

(AB2)33=

1

81

1199054+

2

9

1199053+ 2119905

2+ 8119905

(AB2)34=

4

243

1199054+

8

27

1199053+

8

3

1199052+

32

3

119905

(AB2)41=

1

162

1199053+

1

9

1199052+ 119905 + 4

(AB2)42=

1

81

1199053+

2

9

1199052+ 2119905 + 8

(AB2)43=

1

54

1199053+

1

3

1199052+ 3119905 + 12

(AB2)44=

2

81

1199053+

4

9

1199052+ 4119905 + 16

ABA

= (


)

(76)

where

(ABA)11 =4

81

1199053

(ABA)12 =4

243

1199054+

8

81

1199053

(ABA)13 =2

729

1199055+

8

243

1199054+

4

27

1199053

(ABA)14 =2

6561

1199056+

4

729

1199055+

4

81

1199054+

16

81

1199053

(ABA)21 =2

9

1199052

(ABA)22 =2

27

1199053+

4

9

1199052

(ABA)23 =1

81

1199054+

4

27

1199053+

2

3

1199052

(ABA)24 =1

729

1199055+

2

81

1199054+

2

9

1199053+

8

9

1199052

(ABA)31 =2

3

119905

(ABA)32 =2

9

1199052+

4

3

119905


(ABA)33 =1

27

1199053+

4

9

1199052+ 2119905

(ABA)34 =1

243

1199054+

2

27

1199053+

2

3

1199052+

8

3

119905

(ABA)41 = 1

(ABA)42 =1

3

119905 + 2

(ABA)43 =1

18

1199052+

2

3

119905 + 3

(ABA)44 =1

162

1199053+

1

9

1199052+ 119905 + 4

B3=(

(

(B3)11(B3)12(B3)13(B3)14

(B3)21(B3)22(B3)23(B3)24

(B3)31(B3)32(B3)33(B3)34

(B3)41(B3)42(B3)43(B3)44

)

)

(77)

where

(B3)11=

1

4251528

1199059+

1

118098

1199058+

1

6561

1199057+

11

6561

1199056

+

17

1458

1199055+

4

81

1199054+

8

81

1199053

(B3)12=

1

2125764

1199059+

1

59049

1199058+

2

6561

1199057+

22

6561

1199056

+

17

729

1199055+

8

81

1199054+

16

81

1199053

(B3)13=

1

1417176

1199059+

1

39366

1199058+

1

2187

1199057+

11

2187

1199056

+

17

486

1199055+

4

27

1199054+

8

27

1199053

(B3)14=

1

1062882

1199059+

2

59049

1199058+

4

6561

1199057+

44

6561

1199056

+

34

729

1199055+

16

81

1199054+

32

81

1199053

(B3)21=

1

472392

1199058+

1

13122

1199057+

1

729

1199056+

11

729

1199055

+

17

162

1199054+

4

9

1199053+

8

9

1199052

(B3)22=

1

236196

1199058+

1

6561

1199057+

2

729

1199056+

22

729

1199055

+

17

81

1199054+

8

9

1199053+

16

9

1199052

(B3)23=

1

157464

1199058+

1

4374

1199057+

1

243

1199056+

11

243

1199055

+

17

54

1199054+

4

3

1199053+

8

3

1199052

(B3)24=

1

118098

1199058+

2

6561

1199057+

4

729

1199056+

44

729

1199055

+

34

81

1199054+

16

9

1199053+

32

9

1199052

(B3)31=

1

78732

1199057+

1

2187

1199056+

2

243

1199055+

22

243

1199054

+

17

27

1199053+

8

3

1199052+

16

3

119905

(B3)32=

1

39366

1199057+

2

2187

1199056+

4

243

1199055+

44

243

1199054

+

34

27

1199053+

16

3

1199052+

32

3

119905

(B3)33=

1

26244

1199057+

1

729

1199056+

2

81

1199055+

22

81

1199054

+

17

9

1199053+ 8119905

2+ 16119905

(B3)34=

1

19683

1199057+

4

2187

1199056+

8

243

1199055+

88

243

1199054

+

68

27

1199053+

32

3

1199052+

64

3

119905

(B3)41=

1

26244

1199056+

1

729

1199055+

2

81

1199054+

22

81

1199053

+

17

9

1199052+ 8119905 + 16

(B3)42=

1

13122

1199056+

2

729

1199055+

4

81

1199054+

44

81

1199053

+

34

9

1199052+ 16119905 + 32

(B3)43=

1

8748

1199056+

1

243

1199055+

2

27

1199054+

22

27

1199053

+

17

3

1199052+ 24119905 + 48

(B3)44=

1

6561

1199056+

4

729

1199055+

8

81

1199054+

88

81

1199053

+

68

9

1199052+ 32119905 + 64

B2A

=(

(B2A)11(B2A)

12(B2A)

13(B2A)

14

(B2A)21(B2A)

22(B2A)

23(B2A)

24

(B2A)31(B2A)

32(B2A)

33(B2A)

34

(B2A)41(B2A)

42(B2A)

43(B2A)

44

)

(78)


where

(B2A)

11=

1

26244

1199056+

1

1458

1199055+

1

162

1199054+

2

81

1199053

(B2A)

12=

1

78732

1199057+

2

6561

1199056+

5

1458

1199055+

5

243

1199054+

4

81

1199053

(B2A)

13=

1

472392

1199058+

5

78732

1199057+

2

2187

1199056+

11

1458

1199055

+

17

486

1199054+

2

27

1199053

(B2A)

14=

1

4251528

1199059+

1

118098

1199058+

1

6561

1199057+

11

6561

1199056

+

17

1458

1199055+

4

81

1199054+

8

81

1199053

(B2A)

21=

1

2916

1199055+

1

162

1199054+

1

18

1199053+

2

9

1199052

(B2A)

22=

1

8748

1199056+

2

729

1199055+

5

162

1199054+

5

27

1199053+

4

9

1199052

(B2A)

23=

1

52488

1199057+

5

8748

1199056+

2

243

1199055+

11

162

1199054

+

17

54

1199053+

2

3

1199052

(B2A)

24=

1

472392

1199058+

1

13122

1199057+

1

729

1199056+

11

729

1199055

+

17

162

1199054+

4

9

1199053+

8

9

1199052

(B2A)

31=

1

486

1199054+

1

27

1199053+

1

3

1199052+

4

3

119905

(B2A)

32=

1

1458

1199055+

4

243

1199054+

5

27

1199053+

10

9

1199052+

8

3

119905

(B2A)

33=

1

8748

1199056+

5

1458

1199055+

4

81

1199054+

11

27

1199053+

17

9

1199052+ 4119905

(B2A)

34=

1

78732

1199057+

1

2187

1199056+

2

243

1199055+

22

243

1199054

+

17

27

1199053+

8

3

1199052+

16

3

119905

(B2A)

41=

1

162

1199053+

1

9

1199052+ 119905 + 4

(B2A)

42=

1

486

1199054+

4

81

1199053+

5

9

1199052+

10

3

119905 + 8

(B2A)

43=

1

2916

1199055+

5

486

1199054+

4

27

1199053+

11

9

1199052+

17

3

119905 + 12

(B2A)

44=

1

26244

1199056+

1

729

1199055+

2

81

1199054+

22

81

1199053+

17

9

1199052

+ 8119905 + 16

BA2

=(

(

(BA2)11(BA2

)12(BA2

)13(BA2

)14

(BA2)21(BA2

)22(BA2

)23(BA2

)24

(BA2)31(BA2

)32(BA2

)33(BA2

)34

(BA2)41(BA2

)42(BA2

)43(BA2

)44

)

)

(79)

where

(BA2)11=

1

162

1199053

(BA2)12=

1

243

1199054+

1

81

1199053

(BA2)13=

1

729

1199055+

2

243

1199054+

1

54

1199053

(BA2)14=

2

6561

1199056+

2

729

1199055+

1

81

1199054+

2

81

1199053

(BA2)21=

1

18

1199052

(BA2)22=

1

27

1199053+

1

9

1199052

(BA2)23=

1

81

1199054+

2

27

1199053+

1

6

1199052

(BA2)24=

2

729

1199055+

2

81

1199054+

1

9

1199053+

2

9

1199052

(BA2)31=

1

3

119905

(BA2)32=

2

9

1199052+

2

3

119905

(BA2)33=

2

27

1199053+

4

9

1199052+ 119905

(BA2)34=

4

243

1199054+

4

27

1199053+

2

3

1199052+

4

3

119905

(BA2)41= 1

(BA2)42=

2

3

119905 + 2

(BA2)43=

2

9

1199052+

4

3

119905 + 3

(BA2)44=

4

81

1199053+

4

9

1199052+ 2119905 + 4


BAB

= (


)

(80)

where

(BAB)11 =2

6561

1199056+

2

729

1199055+

1

81

1199054+

2

81

1199053

(BAB)11 =4

6561

1199056+

4

729

1199055+

2

81

1199054+

4

81

1199053

(BAB)11 =2

2187

1199056+

2

243

1199055+

1

27

1199054+

2

27

1199053

(BAB)11 =8

6561

1199056+

8

729

1199055+

4

81

1199054+

8

81

1199053

(BAB)11 =2

729

1199055+

2

81

1199054+

1

9

1199053+

2

9

1199052

(BAB)11 =4

729

1199055+

4

81

1199054+

2

9

1199053+

4

9

1199052

(BAB)11 =2

243

1199055+

2

27

1199054+

1

3

1199053+

2

3

1199052

(BAB)11 =8

729

1199055+

8

81

1199054+

4

9

1199053+

8

9

1199052

(BAB)11 =4

243

1199054+

4

27

1199053+

2

3

1199052+

4

3

119905

(BAB)11 =8

243

1199054+

8

27

1199053+

4

3

1199052+

8

3

119905

(BAB)11 =4

81

1199054+

4

9

1199053+ 2119905

2+ 4119905

(BAB)11 =16

243

1199054+

16

27

1199053+

8

3

1199052+

16

3

119905

(BAB)11 =4

81

1199053+

4

9

1199052+ 2119905 + 4

(BAB)11 =8

81

1199053+

8

9

1199052+ 4119905 + 8

(BAB)11 =4

27

1199053+

4

3

1199052+ 6119905 + 12

(BAB)11 =16

81

1199053+

16

9

1199052+ 8119905 + 16

(81)


(AB)3

0= A

3 (AB)

3

3=B

3

(AB)3

1= (A

2B +BA

2+ABA)

= (

12057211120572121205721312057214

12057221120572221205722312057224

12057231120572321205723312057234

12057241120572421205724312057244

)

(82)

where

12057211=

2

9

1199053

12057212=

4

9

1199053+

5

243

1199054

12057213=

2

3

1199053+

1

243

1199055+

10

243

1199054

12057214=

8

9

1199053+

4

6561

1199056+

2

243

1199055+

5

81

1199054

12057221=

7

9

1199052

12057222=

14

9

1199052+

1

9

1199053

12057223=

7

3

1199052+

2

81

1199054+

2

9

1199053

12057224=

28

9

1199052+

1

243

1199055+

4

81

1199054+

1

3

1199053

12057231= 2119905

12057232= 4119905 +

4

9

1199052

12057233= 6119905 +

1

9

1199053+

8

9

1199052

12057234= 8119905 +

5

243

1199054+

2

9

1199053+

4

3

1199052

12057241= 3

12057242= 6 + 119905

12057243= 9 +

5

18

1199052+ 2119905

12057244= 12 +

1

18

1199053+

5

9

1199052+ 3119905

(AB)3

2= (AB

2+B

2A +BAB)

= (

12057311120573121205731312057314

12057321120573221205732312057324

12057331120573321205733312057334

12057341120573421205734312057344

)

(83)

being

12057311=

17

26244

1199056+

13

1458

1199055+

11

162

1199054+

20

81

1199053

12057312=

10

6561

1199056+

29

1458

1199055+

35

243

1199054+

40

81

1199053+

1

78732

1199057

12057313=

2

729

1199056+

47

1458

1199055+

107

486

1199054+

20

27

1199053

+

1

472392

1199058+

5

78732

1199057


12057314=

1

243

1199056+

65

1458

1199055+

8

27

1199054+

80

81

1199053+

1

4251528

1199059

+

1

118098

1199058+

1

6561

1199057

12057321=

13

2916

1199055+

1

18

1199054+

7

18

1199053+

4

3

1199052

12057322=

8

729

1199055+

7

54

1199054+

23

27

1199053+

8

3

1199052+

1

8748

1199056

12057323=

5

243

1199055+

35

162

1199054+

71

54

1199053+ 4119905

2+

1

52488

1199057+

5

8748

1199056

12057324=

23

729

1199055+

49

162

1199054+

16

9

1199053+

16

3

1199052+

1

472392

1199058

+

1

13122

1199057+

1

729

1199056

12057331=

11

486

1199054+

7

27

1199053+

5

3

1199052+

16

3

119905

12057332=

14

243

1199054+

17

27

1199053+

34

9

1199052+

32

3

119905 +

1

1458

1199055

12057333=

1

9

1199054+

29

27

1199053+

53

9

1199052+ 16119905 +

1

8748

1199056+

5

1458

1199055

12057334=

14

81

1199054+

41

27

1199053+ 8119905

2+

64

3

119905 +

1

78732

1199057

+

1

2187

1199056+

2

243

1199055

12057341=

5

81

1199053+

2

3

1199052+ 4119905 + 12

12057342=

13

81

1199053+

5

3

1199052+

28

3

119905 + 24 +

1

486

1199054

12057343=

17

54

1199053+

26

9

1199052+

44

3

119905 + 36 +

1

2916

1199055+

5

486

1199054

12057344=

40

81

1199053+

37

9

1199052+ 20119905 + 48 +

1

26244

1199056

+

1

729

1199055+

2

81

1199054

(84)



+ + 119909 = sin 119905 (85)

Matrix A

0 1

minus1 minus1


Matrix B

0 1

0 0



119909 (0) = 0 (0) = 1 (86)



2

119905) + radic3119890minus1199052 sin(

radic3

2

119905)

(87)





exp (Bm)

0

((tm)0)(10) ((tm)1)(11)

((tm)0)(10)


Matrix C

0 0

minus1 minus1








exp (Cm)

01




Automatic procedure


Precision

Digits

t

Value of m

18

13

1

1000



exp (A)

06596974858612 05335045275926

minus05335071951242 01261956258061




0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15


minus15

minus1

minus05

0

05

1

15

0 2 4 6 8 10


0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15



for119898 = 1000

Step 119905 119909(119905)

1 01 009516653368102 02 018133052249203 03 025948012999554 04 033058564791315 05 039559200286866 06 045541222101157 07 051092181897118 08 056295408367889 09 0612296198681210 10 06596861707192



119905 |119909119878(119905) minus 119909(119905)| |119909

119877(119905) minus 119909(119905)|

01 00000047853239 0000000167203104 00000712453497 0000035875126707 00002074165163 0000354460586810 00003710415717 0001430787361713 00004547161744 0003825311783616 00003284030320 0008043689956219 00000622406880 0014419715717422 00006216534547 0023032955185125 00011277539714 0033667545654128 00013572764337 0045813471294231 00012347982225 0058707643120834 00008953272672 0071407807759437 00006002231245 0082889689449740 00005492334199 0092155687292443 00007154938930 0098343146001446 00008300921248 0100820433067849 00005487250652 0099260855115152 00002963137375 0093686827986155 00015298326576 0084479859363158 00026838757810 0072355393614361 00032571351779 0058304626062764 00030289419613 0043509187995267 00022189180928 0029236498885170 00013691340100 0016725709670673 00010043530807 0007075055562276 00012821439144 0001140917647379 00018689726696 0000541733952482 00021436005756 0002189733032385 00016148602301 0009108229122588 00002869277587 0019614707845191 00012796703448 0032789720352394 00022771057968 0047474001777797 00021518593816 00623710756305100 00009803998394 00761625070468


0 2 4 6 8 100

002

004

006

008

01

012



5 Conclusions


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1199091= 119909 119909

2= 119909

119899= 119909

(119899minus1) (15)


(119905) = 119860119883 (119905) 119883 (0) = 1198830 (16)


119860 = (



d


1minus119886


119899minus1

)

119883 (119905) = (

1199091 (119905)

1199092 (119905)

119909119899 (119905)

) 1198830= (

11990910

11990920

1199091198990

)

(17)


119861 = (



d


)

119862 = (



d


1minus119886


119899minus1

)

(18)


119890119860asymp (119890

119861119898119890119862119898)

119898

(19)



case

119890119861119898=(

1198920 (119898) 1198921 (


119898)

0 1198920 (119898) 1198921 (



119898)

d

0 0 0 sdot sdot sdot 1198920 (119898)

)

119890119862119898= 119868 minus

1

119886119899minus1

(119890minus119886119899minus1

119898minus 1)119862

=(




d

120596012059611205962sdot sdot sdot 120596

119899minus1

)

(20)


119892119899 (119898) =

1

119899119898119899 (21)

and for any 119894 = 0 119899 minus 2

120596119894=

119886119894

119886119899minus1

(119890minus119886119899minus1

119898minus 1) 120596

119899minus1= 119890


119898 (22)


100381710038171003817100381710038171003817

119890119860minus (119890

119861119898119890119862119898)

119898100381710038171003817100381710038171003817

le

1

2119898

[119861 119862] 119890119861+119862

(23)


100381710038171003817100381710038171003817

119890119860minus (119890

119861119898119890119862119898)

119898100381710038171003817100381710038171003817

le

1

119898

1198901+119886119886 (24)

where 119886 = (1198860 119886

1 119886

119899minus1) Here

119909 = 1199091=10038161003816100381610038161199091

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816119909119899

1003816100381610038161003816 (25)

for any 119909 = (1199091 119909

119899) isin R119899 and 119860 = max

1199091=1119860119909

1=


119860 = (

1198861111988612sdot sdot sdot 119886

1119899

1198862111988622sdot sdot sdot 119886

2119899

d

11988611989911198861198992sdot sdot sdot 119886

119899119899

) (26)


119890119860119905≃ (119890

119861119905119898119890119862119905119898)

119898

(27)


119890119861119905119898= (

1198920 (119905 119898) 1198921 (



119905 119898)

d


0 (119905 119898)

)

119890119862119905119898=(




d

1205960 (119905) 1205961 (


119905)

)

(28)


119892119899 (119905 119898) =

119905119899

119899119898119899 (29)


and for any 119894 = 0 119899 minus 2

120596119894 (119905) =

119886119894

119886119899minus1

(119890minus119886119899minus1

119905119898minus 1) 120596

119899minus1 (119905) = 119890


119905119898

(30)


119890119861119905119898119890119862119905119898= 119890

119861119905119898+ 119867 (119905119898) (31)

where

119867(119905119898) =

1

119886119899minus1

119891 (119905 119898)(

119892119899minus1 (119905 119898)

1198920 (119905 119898)

) (11988601198861sdot sdot sdot 119886

119899minus1)

119891 (119905 119898) = 119890minus119886119899minus1

119905119898minus 1

(32)



119890119860119905≃ (119890

119861119905119898119890119862119905119898)

119898

= (119890119861119905119898+ 119867 (119905119898))

119898

(33)


119867(119905119898) = 120582 (119905 119898) 119869 (119905 119898) (34)

with

120582 (119905 119898) =

1

119886119899minus1

119891 (119905 119898)

119869 (119905 119898) = (

119892119899minus1 (119905 119898)

1198920 (119905 119898)

) (11988601198861sdot sdot sdot 119886

119899minus1)

(35)

Thus

(119890119861119905119898+ 119867 (119905119898))

119898

=

119898

sum

119896=0

120582119896(119905 119898) (119890

119861119905119898 119869 (119905 119898))

119898

119896

(36)


defined as

(119860 119861)119898

119896=

(119898

119896)

sum

119894=1

120590(119898119896)

prod

119895=1


119895)2119861(1+(minus1)

119895)2)

119888119894119895

(37)


119894119895isin Z+cup 0

120590 (119898 119896) =

2119896 + 1 if 2119896 le 1198982 (119898 minus 119896 + 1) if 2119896 gt 119898

(38)


(1) sum120590(119898119896)

119895=1119888119894119895= 119898 for all 119894 = 1 2 (119898119896 )

(2) sum[120590(119898119896)2]



1198942= 0 for all 119894 = 1 2 (119898119896 )

(4) if 119888119894119895= 0 and 119888

119894(119895+2)= 0 then 119888

119894(119895+1)= 0 for all 119894 =

1 2 (119898

119896 ) and 119895 = 1 2 120590(119898 119896) minus 2


119878119899minus1 (119905 119898) = (1198860


119899minus1) 119865 (119905 119898)

119865 (119905 119898) =(

(

(

119905119899minus1

(119899 minus 1) 119905119899minus2119898


2

(119899 minus 1)119905

0119898

119899minus1

)

)

)

(39)


(119890119861119905119898)

119896

=(

1198660119896 (119905 119898) 1198661119896 (



119905 119898)

d


0119896 (119905 119898)

)

(40)

with

119866]119896 (119905 119898) = 119896]119892] (119905 119898) (41)



(119867 (119905 119898))119901= (

119891 (119905 119898)

119886119899minus1 (119899 minus 1)119898

119899minus1)

119901

(119878119899minus1 (119905 119898))

119901minus1


119899minus1)

(42)



(119890119861119905119898)

119896

(119867 (119905 119898))119901= (

119891 (119905 119898)

119886119899minus1 (119899 minus 1)119898

119899minus1)

119901

(119878119899minus1 (119905))

119901minus1


119899minus1)

(43)



Ω =(

(119896 + 1)119899minus1

0 sdot sdot sdot 0


d


) (44)


119867119901(119905 119898) (119890

119861119905119898)

119896

= (

119891 (119905 119898)

119886119899minus1 (119899 minus 1)119898

119899minus1)

119901

times (119878119899minus1 (119905))

119901minus1119865 (119905 119898)119860 (119905 119898)

(45)


119860 (119905119898) = (1198600 (119905 119898) 1198601 (


119860120583 (119905 119898) =

120583

sum

]=0

119886]119896120583minus]119892120583minus]

(46)





119909(119899)+ 119886


1 + 119886

0119909 = 119906 (119905) (47)


119909 (0) = 11990910 119909

(119899minus1)(0) = 1199091198990

(48)

where 1198860 119886

1 119886


10 119909

1198990isin R and

119906 (119905) = sum

]ge0

119905119899]119890120572]119905(120582] cos (120596]119905) + 120583] sin (120596]119905)) (49)


1199091= 119909 119909

2= 119909

119899= 119909

(119899minus1) (50)


(119905) = 119860119883 (119905) + 119861 (119905) 119883 (0) = 1198830 (51)

where

119860 = (



d


1minus119886


119899minus1

)

119861 (119905) = (

0

0

119906 (119905)

) 119883 (119905) = (

1199091 (119905)

1199092 (119905)

119909119899 (119905)

)

1198830= (

11990910

11990920

1199091198990

)

(52)


119883 (119905) = Φ (119905)1198830+ Φ (119905) int

119905

0

exp (119860120591) 119861 (120591) 119889120591 (53)


119898

119896

as

(119890119861119905119898 119869(119905 119898))

119898

119896= (

119901(119898119896)

11990311

(119905) sdot sdot sdot 119901(119898119896)

1199031119899

(119905)

d

119901(119898119896)

1199031198991

(119905) sdot sdot sdot 119901(119898119896)

119903119899119899

(119905)

)

= (119901(119898119896)

119903119894119895

(119905))

119899

119894119895=1

= 119875 (119898 119896) (119905)

(54)

where each 119901(119898119896)

1199031119899



120582119896(119905 119898) = (

1

119886119899minus1

(119890119886119899minus1

119905119898minus 1))

119896

=

119896

sum

]=0

120572]119890120573]119905 (55)

being

120572] = (1

119886119899minus1

)

119896

(

119896

]) (minus1)

]

120573] = minus119886119899minus1

119898

(119896 minus ])

(56)



119890119860119905≃

119898

sum

119896=0

(

120601119896 (119905) 119901

(119898119896)

11990311

(119905) sdot sdot sdot 120601119896 (119905) 119901

(119898119896)

1199031119899

(119905)

d

120601119896 (119905) 119901

(119898119896)

1199031198991

(119905) sdot sdot sdot 120601119896 (119905) 119901

(119898119896)

119903119899119899

(119905)

)

=(

(

119898

sum

119896=0

120601119896 (119905) 119901

(119898119896)

11990311


119898

sum

119896=0

120601119896 (119905) 119901

(119898119896)

1199031119899

(119905)

d

119898

sum

119896=0

120601119896 (119905) 119901

(119898119896)

1199031198991


119898

sum

119896=0

120601119896 (119905) 119901

(119898119896)

119903119899119899

(119905)

)

)

(57)

where

120601119896 (119905) =

119896

sum

]=0

120572]119890120573]119905 (58)


119906 (119905) = 119890120575119905(119876

119886 (119905) cos (120596119905) + 119879119887 (119905) sin (120596119905)) (59)


119887(119905) being



119890minus119860120591119861 (120591)

=

(

(

(

(

(

(

(

(

int

119905

0

(

119898

sum

119896=0

(

119896

sum

]=0

120572]119890120573]120591)119901

(119898119896)

1199031119899

(minus120591)) 119906 (120591) 119889120591

int

119905

0

(

119898

sum

119896=0

(

119896

sum

]=0

120572]119890120573]120591)119901

(119898119896)

1199032119899

(minus120591)) 119906 (120591) 119889120591

int

119905

0

(

119898

sum

119896=0

(

119896

sum

]=0

120572]119890120573]120591)119901

(119898119896)

119903119899119899

(minus120591)) 119906 (120591) 119889120591

)

)

)

)

)

)

)

)

(60)


int

119905

0

(

119898

sum

119896=0

(

119896

sum

]=0

120572]119890120573]120591)119901

(119898119896)

119903119895119899

(minus120591)) 119906 (120591) 119889120591

= sum

120572119895119899

(120578120572119895119899

119862120572119895119899120573119895119899120574119895119899(119905) + ]120572

119895119899

119878120572119895119899120573119895119899120574119895119899(119905))

(61)

where 120572119895119899isin Z+cup 0 120573

119895119899 120574

119895119899isin R 119895 = 1 119899 and

119862119898120573120574 (119905) = int 119905

119898119890120573119905 cos (120574119905) 119889119905

119878119898120573120574 (119905) = int 119905

119898119890120573119905 sin (120574119905) 119889119905

(62)


1198620120573120574 (119905) =

120574

1205732+ 120574

2119890120573119905 sin (120574119905) +

120573

1205732+ 120574

2119890120573119905 cos (120574119905)

1198780120573120574 (119905) =

120573

1205732+ 120574

2119890120573119905 sin (120574119905) minus

120574

1205732+ 120574

2119890120573119905 cos (120574119905)

(63)


119862119898120573120574 (119905) = 119905

1198981198620120573120574 (119905) minus 119898

120574

1205732+ 120574

2119878119898minus1120573120574

minus 119898

120573

1205732+ 120574

2119862119898minus1120573120574 (

119905)

119878119898120573120574 (119905) = 119905

1198981198780120573120574 (119905) minus 119898

120573

1205732+ 120574

2119878119898minus1120573120574

+ 119898

120574

1205732+ 120574

2119862119898minus1120573120574 (

119905)

(64)




119909(4)+ 4 + 3 + 2 + 119909 = 0 (65)


119909 (0) = 1 (0) = 0 (0) = 0 (0) = 0

(66)


(119905) = 119860119883 (119905) (67)


119860 = (

0 1 0 0

0 0 1 0

0 0 0 1


) (68)


119861 = (

0 1 0 0

0 0 1 0

0 0 0 1

0 0 0 0

) 119862 = (

0 0 0 0

0 0 0 0

0 0 0 0


)

(69)


Equation (36)

(119890119861119905119898+ 119867 (119905119898))

119898

=

119898

sum

119896=0

120582119896(119905 119898) (119890

119861119905119898 119869(119905 119898))

119898

119896

(70)


119890119861119905119898=

(

(

(

(

(

(

(

1

119905

119898

1199052

(2119898)

1199053

(6119898)

0 1

119905

119898

1199052

(2119898)

0 0 1

119905

119898

0 0 0 1

)

)

)

)

)

)

)

119867(119905119898) = 120582 (119905 119898) 119869 (119905 119898)

(71)

with

120582 (119905 119898) =

1

4

(eminus4119905119898 minus 1)

119869 (119905 119898) =

(

(

(

(

(

(

1199053

(61198983)

21199053

(61198983)

31199053

(61198983)

41199053

(61198983)

1199052

(21198982)

21199052

(21198982)

31199052

(21198982)

41199052

(21198982)

119905

119898

2119905

119898

3119905

119898

4119905

119898

1 2 3 4

)

)

)

)

)

)

(72)


119890119860119905≃

3

sum

119896=0

120582119896(119905 3) (119890

1198611199053 119869 (119905 3))

3

119896

= A3+ 120582 (A

2B +BA

2+ABA)

+ 1205822(AB

2+B

2A +BAB) + 120582

3B

3

(73)


A = 1198901198611199053 B = 119869 (119905 3) (74)

0 10987654321

05

minus05

0

x1(t

)

t



A3=

(

(

(

(

1 119905

1199052

2

1199053

6

0 1 119905

1199052

2

0 0 1 119905

0 0 0 1

)

)

)

)

A2B =(

(

(

(

1199053

6

1199053

3

1199053

2

21199053

3

1199052

2

1199052 3119905

2

2

21199052

119905 2119905 3119905 4119905

1 2 3 4

)

)

)

)

AB2

=(

(AB2)11(AB2

)12(AB2

)13(AB2

)14

(AB2)21(AB2

)22(AB2

)23(AB2

)24

(AB2)31(AB2

)32(AB2

)33(AB2

)34

(AB2)41(AB2

)42(AB2

)43(AB2

)44

)

(75)

where

(AB2)11=

2

6561

1199056+

4

729

1199055+

4

81

1199054+

16

81

1199053

(AB2)12=

4

6561

1199056+

8

729

1199055+

8

81

1199054+

32

81

1199053


Exponential matrix


Coeff and func

ODE

Initial conditions

ODE order

Initial cond

2

0 1

1 1 1

x t

u(t)

sin(t)

t = 0

a1 a0a2

x10 x20


(AB2)13=

2

2187

1199056+

4

243

1199055+

4

27

1199054+

16

27

1199053

(AB2)14=

8

6561

1199056+

16

729

1199055+

16

81

1199054+

64

81

1199053

(AB2)21=

1

729

1199055+

2

81

1199054+

2

9

1199053+

8

9

1199052

(AB2)22=

2

729

1199055+

4

81

1199054+

4

9

1199053+

16

9

1199052

(AB2)23=

1

243

1199055+

2

27

1199054+

2

3

1199053+

8

3

1199052

(AB2)24=

4

729

1199055+

8

81

1199054+

8

9

1199053+

32

9

1199052

(AB2)31=

1

243

1199054+

2

27

1199053+

2

3

1199052+

8

3

119905

(AB2)32=

2

243

1199054+

4

27

1199053+

4

3

1199052+

16

3

119905

(AB2)33=

1

81

1199054+

2

9

1199053+ 2119905

2+ 8119905

(AB2)34=

4

243

1199054+

8

27

1199053+

8

3

1199052+

32

3

119905

(AB2)41=

1

162

1199053+

1

9

1199052+ 119905 + 4

(AB2)42=

1

81

1199053+

2

9

1199052+ 2119905 + 8

(AB2)43=

1

54

1199053+

1

3

1199052+ 3119905 + 12

(AB2)44=

2

81

1199053+

4

9

1199052+ 4119905 + 16

ABA

= (


)

(76)

where

(ABA)11 =4

81

1199053

(ABA)12 =4

243

1199054+

8

81

1199053

(ABA)13 =2

729

1199055+

8

243

1199054+

4

27

1199053

(ABA)14 =2

6561

1199056+

4

729

1199055+

4

81

1199054+

16

81

1199053

(ABA)21 =2

9

1199052

(ABA)22 =2

27

1199053+

4

9

1199052

(ABA)23 =1

81

1199054+

4

27

1199053+

2

3

1199052

(ABA)24 =1

729

1199055+

2

81

1199054+

2

9

1199053+

8

9

1199052

(ABA)31 =2

3

119905

(ABA)32 =2

9

1199052+

4

3

119905


(ABA)33 =1

27

1199053+

4

9

1199052+ 2119905

(ABA)34 =1

243

1199054+

2

27

1199053+

2

3

1199052+

8

3

119905

(ABA)41 = 1

(ABA)42 =1

3

119905 + 2

(ABA)43 =1

18

1199052+

2

3

119905 + 3

(ABA)44 =1

162

1199053+

1

9

1199052+ 119905 + 4

B3=(

(

(B3)11(B3)12(B3)13(B3)14

(B3)21(B3)22(B3)23(B3)24

(B3)31(B3)32(B3)33(B3)34

(B3)41(B3)42(B3)43(B3)44

)

)

(77)

where

(B3)11=

1

4251528

1199059+

1

118098

1199058+

1

6561

1199057+

11

6561

1199056

+

17

1458

1199055+

4

81

1199054+

8

81

1199053

(B3)12=

1

2125764

1199059+

1

59049

1199058+

2

6561

1199057+

22

6561

1199056

+

17

729

1199055+

8

81

1199054+

16

81

1199053

(B3)13=

1

1417176

1199059+

1

39366

1199058+

1

2187

1199057+

11

2187

1199056

+

17

486

1199055+

4

27

1199054+

8

27

1199053

(B3)14=

1

1062882

1199059+

2

59049

1199058+

4

6561

1199057+

44

6561

1199056

+

34

729

1199055+

16

81

1199054+

32

81

1199053

(B3)21=

1

472392

1199058+

1

13122

1199057+

1

729

1199056+

11

729

1199055

+

17

162

1199054+

4

9

1199053+

8

9

1199052

(B3)22=

1

236196

1199058+

1

6561

1199057+

2

729

1199056+

22

729

1199055

+

17

81

1199054+

8

9

1199053+

16

9

1199052

(B3)23=

1

157464

1199058+

1

4374

1199057+

1

243

1199056+

11

243

1199055

+

17

54

1199054+

4

3

1199053+

8

3

1199052

(B3)24=

1

118098

1199058+

2

6561

1199057+

4

729

1199056+

44

729

1199055

+

34

81

1199054+

16

9

1199053+

32

9

1199052

(B3)31=

1

78732

1199057+

1

2187

1199056+

2

243

1199055+

22

243

1199054

+

17

27

1199053+

8

3

1199052+

16

3

119905

(B3)32=

1

39366

1199057+

2

2187

1199056+

4

243

1199055+

44

243

1199054

+

34

27

1199053+

16

3

1199052+

32

3

119905

(B3)33=

1

26244

1199057+

1

729

1199056+

2

81

1199055+

22

81

1199054

+

17

9

1199053+ 8119905

2+ 16119905

(B3)34=

1

19683

1199057+

4

2187

1199056+

8

243

1199055+

88

243

1199054

+

68

27

1199053+

32

3

1199052+

64

3

119905

(B3)41=

1

26244

1199056+

1

729

1199055+

2

81

1199054+

22

81

1199053

+

17

9

1199052+ 8119905 + 16

(B3)42=

1

13122

1199056+

2

729

1199055+

4

81

1199054+

44

81

1199053

+

34

9

1199052+ 16119905 + 32

(B3)43=

1

8748

1199056+

1

243

1199055+

2

27

1199054+

22

27

1199053

+

17

3

1199052+ 24119905 + 48

(B3)44=

1

6561

1199056+

4

729

1199055+

8

81

1199054+

88

81

1199053

+

68

9

1199052+ 32119905 + 64

B2A

=(

(B2A)11(B2A)

12(B2A)

13(B2A)

14

(B2A)21(B2A)

22(B2A)

23(B2A)

24

(B2A)31(B2A)

32(B2A)

33(B2A)

34

(B2A)41(B2A)

42(B2A)

43(B2A)

44

)

(78)


where

(B2A)

11=

1

26244

1199056+

1

1458

1199055+

1

162

1199054+

2

81

1199053

(B2A)

12=

1

78732

1199057+

2

6561

1199056+

5

1458

1199055+

5

243

1199054+

4

81

1199053

(B2A)

13=

1

472392

1199058+

5

78732

1199057+

2

2187

1199056+

11

1458

1199055

+

17

486

1199054+

2

27

1199053

(B2A)

14=

1

4251528

1199059+

1

118098

1199058+

1

6561

1199057+

11

6561

1199056

+

17

1458

1199055+

4

81

1199054+

8

81

1199053

(B2A)

21=

1

2916

1199055+

1

162

1199054+

1

18

1199053+

2

9

1199052

(B2A)

22=

1

8748

1199056+

2

729

1199055+

5

162

1199054+

5

27

1199053+

4

9

1199052

(B2A)

23=

1

52488

1199057+

5

8748

1199056+

2

243

1199055+

11

162

1199054

+

17

54

1199053+

2

3

1199052

(B2A)

24=

1

472392

1199058+

1

13122

1199057+

1

729

1199056+

11

729

1199055

+

17

162

1199054+

4

9

1199053+

8

9

1199052

(B2A)

31=

1

486

1199054+

1

27

1199053+

1

3

1199052+

4

3

119905

(B2A)

32=

1

1458

1199055+

4

243

1199054+

5

27

1199053+

10

9

1199052+

8

3

119905

(B2A)

33=

1

8748

1199056+

5

1458

1199055+

4

81

1199054+

11

27

1199053+

17

9

1199052+ 4119905

(B2A)

34=

1

78732

1199057+

1

2187

1199056+

2

243

1199055+

22

243

1199054

+

17

27

1199053+

8

3

1199052+

16

3

119905

(B2A)

41=

1

162

1199053+

1

9

1199052+ 119905 + 4

(B2A)

42=

1

486

1199054+

4

81

1199053+

5

9

1199052+

10

3

119905 + 8

(B2A)

43=

1

2916

1199055+

5

486

1199054+

4

27

1199053+

11

9

1199052+

17

3

119905 + 12

(B2A)

44=

1

26244

1199056+

1

729

1199055+

2

81

1199054+

22

81

1199053+

17

9

1199052

+ 8119905 + 16

BA2

=(

(

(BA2)11(BA2

)12(BA2

)13(BA2

)14

(BA2)21(BA2

)22(BA2

)23(BA2

)24

(BA2)31(BA2

)32(BA2

)33(BA2

)34

(BA2)41(BA2

)42(BA2

)43(BA2

)44

)

)

(79)

where

(BA2)11=

1

162

1199053

(BA2)12=

1

243

1199054+

1

81

1199053

(BA2)13=

1

729

1199055+

2

243

1199054+

1

54

1199053

(BA2)14=

2

6561

1199056+

2

729

1199055+

1

81

1199054+

2

81

1199053

(BA2)21=

1

18

1199052

(BA2)22=

1

27

1199053+

1

9

1199052

(BA2)23=

1

81

1199054+

2

27

1199053+

1

6

1199052

(BA2)24=

2

729

1199055+

2

81

1199054+

1

9

1199053+

2

9

1199052

(BA2)31=

1

3

119905

(BA2)32=

2

9

1199052+

2

3

119905

(BA2)33=

2

27

1199053+

4

9

1199052+ 119905

(BA2)34=

4

243

1199054+

4

27

1199053+

2

3

1199052+

4

3

119905

(BA2)41= 1

(BA2)42=

2

3

119905 + 2

(BA2)43=

2

9

1199052+

4

3

119905 + 3

(BA2)44=

4

81

1199053+

4

9

1199052+ 2119905 + 4


BAB

= (


)

(80)

where

(BAB)11 =2

6561

1199056+

2

729

1199055+

1

81

1199054+

2

81

1199053

(BAB)11 =4

6561

1199056+

4

729

1199055+

2

81

1199054+

4

81

1199053

(BAB)11 =2

2187

1199056+

2

243

1199055+

1

27

1199054+

2

27

1199053

(BAB)11 =8

6561

1199056+

8

729

1199055+

4

81

1199054+

8

81

1199053

(BAB)11 =2

729

1199055+

2

81

1199054+

1

9

1199053+

2

9

1199052

(BAB)11 =4

729

1199055+

4

81

1199054+

2

9

1199053+

4

9

1199052

(BAB)11 =2

243

1199055+

2

27

1199054+

1

3

1199053+

2

3

1199052

(BAB)11 =8

729

1199055+

8

81

1199054+

4

9

1199053+

8

9

1199052

(BAB)11 =4

243

1199054+

4

27

1199053+

2

3

1199052+

4

3

119905

(BAB)11 =8

243

1199054+

8

27

1199053+

4

3

1199052+

8

3

119905

(BAB)11 =4

81

1199054+

4

9

1199053+ 2119905

2+ 4119905

(BAB)11 =16

243

1199054+

16

27

1199053+

8

3

1199052+

16

3

119905

(BAB)11 =4

81

1199053+

4

9

1199052+ 2119905 + 4

(BAB)11 =8

81

1199053+

8

9

1199052+ 4119905 + 8

(BAB)11 =4

27

1199053+

4

3

1199052+ 6119905 + 12

(BAB)11 =16

81

1199053+

16

9

1199052+ 8119905 + 16

(81)


(AB)3

0= A

3 (AB)

3

3=B

3

(AB)3

1= (A

2B +BA

2+ABA)

= (

12057211120572121205721312057214

12057221120572221205722312057224

12057231120572321205723312057234

12057241120572421205724312057244

)

(82)

where

12057211=

2

9

1199053

12057212=

4

9

1199053+

5

243

1199054

12057213=

2

3

1199053+

1

243

1199055+

10

243

1199054

12057214=

8

9

1199053+

4

6561

1199056+

2

243

1199055+

5

81

1199054

12057221=

7

9

1199052

12057222=

14

9

1199052+

1

9

1199053

12057223=

7

3

1199052+

2

81

1199054+

2

9

1199053

12057224=

28

9

1199052+

1

243

1199055+

4

81

1199054+

1

3

1199053

12057231= 2119905

12057232= 4119905 +

4

9

1199052

12057233= 6119905 +

1

9

1199053+

8

9

1199052

12057234= 8119905 +

5

243

1199054+

2

9

1199053+

4

3

1199052

12057241= 3

12057242= 6 + 119905

12057243= 9 +

5

18

1199052+ 2119905

12057244= 12 +

1

18

1199053+

5

9

1199052+ 3119905

(AB)3

2= (AB

2+B

2A +BAB)

= (

12057311120573121205731312057314

12057321120573221205732312057324

12057331120573321205733312057334

12057341120573421205734312057344

)

(83)

being

12057311=

17

26244

1199056+

13

1458

1199055+

11

162

1199054+

20

81

1199053

12057312=

10

6561

1199056+

29

1458

1199055+

35

243

1199054+

40

81

1199053+

1

78732

1199057

12057313=

2

729

1199056+

47

1458

1199055+

107

486

1199054+

20

27

1199053

+

1

472392

1199058+

5

78732

1199057


12057314=

1

243

1199056+

65

1458

1199055+

8

27

1199054+

80

81

1199053+

1

4251528

1199059

+

1

118098

1199058+

1

6561

1199057

12057321=

13

2916

1199055+

1

18

1199054+

7

18

1199053+

4

3

1199052

12057322=

8

729

1199055+

7

54

1199054+

23

27

1199053+

8

3

1199052+

1

8748

1199056

12057323=

5

243

1199055+

35

162

1199054+

71

54

1199053+ 4119905

2+

1

52488

1199057+

5

8748

1199056

12057324=

23

729

1199055+

49

162

1199054+

16

9

1199053+

16

3

1199052+

1

472392

1199058

+

1

13122

1199057+

1

729

1199056

12057331=

11

486

1199054+

7

27

1199053+

5

3

1199052+

16

3

119905

12057332=

14

243

1199054+

17

27

1199053+

34

9

1199052+

32

3

119905 +

1

1458

1199055

12057333=

1

9

1199054+

29

27

1199053+

53

9

1199052+ 16119905 +

1

8748

1199056+

5

1458

1199055

12057334=

14

81

1199054+

41

27

1199053+ 8119905

2+

64

3

119905 +

1

78732

1199057

+

1

2187

1199056+

2

243

1199055

12057341=

5

81

1199053+

2

3

1199052+ 4119905 + 12

12057342=

13

81

1199053+

5

3

1199052+

28

3

119905 + 24 +

1

486

1199054

12057343=

17

54

1199053+

26

9

1199052+

44

3

119905 + 36 +

1

2916

1199055+

5

486

1199054

12057344=

40

81

1199053+

37

9

1199052+ 20119905 + 48 +

1

26244

1199056

+

1

729

1199055+

2

81

1199054

(84)



+ + 119909 = sin 119905 (85)

Matrix A

0 1

minus1 minus1


Matrix B

0 1

0 0



119909 (0) = 0 (0) = 1 (86)



2

119905) + radic3119890minus1199052 sin(

radic3

2

119905)

(87)





exp (Bm)

0

((tm)0)(10) ((tm)1)(11)

((tm)0)(10)


Matrix C

0 0

minus1 minus1








exp (Cm)

01




Automatic procedure


Precision

Digits

t

Value of m

18

13

1

1000



exp (A)

06596974858612 05335045275926

minus05335071951242 01261956258061




0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15


minus15

minus1

minus05

0

05

1

15

0 2 4 6 8 10


0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15



for119898 = 1000

Step 119905 119909(119905)

1 01 009516653368102 02 018133052249203 03 025948012999554 04 033058564791315 05 039559200286866 06 045541222101157 07 051092181897118 08 056295408367889 09 0612296198681210 10 06596861707192



119905 |119909119878(119905) minus 119909(119905)| |119909

119877(119905) minus 119909(119905)|

01 00000047853239 0000000167203104 00000712453497 0000035875126707 00002074165163 0000354460586810 00003710415717 0001430787361713 00004547161744 0003825311783616 00003284030320 0008043689956219 00000622406880 0014419715717422 00006216534547 0023032955185125 00011277539714 0033667545654128 00013572764337 0045813471294231 00012347982225 0058707643120834 00008953272672 0071407807759437 00006002231245 0082889689449740 00005492334199 0092155687292443 00007154938930 0098343146001446 00008300921248 0100820433067849 00005487250652 0099260855115152 00002963137375 0093686827986155 00015298326576 0084479859363158 00026838757810 0072355393614361 00032571351779 0058304626062764 00030289419613 0043509187995267 00022189180928 0029236498885170 00013691340100 0016725709670673 00010043530807 0007075055562276 00012821439144 0001140917647379 00018689726696 0000541733952482 00021436005756 0002189733032385 00016148602301 0009108229122588 00002869277587 0019614707845191 00012796703448 0032789720352394 00022771057968 0047474001777797 00021518593816 00623710756305100 00009803998394 00761625070468


0 2 4 6 8 100

002

004

006

008

01

012



5 Conclusions


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










and for any 119894 = 0 119899 minus 2

120596119894 (119905) =

119886119894

119886119899minus1

(119890minus119886119899minus1

119905119898minus 1) 120596

119899minus1 (119905) = 119890


119905119898

(30)


119890119861119905119898119890119862119905119898= 119890

119861119905119898+ 119867 (119905119898) (31)

where

119867(119905119898) =

1

119886119899minus1

119891 (119905 119898)(

119892119899minus1 (119905 119898)

1198920 (119905 119898)

) (11988601198861sdot sdot sdot 119886

119899minus1)

119891 (119905 119898) = 119890minus119886119899minus1

119905119898minus 1

(32)



119890119860119905≃ (119890

119861119905119898119890119862119905119898)

119898

= (119890119861119905119898+ 119867 (119905119898))

119898

(33)


119867(119905119898) = 120582 (119905 119898) 119869 (119905 119898) (34)

with

120582 (119905 119898) =

1

119886119899minus1

119891 (119905 119898)

119869 (119905 119898) = (

119892119899minus1 (119905 119898)

1198920 (119905 119898)

) (11988601198861sdot sdot sdot 119886

119899minus1)

(35)

Thus

(119890119861119905119898+ 119867 (119905119898))

119898

=

119898

sum

119896=0

120582119896(119905 119898) (119890

119861119905119898 119869 (119905 119898))

119898

119896

(36)


defined as

(119860 119861)119898

119896=

(119898

119896)

sum

119894=1

120590(119898119896)

prod

119895=1


119895)2119861(1+(minus1)

119895)2)

119888119894119895

(37)


119894119895isin Z+cup 0

120590 (119898 119896) =

2119896 + 1 if 2119896 le 1198982 (119898 minus 119896 + 1) if 2119896 gt 119898

(38)


(1) sum120590(119898119896)

119895=1119888119894119895= 119898 for all 119894 = 1 2 (119898119896 )

(2) sum[120590(119898119896)2]



1198942= 0 for all 119894 = 1 2 (119898119896 )

(4) if 119888119894119895= 0 and 119888

119894(119895+2)= 0 then 119888

119894(119895+1)= 0 for all 119894 =

1 2 (119898

119896 ) and 119895 = 1 2 120590(119898 119896) minus 2


119878119899minus1 (119905 119898) = (1198860


119899minus1) 119865 (119905 119898)

119865 (119905 119898) =(

(

(

119905119899minus1

(119899 minus 1) 119905119899minus2119898


2

(119899 minus 1)119905

0119898

119899minus1

)

)

)

(39)


(119890119861119905119898)

119896

=(

1198660119896 (119905 119898) 1198661119896 (



119905 119898)

d


0119896 (119905 119898)

)

(40)

with

119866]119896 (119905 119898) = 119896]119892] (119905 119898) (41)



(119867 (119905 119898))119901= (

119891 (119905 119898)

119886119899minus1 (119899 minus 1)119898

119899minus1)

119901

(119878119899minus1 (119905 119898))

119901minus1


119899minus1)

(42)



(119890119861119905119898)

119896

(119867 (119905 119898))119901= (

119891 (119905 119898)

119886119899minus1 (119899 minus 1)119898

119899minus1)

119901

(119878119899minus1 (119905))

119901minus1


119899minus1)

(43)



Ω =(

(119896 + 1)119899minus1

0 sdot sdot sdot 0


d


) (44)


119867119901(119905 119898) (119890

119861119905119898)

119896

= (

119891 (119905 119898)

119886119899minus1 (119899 minus 1)119898

119899minus1)

119901

times (119878119899minus1 (119905))

119901minus1119865 (119905 119898)119860 (119905 119898)

(45)


119860 (119905119898) = (1198600 (119905 119898) 1198601 (


119860120583 (119905 119898) =

120583

sum

]=0

119886]119896120583minus]119892120583minus]

(46)





119909(119899)+ 119886


1 + 119886

0119909 = 119906 (119905) (47)


119909 (0) = 11990910 119909

(119899minus1)(0) = 1199091198990

(48)

where 1198860 119886

1 119886


10 119909

1198990isin R and

119906 (119905) = sum

]ge0

119905119899]119890120572]119905(120582] cos (120596]119905) + 120583] sin (120596]119905)) (49)


1199091= 119909 119909

2= 119909

119899= 119909

(119899minus1) (50)


(119905) = 119860119883 (119905) + 119861 (119905) 119883 (0) = 1198830 (51)

where

119860 = (



d


1minus119886


119899minus1

)

119861 (119905) = (

0

0

119906 (119905)

) 119883 (119905) = (

1199091 (119905)

1199092 (119905)

119909119899 (119905)

)

1198830= (

11990910

11990920

1199091198990

)

(52)


119883 (119905) = Φ (119905)1198830+ Φ (119905) int

119905

0

exp (119860120591) 119861 (120591) 119889120591 (53)


119898

119896

as

(119890119861119905119898 119869(119905 119898))

119898

119896= (

119901(119898119896)

11990311

(119905) sdot sdot sdot 119901(119898119896)

1199031119899

(119905)

d

119901(119898119896)

1199031198991

(119905) sdot sdot sdot 119901(119898119896)

119903119899119899

(119905)

)

= (119901(119898119896)

119903119894119895

(119905))

119899

119894119895=1

= 119875 (119898 119896) (119905)

(54)

where each 119901(119898119896)

1199031119899



120582119896(119905 119898) = (

1

119886119899minus1

(119890119886119899minus1

119905119898minus 1))

119896

=

119896

sum

]=0

120572]119890120573]119905 (55)

being

120572] = (1

119886119899minus1

)

119896

(

119896

]) (minus1)

]

120573] = minus119886119899minus1

119898

(119896 minus ])

(56)



119890119860119905≃

119898

sum

119896=0

(

120601119896 (119905) 119901

(119898119896)

11990311

(119905) sdot sdot sdot 120601119896 (119905) 119901

(119898119896)

1199031119899

(119905)

d

120601119896 (119905) 119901

(119898119896)

1199031198991

(119905) sdot sdot sdot 120601119896 (119905) 119901

(119898119896)

119903119899119899

(119905)

)

=(

(

119898

sum

119896=0

120601119896 (119905) 119901

(119898119896)

11990311


119898

sum

119896=0

120601119896 (119905) 119901

(119898119896)

1199031119899

(119905)

d

119898

sum

119896=0

120601119896 (119905) 119901

(119898119896)

1199031198991


119898

sum

119896=0

120601119896 (119905) 119901

(119898119896)

119903119899119899

(119905)

)

)

(57)

where

120601119896 (119905) =

119896

sum

]=0

120572]119890120573]119905 (58)


119906 (119905) = 119890120575119905(119876

119886 (119905) cos (120596119905) + 119879119887 (119905) sin (120596119905)) (59)


119887(119905) being



119890minus119860120591119861 (120591)

=

(

(

(

(

(

(

(

(

int

119905

0

(

119898

sum

119896=0

(

119896

sum

]=0

120572]119890120573]120591)119901

(119898119896)

1199031119899

(minus120591)) 119906 (120591) 119889120591

int

119905

0

(

119898

sum

119896=0

(

119896

sum

]=0

120572]119890120573]120591)119901

(119898119896)

1199032119899

(minus120591)) 119906 (120591) 119889120591

int

119905

0

(

119898

sum

119896=0

(

119896

sum

]=0

120572]119890120573]120591)119901

(119898119896)

119903119899119899

(minus120591)) 119906 (120591) 119889120591

)

)

)

)

)

)

)

)

(60)


int

119905

0

(

119898

sum

119896=0

(

119896

sum

]=0

120572]119890120573]120591)119901

(119898119896)

119903119895119899

(minus120591)) 119906 (120591) 119889120591

= sum

120572119895119899

(120578120572119895119899

119862120572119895119899120573119895119899120574119895119899(119905) + ]120572

119895119899

119878120572119895119899120573119895119899120574119895119899(119905))

(61)

where 120572119895119899isin Z+cup 0 120573

119895119899 120574

119895119899isin R 119895 = 1 119899 and

119862119898120573120574 (119905) = int 119905

119898119890120573119905 cos (120574119905) 119889119905

119878119898120573120574 (119905) = int 119905

119898119890120573119905 sin (120574119905) 119889119905

(62)


1198620120573120574 (119905) =

120574

1205732+ 120574

2119890120573119905 sin (120574119905) +

120573

1205732+ 120574

2119890120573119905 cos (120574119905)

1198780120573120574 (119905) =

120573

1205732+ 120574

2119890120573119905 sin (120574119905) minus

120574

1205732+ 120574

2119890120573119905 cos (120574119905)

(63)


119862119898120573120574 (119905) = 119905

1198981198620120573120574 (119905) minus 119898

120574

1205732+ 120574

2119878119898minus1120573120574

minus 119898

120573

1205732+ 120574

2119862119898minus1120573120574 (

119905)

119878119898120573120574 (119905) = 119905

1198981198780120573120574 (119905) minus 119898

120573

1205732+ 120574

2119878119898minus1120573120574

+ 119898

120574

1205732+ 120574

2119862119898minus1120573120574 (

119905)

(64)




119909(4)+ 4 + 3 + 2 + 119909 = 0 (65)


119909 (0) = 1 (0) = 0 (0) = 0 (0) = 0

(66)


(119905) = 119860119883 (119905) (67)


119860 = (

0 1 0 0

0 0 1 0

0 0 0 1


) (68)


119861 = (

0 1 0 0

0 0 1 0

0 0 0 1

0 0 0 0

) 119862 = (

0 0 0 0

0 0 0 0

0 0 0 0


)

(69)


Equation (36)

(119890119861119905119898+ 119867 (119905119898))

119898

=

119898

sum

119896=0

120582119896(119905 119898) (119890

119861119905119898 119869(119905 119898))

119898

119896

(70)


119890119861119905119898=

(

(

(

(

(

(

(

1

119905

119898

1199052

(2119898)

1199053

(6119898)

0 1

119905

119898

1199052

(2119898)

0 0 1

119905

119898

0 0 0 1

)

)

)

)

)

)

)

119867(119905119898) = 120582 (119905 119898) 119869 (119905 119898)

(71)

with

120582 (119905 119898) =

1

4

(eminus4119905119898 minus 1)

119869 (119905 119898) =

(

(

(

(

(

(

1199053

(61198983)

21199053

(61198983)

31199053

(61198983)

41199053

(61198983)

1199052

(21198982)

21199052

(21198982)

31199052

(21198982)

41199052

(21198982)

119905

119898

2119905

119898

3119905

119898

4119905

119898

1 2 3 4

)

)

)

)

)

)

(72)


119890119860119905≃

3

sum

119896=0

120582119896(119905 3) (119890

1198611199053 119869 (119905 3))

3

119896

= A3+ 120582 (A

2B +BA

2+ABA)

+ 1205822(AB

2+B

2A +BAB) + 120582

3B

3

(73)


A = 1198901198611199053 B = 119869 (119905 3) (74)

0 10987654321

05

minus05

0

x1(t

)

t



A3=

(

(

(

(

1 119905

1199052

2

1199053

6

0 1 119905

1199052

2

0 0 1 119905

0 0 0 1

)

)

)

)

A2B =(

(

(

(

1199053

6

1199053

3

1199053

2

21199053

3

1199052

2

1199052 3119905

2

2

21199052

119905 2119905 3119905 4119905

1 2 3 4

)

)

)

)

AB2

=(

(AB2)11(AB2

)12(AB2

)13(AB2

)14

(AB2)21(AB2

)22(AB2

)23(AB2

)24

(AB2)31(AB2

)32(AB2

)33(AB2

)34

(AB2)41(AB2

)42(AB2

)43(AB2

)44

)

(75)

where

(AB2)11=

2

6561

1199056+

4

729

1199055+

4

81

1199054+

16

81

1199053

(AB2)12=

4

6561

1199056+

8

729

1199055+

8

81

1199054+

32

81

1199053


Exponential matrix


Coeff and func

ODE

Initial conditions

ODE order

Initial cond

2

0 1

1 1 1

x t

u(t)

sin(t)

t = 0

a1 a0a2

x10 x20


(AB2)13=

2

2187

1199056+

4

243

1199055+

4

27

1199054+

16

27

1199053

(AB2)14=

8

6561

1199056+

16

729

1199055+

16

81

1199054+

64

81

1199053

(AB2)21=

1

729

1199055+

2

81

1199054+

2

9

1199053+

8

9

1199052

(AB2)22=

2

729

1199055+

4

81

1199054+

4

9

1199053+

16

9

1199052

(AB2)23=

1

243

1199055+

2

27

1199054+

2

3

1199053+

8

3

1199052

(AB2)24=

4

729

1199055+

8

81

1199054+

8

9

1199053+

32

9

1199052

(AB2)31=

1

243

1199054+

2

27

1199053+

2

3

1199052+

8

3

119905

(AB2)32=

2

243

1199054+

4

27

1199053+

4

3

1199052+

16

3

119905

(AB2)33=

1

81

1199054+

2

9

1199053+ 2119905

2+ 8119905

(AB2)34=

4

243

1199054+

8

27

1199053+

8

3

1199052+

32

3

119905

(AB2)41=

1

162

1199053+

1

9

1199052+ 119905 + 4

(AB2)42=

1

81

1199053+

2

9

1199052+ 2119905 + 8

(AB2)43=

1

54

1199053+

1

3

1199052+ 3119905 + 12

(AB2)44=

2

81

1199053+

4

9

1199052+ 4119905 + 16

ABA

= (


)

(76)

where

(ABA)11 =4

81

1199053

(ABA)12 =4

243

1199054+

8

81

1199053

(ABA)13 =2

729

1199055+

8

243

1199054+

4

27

1199053

(ABA)14 =2

6561

1199056+

4

729

1199055+

4

81

1199054+

16

81

1199053

(ABA)21 =2

9

1199052

(ABA)22 =2

27

1199053+

4

9

1199052

(ABA)23 =1

81

1199054+

4

27

1199053+

2

3

1199052

(ABA)24 =1

729

1199055+

2

81

1199054+

2

9

1199053+

8

9

1199052

(ABA)31 =2

3

119905

(ABA)32 =2

9

1199052+

4

3

119905


(ABA)33 =1

27

1199053+

4

9

1199052+ 2119905

(ABA)34 =1

243

1199054+

2

27

1199053+

2

3

1199052+

8

3

119905

(ABA)41 = 1

(ABA)42 =1

3

119905 + 2

(ABA)43 =1

18

1199052+

2

3

119905 + 3

(ABA)44 =1

162

1199053+

1

9

1199052+ 119905 + 4

B3=(

(

(B3)11(B3)12(B3)13(B3)14

(B3)21(B3)22(B3)23(B3)24

(B3)31(B3)32(B3)33(B3)34

(B3)41(B3)42(B3)43(B3)44

)

)

(77)

where

(B3)11=

1

4251528

1199059+

1

118098

1199058+

1

6561

1199057+

11

6561

1199056

+

17

1458

1199055+

4

81

1199054+

8

81

1199053

(B3)12=

1

2125764

1199059+

1

59049

1199058+

2

6561

1199057+

22

6561

1199056

+

17

729

1199055+

8

81

1199054+

16

81

1199053

(B3)13=

1

1417176

1199059+

1

39366

1199058+

1

2187

1199057+

11

2187

1199056

+

17

486

1199055+

4

27

1199054+

8

27

1199053

(B3)14=

1

1062882

1199059+

2

59049

1199058+

4

6561

1199057+

44

6561

1199056

+

34

729

1199055+

16

81

1199054+

32

81

1199053

(B3)21=

1

472392

1199058+

1

13122

1199057+

1

729

1199056+

11

729

1199055

+

17

162

1199054+

4

9

1199053+

8

9

1199052

(B3)22=

1

236196

1199058+

1

6561

1199057+

2

729

1199056+

22

729

1199055

+

17

81

1199054+

8

9

1199053+

16

9

1199052

(B3)23=

1

157464

1199058+

1

4374

1199057+

1

243

1199056+

11

243

1199055

+

17

54

1199054+

4

3

1199053+

8

3

1199052

(B3)24=

1

118098

1199058+

2

6561

1199057+

4

729

1199056+

44

729

1199055

+

34

81

1199054+

16

9

1199053+

32

9

1199052

(B3)31=

1

78732

1199057+

1

2187

1199056+

2

243

1199055+

22

243

1199054

+

17

27

1199053+

8

3

1199052+

16

3

119905

(B3)32=

1

39366

1199057+

2

2187

1199056+

4

243

1199055+

44

243

1199054

+

34

27

1199053+

16

3

1199052+

32

3

119905

(B3)33=

1

26244

1199057+

1

729

1199056+

2

81

1199055+

22

81

1199054

+

17

9

1199053+ 8119905

2+ 16119905

(B3)34=

1

19683

1199057+

4

2187

1199056+

8

243

1199055+

88

243

1199054

+

68

27

1199053+

32

3

1199052+

64

3

119905

(B3)41=

1

26244

1199056+

1

729

1199055+

2

81

1199054+

22

81

1199053

+

17

9

1199052+ 8119905 + 16

(B3)42=

1

13122

1199056+

2

729

1199055+

4

81

1199054+

44

81

1199053

+

34

9

1199052+ 16119905 + 32

(B3)43=

1

8748

1199056+

1

243

1199055+

2

27

1199054+

22

27

1199053

+

17

3

1199052+ 24119905 + 48

(B3)44=

1

6561

1199056+

4

729

1199055+

8

81

1199054+

88

81

1199053

+

68

9

1199052+ 32119905 + 64

B2A

=(

(B2A)11(B2A)

12(B2A)

13(B2A)

14

(B2A)21(B2A)

22(B2A)

23(B2A)

24

(B2A)31(B2A)

32(B2A)

33(B2A)

34

(B2A)41(B2A)

42(B2A)

43(B2A)

44

)

(78)


where

(B2A)

11=

1

26244

1199056+

1

1458

1199055+

1

162

1199054+

2

81

1199053

(B2A)

12=

1

78732

1199057+

2

6561

1199056+

5

1458

1199055+

5

243

1199054+

4

81

1199053

(B2A)

13=

1

472392

1199058+

5

78732

1199057+

2

2187

1199056+

11

1458

1199055

+

17

486

1199054+

2

27

1199053

(B2A)

14=

1

4251528

1199059+

1

118098

1199058+

1

6561

1199057+

11

6561

1199056

+

17

1458

1199055+

4

81

1199054+

8

81

1199053

(B2A)

21=

1

2916

1199055+

1

162

1199054+

1

18

1199053+

2

9

1199052

(B2A)

22=

1

8748

1199056+

2

729

1199055+

5

162

1199054+

5

27

1199053+

4

9

1199052

(B2A)

23=

1

52488

1199057+

5

8748

1199056+

2

243

1199055+

11

162

1199054

+

17

54

1199053+

2

3

1199052

(B2A)

24=

1

472392

1199058+

1

13122

1199057+

1

729

1199056+

11

729

1199055

+

17

162

1199054+

4

9

1199053+

8

9

1199052

(B2A)

31=

1

486

1199054+

1

27

1199053+

1

3

1199052+

4

3

119905

(B2A)

32=

1

1458

1199055+

4

243

1199054+

5

27

1199053+

10

9

1199052+

8

3

119905

(B2A)

33=

1

8748

1199056+

5

1458

1199055+

4

81

1199054+

11

27

1199053+

17

9

1199052+ 4119905

(B2A)

34=

1

78732

1199057+

1

2187

1199056+

2

243

1199055+

22

243

1199054

+

17

27

1199053+

8

3

1199052+

16

3

119905

(B2A)

41=

1

162

1199053+

1

9

1199052+ 119905 + 4

(B2A)

42=

1

486

1199054+

4

81

1199053+

5

9

1199052+

10

3

119905 + 8

(B2A)

43=

1

2916

1199055+

5

486

1199054+

4

27

1199053+

11

9

1199052+

17

3

119905 + 12

(B2A)

44=

1

26244

1199056+

1

729

1199055+

2

81

1199054+

22

81

1199053+

17

9

1199052

+ 8119905 + 16

BA2

=(

(

(BA2)11(BA2

)12(BA2

)13(BA2

)14

(BA2)21(BA2

)22(BA2

)23(BA2

)24

(BA2)31(BA2

)32(BA2

)33(BA2

)34

(BA2)41(BA2

)42(BA2

)43(BA2

)44

)

)

(79)

where

(BA2)11=

1

162

1199053

(BA2)12=

1

243

1199054+

1

81

1199053

(BA2)13=

1

729

1199055+

2

243

1199054+

1

54

1199053

(BA2)14=

2

6561

1199056+

2

729

1199055+

1

81

1199054+

2

81

1199053

(BA2)21=

1

18

1199052

(BA2)22=

1

27

1199053+

1

9

1199052

(BA2)23=

1

81

1199054+

2

27

1199053+

1

6

1199052

(BA2)24=

2

729

1199055+

2

81

1199054+

1

9

1199053+

2

9

1199052

(BA2)31=

1

3

119905

(BA2)32=

2

9

1199052+

2

3

119905

(BA2)33=

2

27

1199053+

4

9

1199052+ 119905

(BA2)34=

4

243

1199054+

4

27

1199053+

2

3

1199052+

4

3

119905

(BA2)41= 1

(BA2)42=

2

3

119905 + 2

(BA2)43=

2

9

1199052+

4

3

119905 + 3

(BA2)44=

4

81

1199053+

4

9

1199052+ 2119905 + 4


BAB

= (


)

(80)

where

(BAB)11 =2

6561

1199056+

2

729

1199055+

1

81

1199054+

2

81

1199053

(BAB)11 =4

6561

1199056+

4

729

1199055+

2

81

1199054+

4

81

1199053

(BAB)11 =2

2187

1199056+

2

243

1199055+

1

27

1199054+

2

27

1199053

(BAB)11 =8

6561

1199056+

8

729

1199055+

4

81

1199054+

8

81

1199053

(BAB)11 =2

729

1199055+

2

81

1199054+

1

9

1199053+

2

9

1199052

(BAB)11 =4

729

1199055+

4

81

1199054+

2

9

1199053+

4

9

1199052

(BAB)11 =2

243

1199055+

2

27

1199054+

1

3

1199053+

2

3

1199052

(BAB)11 =8

729

1199055+

8

81

1199054+

4

9

1199053+

8

9

1199052

(BAB)11 =4

243

1199054+

4

27

1199053+

2

3

1199052+

4

3

119905

(BAB)11 =8

243

1199054+

8

27

1199053+

4

3

1199052+

8

3

119905

(BAB)11 =4

81

1199054+

4

9

1199053+ 2119905

2+ 4119905

(BAB)11 =16

243

1199054+

16

27

1199053+

8

3

1199052+

16

3

119905

(BAB)11 =4

81

1199053+

4

9

1199052+ 2119905 + 4

(BAB)11 =8

81

1199053+

8

9

1199052+ 4119905 + 8

(BAB)11 =4

27

1199053+

4

3

1199052+ 6119905 + 12

(BAB)11 =16

81

1199053+

16

9

1199052+ 8119905 + 16

(81)


(AB)3

0= A

3 (AB)

3

3=B

3

(AB)3

1= (A

2B +BA

2+ABA)

= (

12057211120572121205721312057214

12057221120572221205722312057224

12057231120572321205723312057234

12057241120572421205724312057244

)

(82)

where

12057211=

2

9

1199053

12057212=

4

9

1199053+

5

243

1199054

12057213=

2

3

1199053+

1

243

1199055+

10

243

1199054

12057214=

8

9

1199053+

4

6561

1199056+

2

243

1199055+

5

81

1199054

12057221=

7

9

1199052

12057222=

14

9

1199052+

1

9

1199053

12057223=

7

3

1199052+

2

81

1199054+

2

9

1199053

12057224=

28

9

1199052+

1

243

1199055+

4

81

1199054+

1

3

1199053

12057231= 2119905

12057232= 4119905 +

4

9

1199052

12057233= 6119905 +

1

9

1199053+

8

9

1199052

12057234= 8119905 +

5

243

1199054+

2

9

1199053+

4

3

1199052

12057241= 3

12057242= 6 + 119905

12057243= 9 +

5

18

1199052+ 2119905

12057244= 12 +

1

18

1199053+

5

9

1199052+ 3119905

(AB)3

2= (AB

2+B

2A +BAB)

= (

12057311120573121205731312057314

12057321120573221205732312057324

12057331120573321205733312057334

12057341120573421205734312057344

)

(83)

being

12057311=

17

26244

1199056+

13

1458

1199055+

11

162

1199054+

20

81

1199053

12057312=

10

6561

1199056+

29

1458

1199055+

35

243

1199054+

40

81

1199053+

1

78732

1199057

12057313=

2

729

1199056+

47

1458

1199055+

107

486

1199054+

20

27

1199053

+

1

472392

1199058+

5

78732

1199057


12057314=

1

243

1199056+

65

1458

1199055+

8

27

1199054+

80

81

1199053+

1

4251528

1199059

+

1

118098

1199058+

1

6561

1199057

12057321=

13

2916

1199055+

1

18

1199054+

7

18

1199053+

4

3

1199052

12057322=

8

729

1199055+

7

54

1199054+

23

27

1199053+

8

3

1199052+

1

8748

1199056

12057323=

5

243

1199055+

35

162

1199054+

71

54

1199053+ 4119905

2+

1

52488

1199057+

5

8748

1199056

12057324=

23

729

1199055+

49

162

1199054+

16

9

1199053+

16

3

1199052+

1

472392

1199058

+

1

13122

1199057+

1

729

1199056

12057331=

11

486

1199054+

7

27

1199053+

5

3

1199052+

16

3

119905

12057332=

14

243

1199054+

17

27

1199053+

34

9

1199052+

32

3

119905 +

1

1458

1199055

12057333=

1

9

1199054+

29

27

1199053+

53

9

1199052+ 16119905 +

1

8748

1199056+

5

1458

1199055

12057334=

14

81

1199054+

41

27

1199053+ 8119905

2+

64

3

119905 +

1

78732

1199057

+

1

2187

1199056+

2

243

1199055

12057341=

5

81

1199053+

2

3

1199052+ 4119905 + 12

12057342=

13

81

1199053+

5

3

1199052+

28

3

119905 + 24 +

1

486

1199054

12057343=

17

54

1199053+

26

9

1199052+

44

3

119905 + 36 +

1

2916

1199055+

5

486

1199054

12057344=

40

81

1199053+

37

9

1199052+ 20119905 + 48 +

1

26244

1199056

+

1

729

1199055+

2

81

1199054

(84)



+ + 119909 = sin 119905 (85)

Matrix A

0 1

minus1 minus1


Matrix B

0 1

0 0



119909 (0) = 0 (0) = 1 (86)



2

119905) + radic3119890minus1199052 sin(

radic3

2

119905)

(87)





exp (Bm)

0

((tm)0)(10) ((tm)1)(11)

((tm)0)(10)


Matrix C

0 0

minus1 minus1








exp (Cm)

01




Automatic procedure


Precision

Digits

t

Value of m

18

13

1

1000



exp (A)

06596974858612 05335045275926

minus05335071951242 01261956258061




0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15


minus15

minus1

minus05

0

05

1

15

0 2 4 6 8 10


0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15



for119898 = 1000

Step 119905 119909(119905)

1 01 009516653368102 02 018133052249203 03 025948012999554 04 033058564791315 05 039559200286866 06 045541222101157 07 051092181897118 08 056295408367889 09 0612296198681210 10 06596861707192



119905 |119909119878(119905) minus 119909(119905)| |119909

119877(119905) minus 119909(119905)|

01 00000047853239 0000000167203104 00000712453497 0000035875126707 00002074165163 0000354460586810 00003710415717 0001430787361713 00004547161744 0003825311783616 00003284030320 0008043689956219 00000622406880 0014419715717422 00006216534547 0023032955185125 00011277539714 0033667545654128 00013572764337 0045813471294231 00012347982225 0058707643120834 00008953272672 0071407807759437 00006002231245 0082889689449740 00005492334199 0092155687292443 00007154938930 0098343146001446 00008300921248 0100820433067849 00005487250652 0099260855115152 00002963137375 0093686827986155 00015298326576 0084479859363158 00026838757810 0072355393614361 00032571351779 0058304626062764 00030289419613 0043509187995267 00022189180928 0029236498885170 00013691340100 0016725709670673 00010043530807 0007075055562276 00012821439144 0001140917647379 00018689726696 0000541733952482 00021436005756 0002189733032385 00016148602301 0009108229122588 00002869277587 0019614707845191 00012796703448 0032789720352394 00022771057968 0047474001777797 00021518593816 00623710756305100 00009803998394 00761625070468


0 2 4 6 8 100

002

004

006

008

01

012



5 Conclusions


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Ω =(

(119896 + 1)119899minus1

0 sdot sdot sdot 0


d


) (44)


119867119901(119905 119898) (119890

119861119905119898)

119896

= (

119891 (119905 119898)

119886119899minus1 (119899 minus 1)119898

119899minus1)

119901

times (119878119899minus1 (119905))

119901minus1119865 (119905 119898)119860 (119905 119898)

(45)


119860 (119905119898) = (1198600 (119905 119898) 1198601 (


119860120583 (119905 119898) =

120583

sum

]=0

119886]119896120583minus]119892120583minus]

(46)





119909(119899)+ 119886


1 + 119886

0119909 = 119906 (119905) (47)


119909 (0) = 11990910 119909

(119899minus1)(0) = 1199091198990

(48)

where 1198860 119886

1 119886


10 119909

1198990isin R and

119906 (119905) = sum

]ge0

119905119899]119890120572]119905(120582] cos (120596]119905) + 120583] sin (120596]119905)) (49)


1199091= 119909 119909

2= 119909

119899= 119909

(119899minus1) (50)


(119905) = 119860119883 (119905) + 119861 (119905) 119883 (0) = 1198830 (51)

where

119860 = (



d


1minus119886


119899minus1

)

119861 (119905) = (

0

0

119906 (119905)

) 119883 (119905) = (

1199091 (119905)

1199092 (119905)

119909119899 (119905)

)

1198830= (

11990910

11990920

1199091198990

)

(52)


119883 (119905) = Φ (119905)1198830+ Φ (119905) int

119905

0

exp (119860120591) 119861 (120591) 119889120591 (53)


119898

119896

as

(119890119861119905119898 119869(119905 119898))

119898

119896= (

119901(119898119896)

11990311

(119905) sdot sdot sdot 119901(119898119896)

1199031119899

(119905)

d

119901(119898119896)

1199031198991

(119905) sdot sdot sdot 119901(119898119896)

119903119899119899

(119905)

)

= (119901(119898119896)

119903119894119895

(119905))

119899

119894119895=1

= 119875 (119898 119896) (119905)

(54)

where each 119901(119898119896)

1199031119899



120582119896(119905 119898) = (

1

119886119899minus1

(119890119886119899minus1

119905119898minus 1))

119896

=

119896

sum

]=0

120572]119890120573]119905 (55)

being

120572] = (1

119886119899minus1

)

119896

(

119896

]) (minus1)

]

120573] = minus119886119899minus1

119898

(119896 minus ])

(56)



119890119860119905≃

119898

sum

119896=0

(

120601119896 (119905) 119901

(119898119896)

11990311

(119905) sdot sdot sdot 120601119896 (119905) 119901

(119898119896)

1199031119899

(119905)

d

120601119896 (119905) 119901

(119898119896)

1199031198991

(119905) sdot sdot sdot 120601119896 (119905) 119901

(119898119896)

119903119899119899

(119905)

)

=(

(

119898

sum

119896=0

120601119896 (119905) 119901

(119898119896)

11990311


119898

sum

119896=0

120601119896 (119905) 119901

(119898119896)

1199031119899

(119905)

d

119898

sum

119896=0

120601119896 (119905) 119901

(119898119896)

1199031198991


119898

sum

119896=0

120601119896 (119905) 119901

(119898119896)

119903119899119899

(119905)

)

)

(57)

where

120601119896 (119905) =

119896

sum

]=0

120572]119890120573]119905 (58)


119906 (119905) = 119890120575119905(119876

119886 (119905) cos (120596119905) + 119879119887 (119905) sin (120596119905)) (59)


119887(119905) being



119890minus119860120591119861 (120591)

=

(

(

(

(

(

(

(

(

int

119905

0

(

119898

sum

119896=0

(

119896

sum

]=0

120572]119890120573]120591)119901

(119898119896)

1199031119899

(minus120591)) 119906 (120591) 119889120591

int

119905

0

(

119898

sum

119896=0

(

119896

sum

]=0

120572]119890120573]120591)119901

(119898119896)

1199032119899

(minus120591)) 119906 (120591) 119889120591

int

119905

0

(

119898

sum

119896=0

(

119896

sum

]=0

120572]119890120573]120591)119901

(119898119896)

119903119899119899

(minus120591)) 119906 (120591) 119889120591

)

)

)

)

)

)

)

)

(60)


int

119905

0

(

119898

sum

119896=0

(

119896

sum

]=0

120572]119890120573]120591)119901

(119898119896)

119903119895119899

(minus120591)) 119906 (120591) 119889120591

= sum

120572119895119899

(120578120572119895119899

119862120572119895119899120573119895119899120574119895119899(119905) + ]120572

119895119899

119878120572119895119899120573119895119899120574119895119899(119905))

(61)

where 120572119895119899isin Z+cup 0 120573

119895119899 120574

119895119899isin R 119895 = 1 119899 and

119862119898120573120574 (119905) = int 119905

119898119890120573119905 cos (120574119905) 119889119905

119878119898120573120574 (119905) = int 119905

119898119890120573119905 sin (120574119905) 119889119905

(62)


1198620120573120574 (119905) =

120574

1205732+ 120574

2119890120573119905 sin (120574119905) +

120573

1205732+ 120574

2119890120573119905 cos (120574119905)

1198780120573120574 (119905) =

120573

1205732+ 120574

2119890120573119905 sin (120574119905) minus

120574

1205732+ 120574

2119890120573119905 cos (120574119905)

(63)


119862119898120573120574 (119905) = 119905

1198981198620120573120574 (119905) minus 119898

120574

1205732+ 120574

2119878119898minus1120573120574

minus 119898

120573

1205732+ 120574

2119862119898minus1120573120574 (

119905)

119878119898120573120574 (119905) = 119905

1198981198780120573120574 (119905) minus 119898

120573

1205732+ 120574

2119878119898minus1120573120574

+ 119898

120574

1205732+ 120574

2119862119898minus1120573120574 (

119905)

(64)




119909(4)+ 4 + 3 + 2 + 119909 = 0 (65)


119909 (0) = 1 (0) = 0 (0) = 0 (0) = 0

(66)


(119905) = 119860119883 (119905) (67)


119860 = (

0 1 0 0

0 0 1 0

0 0 0 1


) (68)


119861 = (

0 1 0 0

0 0 1 0

0 0 0 1

0 0 0 0

) 119862 = (

0 0 0 0

0 0 0 0

0 0 0 0


)

(69)


Equation (36)

(119890119861119905119898+ 119867 (119905119898))

119898

=

119898

sum

119896=0

120582119896(119905 119898) (119890

119861119905119898 119869(119905 119898))

119898

119896

(70)


119890119861119905119898=

(

(

(

(

(

(

(

1

119905

119898

1199052

(2119898)

1199053

(6119898)

0 1

119905

119898

1199052

(2119898)

0 0 1

119905

119898

0 0 0 1

)

)

)

)

)

)

)

119867(119905119898) = 120582 (119905 119898) 119869 (119905 119898)

(71)

with

120582 (119905 119898) =

1

4

(eminus4119905119898 minus 1)

119869 (119905 119898) =

(

(

(

(

(

(

1199053

(61198983)

21199053

(61198983)

31199053

(61198983)

41199053

(61198983)

1199052

(21198982)

21199052

(21198982)

31199052

(21198982)

41199052

(21198982)

119905

119898

2119905

119898

3119905

119898

4119905

119898

1 2 3 4

)

)

)

)

)

)

(72)


119890119860119905≃

3

sum

119896=0

120582119896(119905 3) (119890

1198611199053 119869 (119905 3))

3

119896

= A3+ 120582 (A

2B +BA

2+ABA)

+ 1205822(AB

2+B

2A +BAB) + 120582

3B

3

(73)


A = 1198901198611199053 B = 119869 (119905 3) (74)

0 10987654321

05

minus05

0

x1(t

)

t



A3=

(

(

(

(

1 119905

1199052

2

1199053

6

0 1 119905

1199052

2

0 0 1 119905

0 0 0 1

)

)

)

)

A2B =(

(

(

(

1199053

6

1199053

3

1199053

2

21199053

3

1199052

2

1199052 3119905

2

2

21199052

119905 2119905 3119905 4119905

1 2 3 4

)

)

)

)

AB2

=(

(AB2)11(AB2

)12(AB2

)13(AB2

)14

(AB2)21(AB2

)22(AB2

)23(AB2

)24

(AB2)31(AB2

)32(AB2

)33(AB2

)34

(AB2)41(AB2

)42(AB2

)43(AB2

)44

)

(75)

where

(AB2)11=

2

6561

1199056+

4

729

1199055+

4

81

1199054+

16

81

1199053

(AB2)12=

4

6561

1199056+

8

729

1199055+

8

81

1199054+

32

81

1199053


Exponential matrix


Coeff and func

ODE

Initial conditions

ODE order

Initial cond

2

0 1

1 1 1

x t

u(t)

sin(t)

t = 0

a1 a0a2

x10 x20


(AB2)13=

2

2187

1199056+

4

243

1199055+

4

27

1199054+

16

27

1199053

(AB2)14=

8

6561

1199056+

16

729

1199055+

16

81

1199054+

64

81

1199053

(AB2)21=

1

729

1199055+

2

81

1199054+

2

9

1199053+

8

9

1199052

(AB2)22=

2

729

1199055+

4

81

1199054+

4

9

1199053+

16

9

1199052

(AB2)23=

1

243

1199055+

2

27

1199054+

2

3

1199053+

8

3

1199052

(AB2)24=

4

729

1199055+

8

81

1199054+

8

9

1199053+

32

9

1199052

(AB2)31=

1

243

1199054+

2

27

1199053+

2

3

1199052+

8

3

119905

(AB2)32=

2

243

1199054+

4

27

1199053+

4

3

1199052+

16

3

119905

(AB2)33=

1

81

1199054+

2

9

1199053+ 2119905

2+ 8119905

(AB2)34=

4

243

1199054+

8

27

1199053+

8

3

1199052+

32

3

119905

(AB2)41=

1

162

1199053+

1

9

1199052+ 119905 + 4

(AB2)42=

1

81

1199053+

2

9

1199052+ 2119905 + 8

(AB2)43=

1

54

1199053+

1

3

1199052+ 3119905 + 12

(AB2)44=

2

81

1199053+

4

9

1199052+ 4119905 + 16

ABA

= (


)

(76)

where

(ABA)11 =4

81

1199053

(ABA)12 =4

243

1199054+

8

81

1199053

(ABA)13 =2

729

1199055+

8

243

1199054+

4

27

1199053

(ABA)14 =2

6561

1199056+

4

729

1199055+

4

81

1199054+

16

81

1199053

(ABA)21 =2

9

1199052

(ABA)22 =2

27

1199053+

4

9

1199052

(ABA)23 =1

81

1199054+

4

27

1199053+

2

3

1199052

(ABA)24 =1

729

1199055+

2

81

1199054+

2

9

1199053+

8

9

1199052

(ABA)31 =2

3

119905

(ABA)32 =2

9

1199052+

4

3

119905


(ABA)33 =1

27

1199053+

4

9

1199052+ 2119905

(ABA)34 =1

243

1199054+

2

27

1199053+

2

3

1199052+

8

3

119905

(ABA)41 = 1

(ABA)42 =1

3

119905 + 2

(ABA)43 =1

18

1199052+

2

3

119905 + 3

(ABA)44 =1

162

1199053+

1

9

1199052+ 119905 + 4

B3=(

(

(B3)11(B3)12(B3)13(B3)14

(B3)21(B3)22(B3)23(B3)24

(B3)31(B3)32(B3)33(B3)34

(B3)41(B3)42(B3)43(B3)44

)

)

(77)

where

(B3)11=

1

4251528

1199059+

1

118098

1199058+

1

6561

1199057+

11

6561

1199056

+

17

1458

1199055+

4

81

1199054+

8

81

1199053

(B3)12=

1

2125764

1199059+

1

59049

1199058+

2

6561

1199057+

22

6561

1199056

+

17

729

1199055+

8

81

1199054+

16

81

1199053

(B3)13=

1

1417176

1199059+

1

39366

1199058+

1

2187

1199057+

11

2187

1199056

+

17

486

1199055+

4

27

1199054+

8

27

1199053

(B3)14=

1

1062882

1199059+

2

59049

1199058+

4

6561

1199057+

44

6561

1199056

+

34

729

1199055+

16

81

1199054+

32

81

1199053

(B3)21=

1

472392

1199058+

1

13122

1199057+

1

729

1199056+

11

729

1199055

+

17

162

1199054+

4

9

1199053+

8

9

1199052

(B3)22=

1

236196

1199058+

1

6561

1199057+

2

729

1199056+

22

729

1199055

+

17

81

1199054+

8

9

1199053+

16

9

1199052

(B3)23=

1

157464

1199058+

1

4374

1199057+

1

243

1199056+

11

243

1199055

+

17

54

1199054+

4

3

1199053+

8

3

1199052

(B3)24=

1

118098

1199058+

2

6561

1199057+

4

729

1199056+

44

729

1199055

+

34

81

1199054+

16

9

1199053+

32

9

1199052

(B3)31=

1

78732

1199057+

1

2187

1199056+

2

243

1199055+

22

243

1199054

+

17

27

1199053+

8

3

1199052+

16

3

119905

(B3)32=

1

39366

1199057+

2

2187

1199056+

4

243

1199055+

44

243

1199054

+

34

27

1199053+

16

3

1199052+

32

3

119905

(B3)33=

1

26244

1199057+

1

729

1199056+

2

81

1199055+

22

81

1199054

+

17

9

1199053+ 8119905

2+ 16119905

(B3)34=

1

19683

1199057+

4

2187

1199056+

8

243

1199055+

88

243

1199054

+

68

27

1199053+

32

3

1199052+

64

3

119905

(B3)41=

1

26244

1199056+

1

729

1199055+

2

81

1199054+

22

81

1199053

+

17

9

1199052+ 8119905 + 16

(B3)42=

1

13122

1199056+

2

729

1199055+

4

81

1199054+

44

81

1199053

+

34

9

1199052+ 16119905 + 32

(B3)43=

1

8748

1199056+

1

243

1199055+

2

27

1199054+

22

27

1199053

+

17

3

1199052+ 24119905 + 48

(B3)44=

1

6561

1199056+

4

729

1199055+

8

81

1199054+

88

81

1199053

+

68

9

1199052+ 32119905 + 64

B2A

=(

(B2A)11(B2A)

12(B2A)

13(B2A)

14

(B2A)21(B2A)

22(B2A)

23(B2A)

24

(B2A)31(B2A)

32(B2A)

33(B2A)

34

(B2A)41(B2A)

42(B2A)

43(B2A)

44

)

(78)


where

(B2A)

11=

1

26244

1199056+

1

1458

1199055+

1

162

1199054+

2

81

1199053

(B2A)

12=

1

78732

1199057+

2

6561

1199056+

5

1458

1199055+

5

243

1199054+

4

81

1199053

(B2A)

13=

1

472392

1199058+

5

78732

1199057+

2

2187

1199056+

11

1458

1199055

+

17

486

1199054+

2

27

1199053

(B2A)

14=

1

4251528

1199059+

1

118098

1199058+

1

6561

1199057+

11

6561

1199056

+

17

1458

1199055+

4

81

1199054+

8

81

1199053

(B2A)

21=

1

2916

1199055+

1

162

1199054+

1

18

1199053+

2

9

1199052

(B2A)

22=

1

8748

1199056+

2

729

1199055+

5

162

1199054+

5

27

1199053+

4

9

1199052

(B2A)

23=

1

52488

1199057+

5

8748

1199056+

2

243

1199055+

11

162

1199054

+

17

54

1199053+

2

3

1199052

(B2A)

24=

1

472392

1199058+

1

13122

1199057+

1

729

1199056+

11

729

1199055

+

17

162

1199054+

4

9

1199053+

8

9

1199052

(B2A)

31=

1

486

1199054+

1

27

1199053+

1

3

1199052+

4

3

119905

(B2A)

32=

1

1458

1199055+

4

243

1199054+

5

27

1199053+

10

9

1199052+

8

3

119905

(B2A)

33=

1

8748

1199056+

5

1458

1199055+

4

81

1199054+

11

27

1199053+

17

9

1199052+ 4119905

(B2A)

34=

1

78732

1199057+

1

2187

1199056+

2

243

1199055+

22

243

1199054

+

17

27

1199053+

8

3

1199052+

16

3

119905

(B2A)

41=

1

162

1199053+

1

9

1199052+ 119905 + 4

(B2A)

42=

1

486

1199054+

4

81

1199053+

5

9

1199052+

10

3

119905 + 8

(B2A)

43=

1

2916

1199055+

5

486

1199054+

4

27

1199053+

11

9

1199052+

17

3

119905 + 12

(B2A)

44=

1

26244

1199056+

1

729

1199055+

2

81

1199054+

22

81

1199053+

17

9

1199052

+ 8119905 + 16

BA2

=(

(

(BA2)11(BA2

)12(BA2

)13(BA2

)14

(BA2)21(BA2

)22(BA2

)23(BA2

)24

(BA2)31(BA2

)32(BA2

)33(BA2

)34

(BA2)41(BA2

)42(BA2

)43(BA2

)44

)

)

(79)

where

(BA2)11=

1

162

1199053

(BA2)12=

1

243

1199054+

1

81

1199053

(BA2)13=

1

729

1199055+

2

243

1199054+

1

54

1199053

(BA2)14=

2

6561

1199056+

2

729

1199055+

1

81

1199054+

2

81

1199053

(BA2)21=

1

18

1199052

(BA2)22=

1

27

1199053+

1

9

1199052

(BA2)23=

1

81

1199054+

2

27

1199053+

1

6

1199052

(BA2)24=

2

729

1199055+

2

81

1199054+

1

9

1199053+

2

9

1199052

(BA2)31=

1

3

119905

(BA2)32=

2

9

1199052+

2

3

119905

(BA2)33=

2

27

1199053+

4

9

1199052+ 119905

(BA2)34=

4

243

1199054+

4

27

1199053+

2

3

1199052+

4

3

119905

(BA2)41= 1

(BA2)42=

2

3

119905 + 2

(BA2)43=

2

9

1199052+

4

3

119905 + 3

(BA2)44=

4

81

1199053+

4

9

1199052+ 2119905 + 4


BAB

= (


)

(80)

where

(BAB)11 =2

6561

1199056+

2

729

1199055+

1

81

1199054+

2

81

1199053

(BAB)11 =4

6561

1199056+

4

729

1199055+

2

81

1199054+

4

81

1199053

(BAB)11 =2

2187

1199056+

2

243

1199055+

1

27

1199054+

2

27

1199053

(BAB)11 =8

6561

1199056+

8

729

1199055+

4

81

1199054+

8

81

1199053

(BAB)11 =2

729

1199055+

2

81

1199054+

1

9

1199053+

2

9

1199052

(BAB)11 =4

729

1199055+

4

81

1199054+

2

9

1199053+

4

9

1199052

(BAB)11 =2

243

1199055+

2

27

1199054+

1

3

1199053+

2

3

1199052

(BAB)11 =8

729

1199055+

8

81

1199054+

4

9

1199053+

8

9

1199052

(BAB)11 =4

243

1199054+

4

27

1199053+

2

3

1199052+

4

3

119905

(BAB)11 =8

243

1199054+

8

27

1199053+

4

3

1199052+

8

3

119905

(BAB)11 =4

81

1199054+

4

9

1199053+ 2119905

2+ 4119905

(BAB)11 =16

243

1199054+

16

27

1199053+

8

3

1199052+

16

3

119905

(BAB)11 =4

81

1199053+

4

9

1199052+ 2119905 + 4

(BAB)11 =8

81

1199053+

8

9

1199052+ 4119905 + 8

(BAB)11 =4

27

1199053+

4

3

1199052+ 6119905 + 12

(BAB)11 =16

81

1199053+

16

9

1199052+ 8119905 + 16

(81)


(AB)3

0= A

3 (AB)

3

3=B

3

(AB)3

1= (A

2B +BA

2+ABA)

= (

12057211120572121205721312057214

12057221120572221205722312057224

12057231120572321205723312057234

12057241120572421205724312057244

)

(82)

where

12057211=

2

9

1199053

12057212=

4

9

1199053+

5

243

1199054

12057213=

2

3

1199053+

1

243

1199055+

10

243

1199054

12057214=

8

9

1199053+

4

6561

1199056+

2

243

1199055+

5

81

1199054

12057221=

7

9

1199052

12057222=

14

9

1199052+

1

9

1199053

12057223=

7

3

1199052+

2

81

1199054+

2

9

1199053

12057224=

28

9

1199052+

1

243

1199055+

4

81

1199054+

1

3

1199053

12057231= 2119905

12057232= 4119905 +

4

9

1199052

12057233= 6119905 +

1

9

1199053+

8

9

1199052

12057234= 8119905 +

5

243

1199054+

2

9

1199053+

4

3

1199052

12057241= 3

12057242= 6 + 119905

12057243= 9 +

5

18

1199052+ 2119905

12057244= 12 +

1

18

1199053+

5

9

1199052+ 3119905

(AB)3

2= (AB

2+B

2A +BAB)

= (

12057311120573121205731312057314

12057321120573221205732312057324

12057331120573321205733312057334

12057341120573421205734312057344

)

(83)

being

12057311=

17

26244

1199056+

13

1458

1199055+

11

162

1199054+

20

81

1199053

12057312=

10

6561

1199056+

29

1458

1199055+

35

243

1199054+

40

81

1199053+

1

78732

1199057

12057313=

2

729

1199056+

47

1458

1199055+

107

486

1199054+

20

27

1199053

+

1

472392

1199058+

5

78732

1199057


12057314=

1

243

1199056+

65

1458

1199055+

8

27

1199054+

80

81

1199053+

1

4251528

1199059

+

1

118098

1199058+

1

6561

1199057

12057321=

13

2916

1199055+

1

18

1199054+

7

18

1199053+

4

3

1199052

12057322=

8

729

1199055+

7

54

1199054+

23

27

1199053+

8

3

1199052+

1

8748

1199056

12057323=

5

243

1199055+

35

162

1199054+

71

54

1199053+ 4119905

2+

1

52488

1199057+

5

8748

1199056

12057324=

23

729

1199055+

49

162

1199054+

16

9

1199053+

16

3

1199052+

1

472392

1199058

+

1

13122

1199057+

1

729

1199056

12057331=

11

486

1199054+

7

27

1199053+

5

3

1199052+

16

3

119905

12057332=

14

243

1199054+

17

27

1199053+

34

9

1199052+

32

3

119905 +

1

1458

1199055

12057333=

1

9

1199054+

29

27

1199053+

53

9

1199052+ 16119905 +

1

8748

1199056+

5

1458

1199055

12057334=

14

81

1199054+

41

27

1199053+ 8119905

2+

64

3

119905 +

1

78732

1199057

+

1

2187

1199056+

2

243

1199055

12057341=

5

81

1199053+

2

3

1199052+ 4119905 + 12

12057342=

13

81

1199053+

5

3

1199052+

28

3

119905 + 24 +

1

486

1199054

12057343=

17

54

1199053+

26

9

1199052+

44

3

119905 + 36 +

1

2916

1199055+

5

486

1199054

12057344=

40

81

1199053+

37

9

1199052+ 20119905 + 48 +

1

26244

1199056

+

1

729

1199055+

2

81

1199054

(84)



+ + 119909 = sin 119905 (85)

Matrix A

0 1

minus1 minus1


Matrix B

0 1

0 0



119909 (0) = 0 (0) = 1 (86)



2

119905) + radic3119890minus1199052 sin(

radic3

2

119905)

(87)





exp (Bm)

0

((tm)0)(10) ((tm)1)(11)

((tm)0)(10)


Matrix C

0 0

minus1 minus1








exp (Cm)

01




Automatic procedure


Precision

Digits

t

Value of m

18

13

1

1000



exp (A)

06596974858612 05335045275926

minus05335071951242 01261956258061




0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15


minus15

minus1

minus05

0

05

1

15

0 2 4 6 8 10


0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15



for119898 = 1000

Step 119905 119909(119905)

1 01 009516653368102 02 018133052249203 03 025948012999554 04 033058564791315 05 039559200286866 06 045541222101157 07 051092181897118 08 056295408367889 09 0612296198681210 10 06596861707192



119905 |119909119878(119905) minus 119909(119905)| |119909

119877(119905) minus 119909(119905)|

01 00000047853239 0000000167203104 00000712453497 0000035875126707 00002074165163 0000354460586810 00003710415717 0001430787361713 00004547161744 0003825311783616 00003284030320 0008043689956219 00000622406880 0014419715717422 00006216534547 0023032955185125 00011277539714 0033667545654128 00013572764337 0045813471294231 00012347982225 0058707643120834 00008953272672 0071407807759437 00006002231245 0082889689449740 00005492334199 0092155687292443 00007154938930 0098343146001446 00008300921248 0100820433067849 00005487250652 0099260855115152 00002963137375 0093686827986155 00015298326576 0084479859363158 00026838757810 0072355393614361 00032571351779 0058304626062764 00030289419613 0043509187995267 00022189180928 0029236498885170 00013691340100 0016725709670673 00010043530807 0007075055562276 00012821439144 0001140917647379 00018689726696 0000541733952482 00021436005756 0002189733032385 00016148602301 0009108229122588 00002869277587 0019614707845191 00012796703448 0032789720352394 00022771057968 0047474001777797 00021518593816 00623710756305100 00009803998394 00761625070468


0 2 4 6 8 100

002

004

006

008

01

012



5 Conclusions


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119890119860119905≃

119898

sum

119896=0

(

120601119896 (119905) 119901

(119898119896)

11990311

(119905) sdot sdot sdot 120601119896 (119905) 119901

(119898119896)

1199031119899

(119905)

d

120601119896 (119905) 119901

(119898119896)

1199031198991

(119905) sdot sdot sdot 120601119896 (119905) 119901

(119898119896)

119903119899119899

(119905)

)

=(

(

119898

sum

119896=0

120601119896 (119905) 119901

(119898119896)

11990311


119898

sum

119896=0

120601119896 (119905) 119901

(119898119896)

1199031119899

(119905)

d

119898

sum

119896=0

120601119896 (119905) 119901

(119898119896)

1199031198991


119898

sum

119896=0

120601119896 (119905) 119901

(119898119896)

119903119899119899

(119905)

)

)

(57)

where

120601119896 (119905) =

119896

sum

]=0

120572]119890120573]119905 (58)


119906 (119905) = 119890120575119905(119876

119886 (119905) cos (120596119905) + 119879119887 (119905) sin (120596119905)) (59)


119887(119905) being



119890minus119860120591119861 (120591)

=

(

(

(

(

(

(

(

(

int

119905

0

(

119898

sum

119896=0

(

119896

sum

]=0

120572]119890120573]120591)119901

(119898119896)

1199031119899

(minus120591)) 119906 (120591) 119889120591

int

119905

0

(

119898

sum

119896=0

(

119896

sum

]=0

120572]119890120573]120591)119901

(119898119896)

1199032119899

(minus120591)) 119906 (120591) 119889120591

int

119905

0

(

119898

sum

119896=0

(

119896

sum

]=0

120572]119890120573]120591)119901

(119898119896)

119903119899119899

(minus120591)) 119906 (120591) 119889120591

)

)

)

)

)

)

)

)

(60)


int

119905

0

(

119898

sum

119896=0

(

119896

sum

]=0

120572]119890120573]120591)119901

(119898119896)

119903119895119899

(minus120591)) 119906 (120591) 119889120591

= sum

120572119895119899

(120578120572119895119899

119862120572119895119899120573119895119899120574119895119899(119905) + ]120572

119895119899

119878120572119895119899120573119895119899120574119895119899(119905))

(61)

where 120572119895119899isin Z+cup 0 120573

119895119899 120574

119895119899isin R 119895 = 1 119899 and

119862119898120573120574 (119905) = int 119905

119898119890120573119905 cos (120574119905) 119889119905

119878119898120573120574 (119905) = int 119905

119898119890120573119905 sin (120574119905) 119889119905

(62)


1198620120573120574 (119905) =

120574

1205732+ 120574

2119890120573119905 sin (120574119905) +

120573

1205732+ 120574

2119890120573119905 cos (120574119905)

1198780120573120574 (119905) =

120573

1205732+ 120574

2119890120573119905 sin (120574119905) minus

120574

1205732+ 120574

2119890120573119905 cos (120574119905)

(63)


119862119898120573120574 (119905) = 119905

1198981198620120573120574 (119905) minus 119898

120574

1205732+ 120574

2119878119898minus1120573120574

minus 119898

120573

1205732+ 120574

2119862119898minus1120573120574 (

119905)

119878119898120573120574 (119905) = 119905

1198981198780120573120574 (119905) minus 119898

120573

1205732+ 120574

2119878119898minus1120573120574

+ 119898

120574

1205732+ 120574

2119862119898minus1120573120574 (

119905)

(64)




119909(4)+ 4 + 3 + 2 + 119909 = 0 (65)


119909 (0) = 1 (0) = 0 (0) = 0 (0) = 0

(66)


(119905) = 119860119883 (119905) (67)


119860 = (

0 1 0 0

0 0 1 0

0 0 0 1


) (68)


119861 = (

0 1 0 0

0 0 1 0

0 0 0 1

0 0 0 0

) 119862 = (

0 0 0 0

0 0 0 0

0 0 0 0


)

(69)


Equation (36)

(119890119861119905119898+ 119867 (119905119898))

119898

=

119898

sum

119896=0

120582119896(119905 119898) (119890

119861119905119898 119869(119905 119898))

119898

119896

(70)


119890119861119905119898=

(

(

(

(

(

(

(

1

119905

119898

1199052

(2119898)

1199053

(6119898)

0 1

119905

119898

1199052

(2119898)

0 0 1

119905

119898

0 0 0 1

)

)

)

)

)

)

)

119867(119905119898) = 120582 (119905 119898) 119869 (119905 119898)

(71)

with

120582 (119905 119898) =

1

4

(eminus4119905119898 minus 1)

119869 (119905 119898) =

(

(

(

(

(

(

1199053

(61198983)

21199053

(61198983)

31199053

(61198983)

41199053

(61198983)

1199052

(21198982)

21199052

(21198982)

31199052

(21198982)

41199052

(21198982)

119905

119898

2119905

119898

3119905

119898

4119905

119898

1 2 3 4

)

)

)

)

)

)

(72)


119890119860119905≃

3

sum

119896=0

120582119896(119905 3) (119890

1198611199053 119869 (119905 3))

3

119896

= A3+ 120582 (A

2B +BA

2+ABA)

+ 1205822(AB

2+B

2A +BAB) + 120582

3B

3

(73)


A = 1198901198611199053 B = 119869 (119905 3) (74)

0 10987654321

05

minus05

0

x1(t

)

t



A3=

(

(

(

(

1 119905

1199052

2

1199053

6

0 1 119905

1199052

2

0 0 1 119905

0 0 0 1

)

)

)

)

A2B =(

(

(

(

1199053

6

1199053

3

1199053

2

21199053

3

1199052

2

1199052 3119905

2

2

21199052

119905 2119905 3119905 4119905

1 2 3 4

)

)

)

)

AB2

=(

(AB2)11(AB2

)12(AB2

)13(AB2

)14

(AB2)21(AB2

)22(AB2

)23(AB2

)24

(AB2)31(AB2

)32(AB2

)33(AB2

)34

(AB2)41(AB2

)42(AB2

)43(AB2

)44

)

(75)

where

(AB2)11=

2

6561

1199056+

4

729

1199055+

4

81

1199054+

16

81

1199053

(AB2)12=

4

6561

1199056+

8

729

1199055+

8

81

1199054+

32

81

1199053


Exponential matrix


Coeff and func

ODE

Initial conditions

ODE order

Initial cond

2

0 1

1 1 1

x t

u(t)

sin(t)

t = 0

a1 a0a2

x10 x20


(AB2)13=

2

2187

1199056+

4

243

1199055+

4

27

1199054+

16

27

1199053

(AB2)14=

8

6561

1199056+

16

729

1199055+

16

81

1199054+

64

81

1199053

(AB2)21=

1

729

1199055+

2

81

1199054+

2

9

1199053+

8

9

1199052

(AB2)22=

2

729

1199055+

4

81

1199054+

4

9

1199053+

16

9

1199052

(AB2)23=

1

243

1199055+

2

27

1199054+

2

3

1199053+

8

3

1199052

(AB2)24=

4

729

1199055+

8

81

1199054+

8

9

1199053+

32

9

1199052

(AB2)31=

1

243

1199054+

2

27

1199053+

2

3

1199052+

8

3

119905

(AB2)32=

2

243

1199054+

4

27

1199053+

4

3

1199052+

16

3

119905

(AB2)33=

1

81

1199054+

2

9

1199053+ 2119905

2+ 8119905

(AB2)34=

4

243

1199054+

8

27

1199053+

8

3

1199052+

32

3

119905

(AB2)41=

1

162

1199053+

1

9

1199052+ 119905 + 4

(AB2)42=

1

81

1199053+

2

9

1199052+ 2119905 + 8

(AB2)43=

1

54

1199053+

1

3

1199052+ 3119905 + 12

(AB2)44=

2

81

1199053+

4

9

1199052+ 4119905 + 16

ABA

= (


)

(76)

where

(ABA)11 =4

81

1199053

(ABA)12 =4

243

1199054+

8

81

1199053

(ABA)13 =2

729

1199055+

8

243

1199054+

4

27

1199053

(ABA)14 =2

6561

1199056+

4

729

1199055+

4

81

1199054+

16

81

1199053

(ABA)21 =2

9

1199052

(ABA)22 =2

27

1199053+

4

9

1199052

(ABA)23 =1

81

1199054+

4

27

1199053+

2

3

1199052

(ABA)24 =1

729

1199055+

2

81

1199054+

2

9

1199053+

8

9

1199052

(ABA)31 =2

3

119905

(ABA)32 =2

9

1199052+

4

3

119905


(ABA)33 =1

27

1199053+

4

9

1199052+ 2119905

(ABA)34 =1

243

1199054+

2

27

1199053+

2

3

1199052+

8

3

119905

(ABA)41 = 1

(ABA)42 =1

3

119905 + 2

(ABA)43 =1

18

1199052+

2

3

119905 + 3

(ABA)44 =1

162

1199053+

1

9

1199052+ 119905 + 4

B3=(

(

(B3)11(B3)12(B3)13(B3)14

(B3)21(B3)22(B3)23(B3)24

(B3)31(B3)32(B3)33(B3)34

(B3)41(B3)42(B3)43(B3)44

)

)

(77)

where

(B3)11=

1

4251528

1199059+

1

118098

1199058+

1

6561

1199057+

11

6561

1199056

+

17

1458

1199055+

4

81

1199054+

8

81

1199053

(B3)12=

1

2125764

1199059+

1

59049

1199058+

2

6561

1199057+

22

6561

1199056

+

17

729

1199055+

8

81

1199054+

16

81

1199053

(B3)13=

1

1417176

1199059+

1

39366

1199058+

1

2187

1199057+

11

2187

1199056

+

17

486

1199055+

4

27

1199054+

8

27

1199053

(B3)14=

1

1062882

1199059+

2

59049

1199058+

4

6561

1199057+

44

6561

1199056

+

34

729

1199055+

16

81

1199054+

32

81

1199053

(B3)21=

1

472392

1199058+

1

13122

1199057+

1

729

1199056+

11

729

1199055

+

17

162

1199054+

4

9

1199053+

8

9

1199052

(B3)22=

1

236196

1199058+

1

6561

1199057+

2

729

1199056+

22

729

1199055

+

17

81

1199054+

8

9

1199053+

16

9

1199052

(B3)23=

1

157464

1199058+

1

4374

1199057+

1

243

1199056+

11

243

1199055

+

17

54

1199054+

4

3

1199053+

8

3

1199052

(B3)24=

1

118098

1199058+

2

6561

1199057+

4

729

1199056+

44

729

1199055

+

34

81

1199054+

16

9

1199053+

32

9

1199052

(B3)31=

1

78732

1199057+

1

2187

1199056+

2

243

1199055+

22

243

1199054

+

17

27

1199053+

8

3

1199052+

16

3

119905

(B3)32=

1

39366

1199057+

2

2187

1199056+

4

243

1199055+

44

243

1199054

+

34

27

1199053+

16

3

1199052+

32

3

119905

(B3)33=

1

26244

1199057+

1

729

1199056+

2

81

1199055+

22

81

1199054

+

17

9

1199053+ 8119905

2+ 16119905

(B3)34=

1

19683

1199057+

4

2187

1199056+

8

243

1199055+

88

243

1199054

+

68

27

1199053+

32

3

1199052+

64

3

119905

(B3)41=

1

26244

1199056+

1

729

1199055+

2

81

1199054+

22

81

1199053

+

17

9

1199052+ 8119905 + 16

(B3)42=

1

13122

1199056+

2

729

1199055+

4

81

1199054+

44

81

1199053

+

34

9

1199052+ 16119905 + 32

(B3)43=

1

8748

1199056+

1

243

1199055+

2

27

1199054+

22

27

1199053

+

17

3

1199052+ 24119905 + 48

(B3)44=

1

6561

1199056+

4

729

1199055+

8

81

1199054+

88

81

1199053

+

68

9

1199052+ 32119905 + 64

B2A

=(

(B2A)11(B2A)

12(B2A)

13(B2A)

14

(B2A)21(B2A)

22(B2A)

23(B2A)

24

(B2A)31(B2A)

32(B2A)

33(B2A)

34

(B2A)41(B2A)

42(B2A)

43(B2A)

44

)

(78)


where

(B2A)

11=

1

26244

1199056+

1

1458

1199055+

1

162

1199054+

2

81

1199053

(B2A)

12=

1

78732

1199057+

2

6561

1199056+

5

1458

1199055+

5

243

1199054+

4

81

1199053

(B2A)

13=

1

472392

1199058+

5

78732

1199057+

2

2187

1199056+

11

1458

1199055

+

17

486

1199054+

2

27

1199053

(B2A)

14=

1

4251528

1199059+

1

118098

1199058+

1

6561

1199057+

11

6561

1199056

+

17

1458

1199055+

4

81

1199054+

8

81

1199053

(B2A)

21=

1

2916

1199055+

1

162

1199054+

1

18

1199053+

2

9

1199052

(B2A)

22=

1

8748

1199056+

2

729

1199055+

5

162

1199054+

5

27

1199053+

4

9

1199052

(B2A)

23=

1

52488

1199057+

5

8748

1199056+

2

243

1199055+

11

162

1199054

+

17

54

1199053+

2

3

1199052

(B2A)

24=

1

472392

1199058+

1

13122

1199057+

1

729

1199056+

11

729

1199055

+

17

162

1199054+

4

9

1199053+

8

9

1199052

(B2A)

31=

1

486

1199054+

1

27

1199053+

1

3

1199052+

4

3

119905

(B2A)

32=

1

1458

1199055+

4

243

1199054+

5

27

1199053+

10

9

1199052+

8

3

119905

(B2A)

33=

1

8748

1199056+

5

1458

1199055+

4

81

1199054+

11

27

1199053+

17

9

1199052+ 4119905

(B2A)

34=

1

78732

1199057+

1

2187

1199056+

2

243

1199055+

22

243

1199054

+

17

27

1199053+

8

3

1199052+

16

3

119905

(B2A)

41=

1

162

1199053+

1

9

1199052+ 119905 + 4

(B2A)

42=

1

486

1199054+

4

81

1199053+

5

9

1199052+

10

3

119905 + 8

(B2A)

43=

1

2916

1199055+

5

486

1199054+

4

27

1199053+

11

9

1199052+

17

3

119905 + 12

(B2A)

44=

1

26244

1199056+

1

729

1199055+

2

81

1199054+

22

81

1199053+

17

9

1199052

+ 8119905 + 16

BA2

=(

(

(BA2)11(BA2

)12(BA2

)13(BA2

)14

(BA2)21(BA2

)22(BA2

)23(BA2

)24

(BA2)31(BA2

)32(BA2

)33(BA2

)34

(BA2)41(BA2

)42(BA2

)43(BA2

)44

)

)

(79)

where

(BA2)11=

1

162

1199053

(BA2)12=

1

243

1199054+

1

81

1199053

(BA2)13=

1

729

1199055+

2

243

1199054+

1

54

1199053

(BA2)14=

2

6561

1199056+

2

729

1199055+

1

81

1199054+

2

81

1199053

(BA2)21=

1

18

1199052

(BA2)22=

1

27

1199053+

1

9

1199052

(BA2)23=

1

81

1199054+

2

27

1199053+

1

6

1199052

(BA2)24=

2

729

1199055+

2

81

1199054+

1

9

1199053+

2

9

1199052

(BA2)31=

1

3

119905

(BA2)32=

2

9

1199052+

2

3

119905

(BA2)33=

2

27

1199053+

4

9

1199052+ 119905

(BA2)34=

4

243

1199054+

4

27

1199053+

2

3

1199052+

4

3

119905

(BA2)41= 1

(BA2)42=

2

3

119905 + 2

(BA2)43=

2

9

1199052+

4

3

119905 + 3

(BA2)44=

4

81

1199053+

4

9

1199052+ 2119905 + 4


BAB

= (


)

(80)

where

(BAB)11 =2

6561

1199056+

2

729

1199055+

1

81

1199054+

2

81

1199053

(BAB)11 =4

6561

1199056+

4

729

1199055+

2

81

1199054+

4

81

1199053

(BAB)11 =2

2187

1199056+

2

243

1199055+

1

27

1199054+

2

27

1199053

(BAB)11 =8

6561

1199056+

8

729

1199055+

4

81

1199054+

8

81

1199053

(BAB)11 =2

729

1199055+

2

81

1199054+

1

9

1199053+

2

9

1199052

(BAB)11 =4

729

1199055+

4

81

1199054+

2

9

1199053+

4

9

1199052

(BAB)11 =2

243

1199055+

2

27

1199054+

1

3

1199053+

2

3

1199052

(BAB)11 =8

729

1199055+

8

81

1199054+

4

9

1199053+

8

9

1199052

(BAB)11 =4

243

1199054+

4

27

1199053+

2

3

1199052+

4

3

119905

(BAB)11 =8

243

1199054+

8

27

1199053+

4

3

1199052+

8

3

119905

(BAB)11 =4

81

1199054+

4

9

1199053+ 2119905

2+ 4119905

(BAB)11 =16

243

1199054+

16

27

1199053+

8

3

1199052+

16

3

119905

(BAB)11 =4

81

1199053+

4

9

1199052+ 2119905 + 4

(BAB)11 =8

81

1199053+

8

9

1199052+ 4119905 + 8

(BAB)11 =4

27

1199053+

4

3

1199052+ 6119905 + 12

(BAB)11 =16

81

1199053+

16

9

1199052+ 8119905 + 16

(81)


(AB)3

0= A

3 (AB)

3

3=B

3

(AB)3

1= (A

2B +BA

2+ABA)

= (

12057211120572121205721312057214

12057221120572221205722312057224

12057231120572321205723312057234

12057241120572421205724312057244

)

(82)

where

12057211=

2

9

1199053

12057212=

4

9

1199053+

5

243

1199054

12057213=

2

3

1199053+

1

243

1199055+

10

243

1199054

12057214=

8

9

1199053+

4

6561

1199056+

2

243

1199055+

5

81

1199054

12057221=

7

9

1199052

12057222=

14

9

1199052+

1

9

1199053

12057223=

7

3

1199052+

2

81

1199054+

2

9

1199053

12057224=

28

9

1199052+

1

243

1199055+

4

81

1199054+

1

3

1199053

12057231= 2119905

12057232= 4119905 +

4

9

1199052

12057233= 6119905 +

1

9

1199053+

8

9

1199052

12057234= 8119905 +

5

243

1199054+

2

9

1199053+

4

3

1199052

12057241= 3

12057242= 6 + 119905

12057243= 9 +

5

18

1199052+ 2119905

12057244= 12 +

1

18

1199053+

5

9

1199052+ 3119905

(AB)3

2= (AB

2+B

2A +BAB)

= (

12057311120573121205731312057314

12057321120573221205732312057324

12057331120573321205733312057334

12057341120573421205734312057344

)

(83)

being

12057311=

17

26244

1199056+

13

1458

1199055+

11

162

1199054+

20

81

1199053

12057312=

10

6561

1199056+

29

1458

1199055+

35

243

1199054+

40

81

1199053+

1

78732

1199057

12057313=

2

729

1199056+

47

1458

1199055+

107

486

1199054+

20

27

1199053

+

1

472392

1199058+

5

78732

1199057


12057314=

1

243

1199056+

65

1458

1199055+

8

27

1199054+

80

81

1199053+

1

4251528

1199059

+

1

118098

1199058+

1

6561

1199057

12057321=

13

2916

1199055+

1

18

1199054+

7

18

1199053+

4

3

1199052

12057322=

8

729

1199055+

7

54

1199054+

23

27

1199053+

8

3

1199052+

1

8748

1199056

12057323=

5

243

1199055+

35

162

1199054+

71

54

1199053+ 4119905

2+

1

52488

1199057+

5

8748

1199056

12057324=

23

729

1199055+

49

162

1199054+

16

9

1199053+

16

3

1199052+

1

472392

1199058

+

1

13122

1199057+

1

729

1199056

12057331=

11

486

1199054+

7

27

1199053+

5

3

1199052+

16

3

119905

12057332=

14

243

1199054+

17

27

1199053+

34

9

1199052+

32

3

119905 +

1

1458

1199055

12057333=

1

9

1199054+

29

27

1199053+

53

9

1199052+ 16119905 +

1

8748

1199056+

5

1458

1199055

12057334=

14

81

1199054+

41

27

1199053+ 8119905

2+

64

3

119905 +

1

78732

1199057

+

1

2187

1199056+

2

243

1199055

12057341=

5

81

1199053+

2

3

1199052+ 4119905 + 12

12057342=

13

81

1199053+

5

3

1199052+

28

3

119905 + 24 +

1

486

1199054

12057343=

17

54

1199053+

26

9

1199052+

44

3

119905 + 36 +

1

2916

1199055+

5

486

1199054

12057344=

40

81

1199053+

37

9

1199052+ 20119905 + 48 +

1

26244

1199056

+

1

729

1199055+

2

81

1199054

(84)



+ + 119909 = sin 119905 (85)

Matrix A

0 1

minus1 minus1


Matrix B

0 1

0 0



119909 (0) = 0 (0) = 1 (86)



2

119905) + radic3119890minus1199052 sin(

radic3

2

119905)

(87)





exp (Bm)

0

((tm)0)(10) ((tm)1)(11)

((tm)0)(10)


Matrix C

0 0

minus1 minus1








exp (Cm)

01




Automatic procedure


Precision

Digits

t

Value of m

18

13

1

1000



exp (A)

06596974858612 05335045275926

minus05335071951242 01261956258061




0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15


minus15

minus1

minus05

0

05

1

15

0 2 4 6 8 10


0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15



for119898 = 1000

Step 119905 119909(119905)

1 01 009516653368102 02 018133052249203 03 025948012999554 04 033058564791315 05 039559200286866 06 045541222101157 07 051092181897118 08 056295408367889 09 0612296198681210 10 06596861707192



119905 |119909119878(119905) minus 119909(119905)| |119909

119877(119905) minus 119909(119905)|

01 00000047853239 0000000167203104 00000712453497 0000035875126707 00002074165163 0000354460586810 00003710415717 0001430787361713 00004547161744 0003825311783616 00003284030320 0008043689956219 00000622406880 0014419715717422 00006216534547 0023032955185125 00011277539714 0033667545654128 00013572764337 0045813471294231 00012347982225 0058707643120834 00008953272672 0071407807759437 00006002231245 0082889689449740 00005492334199 0092155687292443 00007154938930 0098343146001446 00008300921248 0100820433067849 00005487250652 0099260855115152 00002963137375 0093686827986155 00015298326576 0084479859363158 00026838757810 0072355393614361 00032571351779 0058304626062764 00030289419613 0043509187995267 00022189180928 0029236498885170 00013691340100 0016725709670673 00010043530807 0007075055562276 00012821439144 0001140917647379 00018689726696 0000541733952482 00021436005756 0002189733032385 00016148602301 0009108229122588 00002869277587 0019614707845191 00012796703448 0032789720352394 00022771057968 0047474001777797 00021518593816 00623710756305100 00009803998394 00761625070468


0 2 4 6 8 100

002

004

006

008

01

012



5 Conclusions


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Equation (36)

(119890119861119905119898+ 119867 (119905119898))

119898

=

119898

sum

119896=0

120582119896(119905 119898) (119890

119861119905119898 119869(119905 119898))

119898

119896

(70)


119890119861119905119898=

(

(

(

(

(

(

(

1

119905

119898

1199052

(2119898)

1199053

(6119898)

0 1

119905

119898

1199052

(2119898)

0 0 1

119905

119898

0 0 0 1

)

)

)

)

)

)

)

119867(119905119898) = 120582 (119905 119898) 119869 (119905 119898)

(71)

with

120582 (119905 119898) =

1

4

(eminus4119905119898 minus 1)

119869 (119905 119898) =

(

(

(

(

(

(

1199053

(61198983)

21199053

(61198983)

31199053

(61198983)

41199053

(61198983)

1199052

(21198982)

21199052

(21198982)

31199052

(21198982)

41199052

(21198982)

119905

119898

2119905

119898

3119905

119898

4119905

119898

1 2 3 4

)

)

)

)

)

)

(72)


119890119860119905≃

3

sum

119896=0

120582119896(119905 3) (119890

1198611199053 119869 (119905 3))

3

119896

= A3+ 120582 (A

2B +BA

2+ABA)

+ 1205822(AB

2+B

2A +BAB) + 120582

3B

3

(73)


A = 1198901198611199053 B = 119869 (119905 3) (74)

0 10987654321

05

minus05

0

x1(t

)

t



A3=

(

(

(

(

1 119905

1199052

2

1199053

6

0 1 119905

1199052

2

0 0 1 119905

0 0 0 1

)

)

)

)

A2B =(

(

(

(

1199053

6

1199053

3

1199053

2

21199053

3

1199052

2

1199052 3119905

2

2

21199052

119905 2119905 3119905 4119905

1 2 3 4

)

)

)

)

AB2

=(

(AB2)11(AB2

)12(AB2

)13(AB2

)14

(AB2)21(AB2

)22(AB2

)23(AB2

)24

(AB2)31(AB2

)32(AB2

)33(AB2

)34

(AB2)41(AB2

)42(AB2

)43(AB2

)44

)

(75)

where

(AB2)11=

2

6561

1199056+

4

729

1199055+

4

81

1199054+

16

81

1199053

(AB2)12=

4

6561

1199056+

8

729

1199055+

8

81

1199054+

32

81

1199053


Exponential matrix


Coeff and func

ODE

Initial conditions

ODE order

Initial cond

2

0 1

1 1 1

x t

u(t)

sin(t)

t = 0

a1 a0a2

x10 x20


(AB2)13=

2

2187

1199056+

4

243

1199055+

4

27

1199054+

16

27

1199053

(AB2)14=

8

6561

1199056+

16

729

1199055+

16

81

1199054+

64

81

1199053

(AB2)21=

1

729

1199055+

2

81

1199054+

2

9

1199053+

8

9

1199052

(AB2)22=

2

729

1199055+

4

81

1199054+

4

9

1199053+

16

9

1199052

(AB2)23=

1

243

1199055+

2

27

1199054+

2

3

1199053+

8

3

1199052

(AB2)24=

4

729

1199055+

8

81

1199054+

8

9

1199053+

32

9

1199052

(AB2)31=

1

243

1199054+

2

27

1199053+

2

3

1199052+

8

3

119905

(AB2)32=

2

243

1199054+

4

27

1199053+

4

3

1199052+

16

3

119905

(AB2)33=

1

81

1199054+

2

9

1199053+ 2119905

2+ 8119905

(AB2)34=

4

243

1199054+

8

27

1199053+

8

3

1199052+

32

3

119905

(AB2)41=

1

162

1199053+

1

9

1199052+ 119905 + 4

(AB2)42=

1

81

1199053+

2

9

1199052+ 2119905 + 8

(AB2)43=

1

54

1199053+

1

3

1199052+ 3119905 + 12

(AB2)44=

2

81

1199053+

4

9

1199052+ 4119905 + 16

ABA

= (


)

(76)

where

(ABA)11 =4

81

1199053

(ABA)12 =4

243

1199054+

8

81

1199053

(ABA)13 =2

729

1199055+

8

243

1199054+

4

27

1199053

(ABA)14 =2

6561

1199056+

4

729

1199055+

4

81

1199054+

16

81

1199053

(ABA)21 =2

9

1199052

(ABA)22 =2

27

1199053+

4

9

1199052

(ABA)23 =1

81

1199054+

4

27

1199053+

2

3

1199052

(ABA)24 =1

729

1199055+

2

81

1199054+

2

9

1199053+

8

9

1199052

(ABA)31 =2

3

119905

(ABA)32 =2

9

1199052+

4

3

119905


(ABA)33 =1

27

1199053+

4

9

1199052+ 2119905

(ABA)34 =1

243

1199054+

2

27

1199053+

2

3

1199052+

8

3

119905

(ABA)41 = 1

(ABA)42 =1

3

119905 + 2

(ABA)43 =1

18

1199052+

2

3

119905 + 3

(ABA)44 =1

162

1199053+

1

9

1199052+ 119905 + 4

B3=(

(

(B3)11(B3)12(B3)13(B3)14

(B3)21(B3)22(B3)23(B3)24

(B3)31(B3)32(B3)33(B3)34

(B3)41(B3)42(B3)43(B3)44

)

)

(77)

where

(B3)11=

1

4251528

1199059+

1

118098

1199058+

1

6561

1199057+

11

6561

1199056

+

17

1458

1199055+

4

81

1199054+

8

81

1199053

(B3)12=

1

2125764

1199059+

1

59049

1199058+

2

6561

1199057+

22

6561

1199056

+

17

729

1199055+

8

81

1199054+

16

81

1199053

(B3)13=

1

1417176

1199059+

1

39366

1199058+

1

2187

1199057+

11

2187

1199056

+

17

486

1199055+

4

27

1199054+

8

27

1199053

(B3)14=

1

1062882

1199059+

2

59049

1199058+

4

6561

1199057+

44

6561

1199056

+

34

729

1199055+

16

81

1199054+

32

81

1199053

(B3)21=

1

472392

1199058+

1

13122

1199057+

1

729

1199056+

11

729

1199055

+

17

162

1199054+

4

9

1199053+

8

9

1199052

(B3)22=

1

236196

1199058+

1

6561

1199057+

2

729

1199056+

22

729

1199055

+

17

81

1199054+

8

9

1199053+

16

9

1199052

(B3)23=

1

157464

1199058+

1

4374

1199057+

1

243

1199056+

11

243

1199055

+

17

54

1199054+

4

3

1199053+

8

3

1199052

(B3)24=

1

118098

1199058+

2

6561

1199057+

4

729

1199056+

44

729

1199055

+

34

81

1199054+

16

9

1199053+

32

9

1199052

(B3)31=

1

78732

1199057+

1

2187

1199056+

2

243

1199055+

22

243

1199054

+

17

27

1199053+

8

3

1199052+

16

3

119905

(B3)32=

1

39366

1199057+

2

2187

1199056+

4

243

1199055+

44

243

1199054

+

34

27

1199053+

16

3

1199052+

32

3

119905

(B3)33=

1

26244

1199057+

1

729

1199056+

2

81

1199055+

22

81

1199054

+

17

9

1199053+ 8119905

2+ 16119905

(B3)34=

1

19683

1199057+

4

2187

1199056+

8

243

1199055+

88

243

1199054

+

68

27

1199053+

32

3

1199052+

64

3

119905

(B3)41=

1

26244

1199056+

1

729

1199055+

2

81

1199054+

22

81

1199053

+

17

9

1199052+ 8119905 + 16

(B3)42=

1

13122

1199056+

2

729

1199055+

4

81

1199054+

44

81

1199053

+

34

9

1199052+ 16119905 + 32

(B3)43=

1

8748

1199056+

1

243

1199055+

2

27

1199054+

22

27

1199053

+

17

3

1199052+ 24119905 + 48

(B3)44=

1

6561

1199056+

4

729

1199055+

8

81

1199054+

88

81

1199053

+

68

9

1199052+ 32119905 + 64

B2A

=(

(B2A)11(B2A)

12(B2A)

13(B2A)

14

(B2A)21(B2A)

22(B2A)

23(B2A)

24

(B2A)31(B2A)

32(B2A)

33(B2A)

34

(B2A)41(B2A)

42(B2A)

43(B2A)

44

)

(78)


where

(B2A)

11=

1

26244

1199056+

1

1458

1199055+

1

162

1199054+

2

81

1199053

(B2A)

12=

1

78732

1199057+

2

6561

1199056+

5

1458

1199055+

5

243

1199054+

4

81

1199053

(B2A)

13=

1

472392

1199058+

5

78732

1199057+

2

2187

1199056+

11

1458

1199055

+

17

486

1199054+

2

27

1199053

(B2A)

14=

1

4251528

1199059+

1

118098

1199058+

1

6561

1199057+

11

6561

1199056

+

17

1458

1199055+

4

81

1199054+

8

81

1199053

(B2A)

21=

1

2916

1199055+

1

162

1199054+

1

18

1199053+

2

9

1199052

(B2A)

22=

1

8748

1199056+

2

729

1199055+

5

162

1199054+

5

27

1199053+

4

9

1199052

(B2A)

23=

1

52488

1199057+

5

8748

1199056+

2

243

1199055+

11

162

1199054

+

17

54

1199053+

2

3

1199052

(B2A)

24=

1

472392

1199058+

1

13122

1199057+

1

729

1199056+

11

729

1199055

+

17

162

1199054+

4

9

1199053+

8

9

1199052

(B2A)

31=

1

486

1199054+

1

27

1199053+

1

3

1199052+

4

3

119905

(B2A)

32=

1

1458

1199055+

4

243

1199054+

5

27

1199053+

10

9

1199052+

8

3

119905

(B2A)

33=

1

8748

1199056+

5

1458

1199055+

4

81

1199054+

11

27

1199053+

17

9

1199052+ 4119905

(B2A)

34=

1

78732

1199057+

1

2187

1199056+

2

243

1199055+

22

243

1199054

+

17

27

1199053+

8

3

1199052+

16

3

119905

(B2A)

41=

1

162

1199053+

1

9

1199052+ 119905 + 4

(B2A)

42=

1

486

1199054+

4

81

1199053+

5

9

1199052+

10

3

119905 + 8

(B2A)

43=

1

2916

1199055+

5

486

1199054+

4

27

1199053+

11

9

1199052+

17

3

119905 + 12

(B2A)

44=

1

26244

1199056+

1

729

1199055+

2

81

1199054+

22

81

1199053+

17

9

1199052

+ 8119905 + 16

BA2

=(

(

(BA2)11(BA2

)12(BA2

)13(BA2

)14

(BA2)21(BA2

)22(BA2

)23(BA2

)24

(BA2)31(BA2

)32(BA2

)33(BA2

)34

(BA2)41(BA2

)42(BA2

)43(BA2

)44

)

)

(79)

where

(BA2)11=

1

162

1199053

(BA2)12=

1

243

1199054+

1

81

1199053

(BA2)13=

1

729

1199055+

2

243

1199054+

1

54

1199053

(BA2)14=

2

6561

1199056+

2

729

1199055+

1

81

1199054+

2

81

1199053

(BA2)21=

1

18

1199052

(BA2)22=

1

27

1199053+

1

9

1199052

(BA2)23=

1

81

1199054+

2

27

1199053+

1

6

1199052

(BA2)24=

2

729

1199055+

2

81

1199054+

1

9

1199053+

2

9

1199052

(BA2)31=

1

3

119905

(BA2)32=

2

9

1199052+

2

3

119905

(BA2)33=

2

27

1199053+

4

9

1199052+ 119905

(BA2)34=

4

243

1199054+

4

27

1199053+

2

3

1199052+

4

3

119905

(BA2)41= 1

(BA2)42=

2

3

119905 + 2

(BA2)43=

2

9

1199052+

4

3

119905 + 3

(BA2)44=

4

81

1199053+

4

9

1199052+ 2119905 + 4


BAB

= (


)

(80)

where

(BAB)11 =2

6561

1199056+

2

729

1199055+

1

81

1199054+

2

81

1199053

(BAB)11 =4

6561

1199056+

4

729

1199055+

2

81

1199054+

4

81

1199053

(BAB)11 =2

2187

1199056+

2

243

1199055+

1

27

1199054+

2

27

1199053

(BAB)11 =8

6561

1199056+

8

729

1199055+

4

81

1199054+

8

81

1199053

(BAB)11 =2

729

1199055+

2

81

1199054+

1

9

1199053+

2

9

1199052

(BAB)11 =4

729

1199055+

4

81

1199054+

2

9

1199053+

4

9

1199052

(BAB)11 =2

243

1199055+

2

27

1199054+

1

3

1199053+

2

3

1199052

(BAB)11 =8

729

1199055+

8

81

1199054+

4

9

1199053+

8

9

1199052

(BAB)11 =4

243

1199054+

4

27

1199053+

2

3

1199052+

4

3

119905

(BAB)11 =8

243

1199054+

8

27

1199053+

4

3

1199052+

8

3

119905

(BAB)11 =4

81

1199054+

4

9

1199053+ 2119905

2+ 4119905

(BAB)11 =16

243

1199054+

16

27

1199053+

8

3

1199052+

16

3

119905

(BAB)11 =4

81

1199053+

4

9

1199052+ 2119905 + 4

(BAB)11 =8

81

1199053+

8

9

1199052+ 4119905 + 8

(BAB)11 =4

27

1199053+

4

3

1199052+ 6119905 + 12

(BAB)11 =16

81

1199053+

16

9

1199052+ 8119905 + 16

(81)


(AB)3

0= A

3 (AB)

3

3=B

3

(AB)3

1= (A

2B +BA

2+ABA)

= (

12057211120572121205721312057214

12057221120572221205722312057224

12057231120572321205723312057234

12057241120572421205724312057244

)

(82)

where

12057211=

2

9

1199053

12057212=

4

9

1199053+

5

243

1199054

12057213=

2

3

1199053+

1

243

1199055+

10

243

1199054

12057214=

8

9

1199053+

4

6561

1199056+

2

243

1199055+

5

81

1199054

12057221=

7

9

1199052

12057222=

14

9

1199052+

1

9

1199053

12057223=

7

3

1199052+

2

81

1199054+

2

9

1199053

12057224=

28

9

1199052+

1

243

1199055+

4

81

1199054+

1

3

1199053

12057231= 2119905

12057232= 4119905 +

4

9

1199052

12057233= 6119905 +

1

9

1199053+

8

9

1199052

12057234= 8119905 +

5

243

1199054+

2

9

1199053+

4

3

1199052

12057241= 3

12057242= 6 + 119905

12057243= 9 +

5

18

1199052+ 2119905

12057244= 12 +

1

18

1199053+

5

9

1199052+ 3119905

(AB)3

2= (AB

2+B

2A +BAB)

= (

12057311120573121205731312057314

12057321120573221205732312057324

12057331120573321205733312057334

12057341120573421205734312057344

)

(83)

being

12057311=

17

26244

1199056+

13

1458

1199055+

11

162

1199054+

20

81

1199053

12057312=

10

6561

1199056+

29

1458

1199055+

35

243

1199054+

40

81

1199053+

1

78732

1199057

12057313=

2

729

1199056+

47

1458

1199055+

107

486

1199054+

20

27

1199053

+

1

472392

1199058+

5

78732

1199057


12057314=

1

243

1199056+

65

1458

1199055+

8

27

1199054+

80

81

1199053+

1

4251528

1199059

+

1

118098

1199058+

1

6561

1199057

12057321=

13

2916

1199055+

1

18

1199054+

7

18

1199053+

4

3

1199052

12057322=

8

729

1199055+

7

54

1199054+

23

27

1199053+

8

3

1199052+

1

8748

1199056

12057323=

5

243

1199055+

35

162

1199054+

71

54

1199053+ 4119905

2+

1

52488

1199057+

5

8748

1199056

12057324=

23

729

1199055+

49

162

1199054+

16

9

1199053+

16

3

1199052+

1

472392

1199058

+

1

13122

1199057+

1

729

1199056

12057331=

11

486

1199054+

7

27

1199053+

5

3

1199052+

16

3

119905

12057332=

14

243

1199054+

17

27

1199053+

34

9

1199052+

32

3

119905 +

1

1458

1199055

12057333=

1

9

1199054+

29

27

1199053+

53

9

1199052+ 16119905 +

1

8748

1199056+

5

1458

1199055

12057334=

14

81

1199054+

41

27

1199053+ 8119905

2+

64

3

119905 +

1

78732

1199057

+

1

2187

1199056+

2

243

1199055

12057341=

5

81

1199053+

2

3

1199052+ 4119905 + 12

12057342=

13

81

1199053+

5

3

1199052+

28

3

119905 + 24 +

1

486

1199054

12057343=

17

54

1199053+

26

9

1199052+

44

3

119905 + 36 +

1

2916

1199055+

5

486

1199054

12057344=

40

81

1199053+

37

9

1199052+ 20119905 + 48 +

1

26244

1199056

+

1

729

1199055+

2

81

1199054

(84)



+ + 119909 = sin 119905 (85)

Matrix A

0 1

minus1 minus1


Matrix B

0 1

0 0



119909 (0) = 0 (0) = 1 (86)



2

119905) + radic3119890minus1199052 sin(

radic3

2

119905)

(87)





exp (Bm)

0

((tm)0)(10) ((tm)1)(11)

((tm)0)(10)


Matrix C

0 0

minus1 minus1








exp (Cm)

01




Automatic procedure


Precision

Digits

t

Value of m

18

13

1

1000



exp (A)

06596974858612 05335045275926

minus05335071951242 01261956258061




0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15


minus15

minus1

minus05

0

05

1

15

0 2 4 6 8 10


0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15



for119898 = 1000

Step 119905 119909(119905)

1 01 009516653368102 02 018133052249203 03 025948012999554 04 033058564791315 05 039559200286866 06 045541222101157 07 051092181897118 08 056295408367889 09 0612296198681210 10 06596861707192



119905 |119909119878(119905) minus 119909(119905)| |119909

119877(119905) minus 119909(119905)|

01 00000047853239 0000000167203104 00000712453497 0000035875126707 00002074165163 0000354460586810 00003710415717 0001430787361713 00004547161744 0003825311783616 00003284030320 0008043689956219 00000622406880 0014419715717422 00006216534547 0023032955185125 00011277539714 0033667545654128 00013572764337 0045813471294231 00012347982225 0058707643120834 00008953272672 0071407807759437 00006002231245 0082889689449740 00005492334199 0092155687292443 00007154938930 0098343146001446 00008300921248 0100820433067849 00005487250652 0099260855115152 00002963137375 0093686827986155 00015298326576 0084479859363158 00026838757810 0072355393614361 00032571351779 0058304626062764 00030289419613 0043509187995267 00022189180928 0029236498885170 00013691340100 0016725709670673 00010043530807 0007075055562276 00012821439144 0001140917647379 00018689726696 0000541733952482 00021436005756 0002189733032385 00016148602301 0009108229122588 00002869277587 0019614707845191 00012796703448 0032789720352394 00022771057968 0047474001777797 00021518593816 00623710756305100 00009803998394 00761625070468


0 2 4 6 8 100

002

004

006

008

01

012



5 Conclusions


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Exponential matrix


Coeff and func

ODE

Initial conditions

ODE order

Initial cond

2

0 1

1 1 1

x t

u(t)

sin(t)

t = 0

a1 a0a2

x10 x20


(AB2)13=

2

2187

1199056+

4

243

1199055+

4

27

1199054+

16

27

1199053

(AB2)14=

8

6561

1199056+

16

729

1199055+

16

81

1199054+

64

81

1199053

(AB2)21=

1

729

1199055+

2

81

1199054+

2

9

1199053+

8

9

1199052

(AB2)22=

2

729

1199055+

4

81

1199054+

4

9

1199053+

16

9

1199052

(AB2)23=

1

243

1199055+

2

27

1199054+

2

3

1199053+

8

3

1199052

(AB2)24=

4

729

1199055+

8

81

1199054+

8

9

1199053+

32

9

1199052

(AB2)31=

1

243

1199054+

2

27

1199053+

2

3

1199052+

8

3

119905

(AB2)32=

2

243

1199054+

4

27

1199053+

4

3

1199052+

16

3

119905

(AB2)33=

1

81

1199054+

2

9

1199053+ 2119905

2+ 8119905

(AB2)34=

4

243

1199054+

8

27

1199053+

8

3

1199052+

32

3

119905

(AB2)41=

1

162

1199053+

1

9

1199052+ 119905 + 4

(AB2)42=

1

81

1199053+

2

9

1199052+ 2119905 + 8

(AB2)43=

1

54

1199053+

1

3

1199052+ 3119905 + 12

(AB2)44=

2

81

1199053+

4

9

1199052+ 4119905 + 16

ABA

= (


)

(76)

where

(ABA)11 =4

81

1199053

(ABA)12 =4

243

1199054+

8

81

1199053

(ABA)13 =2

729

1199055+

8

243

1199054+

4

27

1199053

(ABA)14 =2

6561

1199056+

4

729

1199055+

4

81

1199054+

16

81

1199053

(ABA)21 =2

9

1199052

(ABA)22 =2

27

1199053+

4

9

1199052

(ABA)23 =1

81

1199054+

4

27

1199053+

2

3

1199052

(ABA)24 =1

729

1199055+

2

81

1199054+

2

9

1199053+

8

9

1199052

(ABA)31 =2

3

119905

(ABA)32 =2

9

1199052+

4

3

119905


(ABA)33 =1

27

1199053+

4

9

1199052+ 2119905

(ABA)34 =1

243

1199054+

2

27

1199053+

2

3

1199052+

8

3

119905

(ABA)41 = 1

(ABA)42 =1

3

119905 + 2

(ABA)43 =1

18

1199052+

2

3

119905 + 3

(ABA)44 =1

162

1199053+

1

9

1199052+ 119905 + 4

B3=(

(

(B3)11(B3)12(B3)13(B3)14

(B3)21(B3)22(B3)23(B3)24

(B3)31(B3)32(B3)33(B3)34

(B3)41(B3)42(B3)43(B3)44

)

)

(77)

where

(B3)11=

1

4251528

1199059+

1

118098

1199058+

1

6561

1199057+

11

6561

1199056

+

17

1458

1199055+

4

81

1199054+

8

81

1199053

(B3)12=

1

2125764

1199059+

1

59049

1199058+

2

6561

1199057+

22

6561

1199056

+

17

729

1199055+

8

81

1199054+

16

81

1199053

(B3)13=

1

1417176

1199059+

1

39366

1199058+

1

2187

1199057+

11

2187

1199056

+

17

486

1199055+

4

27

1199054+

8

27

1199053

(B3)14=

1

1062882

1199059+

2

59049

1199058+

4

6561

1199057+

44

6561

1199056

+

34

729

1199055+

16

81

1199054+

32

81

1199053

(B3)21=

1

472392

1199058+

1

13122

1199057+

1

729

1199056+

11

729

1199055

+

17

162

1199054+

4

9

1199053+

8

9

1199052

(B3)22=

1

236196

1199058+

1

6561

1199057+

2

729

1199056+

22

729

1199055

+

17

81

1199054+

8

9

1199053+

16

9

1199052

(B3)23=

1

157464

1199058+

1

4374

1199057+

1

243

1199056+

11

243

1199055

+

17

54

1199054+

4

3

1199053+

8

3

1199052

(B3)24=

1

118098

1199058+

2

6561

1199057+

4

729

1199056+

44

729

1199055

+

34

81

1199054+

16

9

1199053+

32

9

1199052

(B3)31=

1

78732

1199057+

1

2187

1199056+

2

243

1199055+

22

243

1199054

+

17

27

1199053+

8

3

1199052+

16

3

119905

(B3)32=

1

39366

1199057+

2

2187

1199056+

4

243

1199055+

44

243

1199054

+

34

27

1199053+

16

3

1199052+

32

3

119905

(B3)33=

1

26244

1199057+

1

729

1199056+

2

81

1199055+

22

81

1199054

+

17

9

1199053+ 8119905

2+ 16119905

(B3)34=

1

19683

1199057+

4

2187

1199056+

8

243

1199055+

88

243

1199054

+

68

27

1199053+

32

3

1199052+

64

3

119905

(B3)41=

1

26244

1199056+

1

729

1199055+

2

81

1199054+

22

81

1199053

+

17

9

1199052+ 8119905 + 16

(B3)42=

1

13122

1199056+

2

729

1199055+

4

81

1199054+

44

81

1199053

+

34

9

1199052+ 16119905 + 32

(B3)43=

1

8748

1199056+

1

243

1199055+

2

27

1199054+

22

27

1199053

+

17

3

1199052+ 24119905 + 48

(B3)44=

1

6561

1199056+

4

729

1199055+

8

81

1199054+

88

81

1199053

+

68

9

1199052+ 32119905 + 64

B2A

=(

(B2A)11(B2A)

12(B2A)

13(B2A)

14

(B2A)21(B2A)

22(B2A)

23(B2A)

24

(B2A)31(B2A)

32(B2A)

33(B2A)

34

(B2A)41(B2A)

42(B2A)

43(B2A)

44

)

(78)


where

(B2A)

11=

1

26244

1199056+

1

1458

1199055+

1

162

1199054+

2

81

1199053

(B2A)

12=

1

78732

1199057+

2

6561

1199056+

5

1458

1199055+

5

243

1199054+

4

81

1199053

(B2A)

13=

1

472392

1199058+

5

78732

1199057+

2

2187

1199056+

11

1458

1199055

+

17

486

1199054+

2

27

1199053

(B2A)

14=

1

4251528

1199059+

1

118098

1199058+

1

6561

1199057+

11

6561

1199056

+

17

1458

1199055+

4

81

1199054+

8

81

1199053

(B2A)

21=

1

2916

1199055+

1

162

1199054+

1

18

1199053+

2

9

1199052

(B2A)

22=

1

8748

1199056+

2

729

1199055+

5

162

1199054+

5

27

1199053+

4

9

1199052

(B2A)

23=

1

52488

1199057+

5

8748

1199056+

2

243

1199055+

11

162

1199054

+

17

54

1199053+

2

3

1199052

(B2A)

24=

1

472392

1199058+

1

13122

1199057+

1

729

1199056+

11

729

1199055

+

17

162

1199054+

4

9

1199053+

8

9

1199052

(B2A)

31=

1

486

1199054+

1

27

1199053+

1

3

1199052+

4

3

119905

(B2A)

32=

1

1458

1199055+

4

243

1199054+

5

27

1199053+

10

9

1199052+

8

3

119905

(B2A)

33=

1

8748

1199056+

5

1458

1199055+

4

81

1199054+

11

27

1199053+

17

9

1199052+ 4119905

(B2A)

34=

1

78732

1199057+

1

2187

1199056+

2

243

1199055+

22

243

1199054

+

17

27

1199053+

8

3

1199052+

16

3

119905

(B2A)

41=

1

162

1199053+

1

9

1199052+ 119905 + 4

(B2A)

42=

1

486

1199054+

4

81

1199053+

5

9

1199052+

10

3

119905 + 8

(B2A)

43=

1

2916

1199055+

5

486

1199054+

4

27

1199053+

11

9

1199052+

17

3

119905 + 12

(B2A)

44=

1

26244

1199056+

1

729

1199055+

2

81

1199054+

22

81

1199053+

17

9

1199052

+ 8119905 + 16

BA2

=(

(

(BA2)11(BA2

)12(BA2

)13(BA2

)14

(BA2)21(BA2

)22(BA2

)23(BA2

)24

(BA2)31(BA2

)32(BA2

)33(BA2

)34

(BA2)41(BA2

)42(BA2

)43(BA2

)44

)

)

(79)

where

(BA2)11=

1

162

1199053

(BA2)12=

1

243

1199054+

1

81

1199053

(BA2)13=

1

729

1199055+

2

243

1199054+

1

54

1199053

(BA2)14=

2

6561

1199056+

2

729

1199055+

1

81

1199054+

2

81

1199053

(BA2)21=

1

18

1199052

(BA2)22=

1

27

1199053+

1

9

1199052

(BA2)23=

1

81

1199054+

2

27

1199053+

1

6

1199052

(BA2)24=

2

729

1199055+

2

81

1199054+

1

9

1199053+

2

9

1199052

(BA2)31=

1

3

119905

(BA2)32=

2

9

1199052+

2

3

119905

(BA2)33=

2

27

1199053+

4

9

1199052+ 119905

(BA2)34=

4

243

1199054+

4

27

1199053+

2

3

1199052+

4

3

119905

(BA2)41= 1

(BA2)42=

2

3

119905 + 2

(BA2)43=

2

9

1199052+

4

3

119905 + 3

(BA2)44=

4

81

1199053+

4

9

1199052+ 2119905 + 4


BAB

= (


)

(80)

where

(BAB)11 =2

6561

1199056+

2

729

1199055+

1

81

1199054+

2

81

1199053

(BAB)11 =4

6561

1199056+

4

729

1199055+

2

81

1199054+

4

81

1199053

(BAB)11 =2

2187

1199056+

2

243

1199055+

1

27

1199054+

2

27

1199053

(BAB)11 =8

6561

1199056+

8

729

1199055+

4

81

1199054+

8

81

1199053

(BAB)11 =2

729

1199055+

2

81

1199054+

1

9

1199053+

2

9

1199052

(BAB)11 =4

729

1199055+

4

81

1199054+

2

9

1199053+

4

9

1199052

(BAB)11 =2

243

1199055+

2

27

1199054+

1

3

1199053+

2

3

1199052

(BAB)11 =8

729

1199055+

8

81

1199054+

4

9

1199053+

8

9

1199052

(BAB)11 =4

243

1199054+

4

27

1199053+

2

3

1199052+

4

3

119905

(BAB)11 =8

243

1199054+

8

27

1199053+

4

3

1199052+

8

3

119905

(BAB)11 =4

81

1199054+

4

9

1199053+ 2119905

2+ 4119905

(BAB)11 =16

243

1199054+

16

27

1199053+

8

3

1199052+

16

3

119905

(BAB)11 =4

81

1199053+

4

9

1199052+ 2119905 + 4

(BAB)11 =8

81

1199053+

8

9

1199052+ 4119905 + 8

(BAB)11 =4

27

1199053+

4

3

1199052+ 6119905 + 12

(BAB)11 =16

81

1199053+

16

9

1199052+ 8119905 + 16

(81)


(AB)3

0= A

3 (AB)

3

3=B

3

(AB)3

1= (A

2B +BA

2+ABA)

= (

12057211120572121205721312057214

12057221120572221205722312057224

12057231120572321205723312057234

12057241120572421205724312057244

)

(82)

where

12057211=

2

9

1199053

12057212=

4

9

1199053+

5

243

1199054

12057213=

2

3

1199053+

1

243

1199055+

10

243

1199054

12057214=

8

9

1199053+

4

6561

1199056+

2

243

1199055+

5

81

1199054

12057221=

7

9

1199052

12057222=

14

9

1199052+

1

9

1199053

12057223=

7

3

1199052+

2

81

1199054+

2

9

1199053

12057224=

28

9

1199052+

1

243

1199055+

4

81

1199054+

1

3

1199053

12057231= 2119905

12057232= 4119905 +

4

9

1199052

12057233= 6119905 +

1

9

1199053+

8

9

1199052

12057234= 8119905 +

5

243

1199054+

2

9

1199053+

4

3

1199052

12057241= 3

12057242= 6 + 119905

12057243= 9 +

5

18

1199052+ 2119905

12057244= 12 +

1

18

1199053+

5

9

1199052+ 3119905

(AB)3

2= (AB

2+B

2A +BAB)

= (

12057311120573121205731312057314

12057321120573221205732312057324

12057331120573321205733312057334

12057341120573421205734312057344

)

(83)

being

12057311=

17

26244

1199056+

13

1458

1199055+

11

162

1199054+

20

81

1199053

12057312=

10

6561

1199056+

29

1458

1199055+

35

243

1199054+

40

81

1199053+

1

78732

1199057

12057313=

2

729

1199056+

47

1458

1199055+

107

486

1199054+

20

27

1199053

+

1

472392

1199058+

5

78732

1199057


12057314=

1

243

1199056+

65

1458

1199055+

8

27

1199054+

80

81

1199053+

1

4251528

1199059

+

1

118098

1199058+

1

6561

1199057

12057321=

13

2916

1199055+

1

18

1199054+

7

18

1199053+

4

3

1199052

12057322=

8

729

1199055+

7

54

1199054+

23

27

1199053+

8

3

1199052+

1

8748

1199056

12057323=

5

243

1199055+

35

162

1199054+

71

54

1199053+ 4119905

2+

1

52488

1199057+

5

8748

1199056

12057324=

23

729

1199055+

49

162

1199054+

16

9

1199053+

16

3

1199052+

1

472392

1199058

+

1

13122

1199057+

1

729

1199056

12057331=

11

486

1199054+

7

27

1199053+

5

3

1199052+

16

3

119905

12057332=

14

243

1199054+

17

27

1199053+

34

9

1199052+

32

3

119905 +

1

1458

1199055

12057333=

1

9

1199054+

29

27

1199053+

53

9

1199052+ 16119905 +

1

8748

1199056+

5

1458

1199055

12057334=

14

81

1199054+

41

27

1199053+ 8119905

2+

64

3

119905 +

1

78732

1199057

+

1

2187

1199056+

2

243

1199055

12057341=

5

81

1199053+

2

3

1199052+ 4119905 + 12

12057342=

13

81

1199053+

5

3

1199052+

28

3

119905 + 24 +

1

486

1199054

12057343=

17

54

1199053+

26

9

1199052+

44

3

119905 + 36 +

1

2916

1199055+

5

486

1199054

12057344=

40

81

1199053+

37

9

1199052+ 20119905 + 48 +

1

26244

1199056

+

1

729

1199055+

2

81

1199054

(84)



+ + 119909 = sin 119905 (85)

Matrix A

0 1

minus1 minus1


Matrix B

0 1

0 0



119909 (0) = 0 (0) = 1 (86)



2

119905) + radic3119890minus1199052 sin(

radic3

2

119905)

(87)





exp (Bm)

0

((tm)0)(10) ((tm)1)(11)

((tm)0)(10)


Matrix C

0 0

minus1 minus1








exp (Cm)

01




Automatic procedure


Precision

Digits

t

Value of m

18

13

1

1000



exp (A)

06596974858612 05335045275926

minus05335071951242 01261956258061




0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15


minus15

minus1

minus05

0

05

1

15

0 2 4 6 8 10


0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15



for119898 = 1000

Step 119905 119909(119905)

1 01 009516653368102 02 018133052249203 03 025948012999554 04 033058564791315 05 039559200286866 06 045541222101157 07 051092181897118 08 056295408367889 09 0612296198681210 10 06596861707192



119905 |119909119878(119905) minus 119909(119905)| |119909

119877(119905) minus 119909(119905)|

01 00000047853239 0000000167203104 00000712453497 0000035875126707 00002074165163 0000354460586810 00003710415717 0001430787361713 00004547161744 0003825311783616 00003284030320 0008043689956219 00000622406880 0014419715717422 00006216534547 0023032955185125 00011277539714 0033667545654128 00013572764337 0045813471294231 00012347982225 0058707643120834 00008953272672 0071407807759437 00006002231245 0082889689449740 00005492334199 0092155687292443 00007154938930 0098343146001446 00008300921248 0100820433067849 00005487250652 0099260855115152 00002963137375 0093686827986155 00015298326576 0084479859363158 00026838757810 0072355393614361 00032571351779 0058304626062764 00030289419613 0043509187995267 00022189180928 0029236498885170 00013691340100 0016725709670673 00010043530807 0007075055562276 00012821439144 0001140917647379 00018689726696 0000541733952482 00021436005756 0002189733032385 00016148602301 0009108229122588 00002869277587 0019614707845191 00012796703448 0032789720352394 00022771057968 0047474001777797 00021518593816 00623710756305100 00009803998394 00761625070468


0 2 4 6 8 100

002

004

006

008

01

012



5 Conclusions


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(ABA)33 =1

27

1199053+

4

9

1199052+ 2119905

(ABA)34 =1

243

1199054+

2

27

1199053+

2

3

1199052+

8

3

119905

(ABA)41 = 1

(ABA)42 =1

3

119905 + 2

(ABA)43 =1

18

1199052+

2

3

119905 + 3

(ABA)44 =1

162

1199053+

1

9

1199052+ 119905 + 4

B3=(

(

(B3)11(B3)12(B3)13(B3)14

(B3)21(B3)22(B3)23(B3)24

(B3)31(B3)32(B3)33(B3)34

(B3)41(B3)42(B3)43(B3)44

)

)

(77)

where

(B3)11=

1

4251528

1199059+

1

118098

1199058+

1

6561

1199057+

11

6561

1199056

+

17

1458

1199055+

4

81

1199054+

8

81

1199053

(B3)12=

1

2125764

1199059+

1

59049

1199058+

2

6561

1199057+

22

6561

1199056

+

17

729

1199055+

8

81

1199054+

16

81

1199053

(B3)13=

1

1417176

1199059+

1

39366

1199058+

1

2187

1199057+

11

2187

1199056

+

17

486

1199055+

4

27

1199054+

8

27

1199053

(B3)14=

1

1062882

1199059+

2

59049

1199058+

4

6561

1199057+

44

6561

1199056

+

34

729

1199055+

16

81

1199054+

32

81

1199053

(B3)21=

1

472392

1199058+

1

13122

1199057+

1

729

1199056+

11

729

1199055

+

17

162

1199054+

4

9

1199053+

8

9

1199052

(B3)22=

1

236196

1199058+

1

6561

1199057+

2

729

1199056+

22

729

1199055

+

17

81

1199054+

8

9

1199053+

16

9

1199052

(B3)23=

1

157464

1199058+

1

4374

1199057+

1

243

1199056+

11

243

1199055

+

17

54

1199054+

4

3

1199053+

8

3

1199052

(B3)24=

1

118098

1199058+

2

6561

1199057+

4

729

1199056+

44

729

1199055

+

34

81

1199054+

16

9

1199053+

32

9

1199052

(B3)31=

1

78732

1199057+

1

2187

1199056+

2

243

1199055+

22

243

1199054

+

17

27

1199053+

8

3

1199052+

16

3

119905

(B3)32=

1

39366

1199057+

2

2187

1199056+

4

243

1199055+

44

243

1199054

+

34

27

1199053+

16

3

1199052+

32

3

119905

(B3)33=

1

26244

1199057+

1

729

1199056+

2

81

1199055+

22

81

1199054

+

17

9

1199053+ 8119905

2+ 16119905

(B3)34=

1

19683

1199057+

4

2187

1199056+

8

243

1199055+

88

243

1199054

+

68

27

1199053+

32

3

1199052+

64

3

119905

(B3)41=

1

26244

1199056+

1

729

1199055+

2

81

1199054+

22

81

1199053

+

17

9

1199052+ 8119905 + 16

(B3)42=

1

13122

1199056+

2

729

1199055+

4

81

1199054+

44

81

1199053

+

34

9

1199052+ 16119905 + 32

(B3)43=

1

8748

1199056+

1

243

1199055+

2

27

1199054+

22

27

1199053

+

17

3

1199052+ 24119905 + 48

(B3)44=

1

6561

1199056+

4

729

1199055+

8

81

1199054+

88

81

1199053

+

68

9

1199052+ 32119905 + 64

B2A

=(

(B2A)11(B2A)

12(B2A)

13(B2A)

14

(B2A)21(B2A)

22(B2A)

23(B2A)

24

(B2A)31(B2A)

32(B2A)

33(B2A)

34

(B2A)41(B2A)

42(B2A)

43(B2A)

44

)

(78)


where

(B2A)

11=

1

26244

1199056+

1

1458

1199055+

1

162

1199054+

2

81

1199053

(B2A)

12=

1

78732

1199057+

2

6561

1199056+

5

1458

1199055+

5

243

1199054+

4

81

1199053

(B2A)

13=

1

472392

1199058+

5

78732

1199057+

2

2187

1199056+

11

1458

1199055

+

17

486

1199054+

2

27

1199053

(B2A)

14=

1

4251528

1199059+

1

118098

1199058+

1

6561

1199057+

11

6561

1199056

+

17

1458

1199055+

4

81

1199054+

8

81

1199053

(B2A)

21=

1

2916

1199055+

1

162

1199054+

1

18

1199053+

2

9

1199052

(B2A)

22=

1

8748

1199056+

2

729

1199055+

5

162

1199054+

5

27

1199053+

4

9

1199052

(B2A)

23=

1

52488

1199057+

5

8748

1199056+

2

243

1199055+

11

162

1199054

+

17

54

1199053+

2

3

1199052

(B2A)

24=

1

472392

1199058+

1

13122

1199057+

1

729

1199056+

11

729

1199055

+

17

162

1199054+

4

9

1199053+

8

9

1199052

(B2A)

31=

1

486

1199054+

1

27

1199053+

1

3

1199052+

4

3

119905

(B2A)

32=

1

1458

1199055+

4

243

1199054+

5

27

1199053+

10

9

1199052+

8

3

119905

(B2A)

33=

1

8748

1199056+

5

1458

1199055+

4

81

1199054+

11

27

1199053+

17

9

1199052+ 4119905

(B2A)

34=

1

78732

1199057+

1

2187

1199056+

2

243

1199055+

22

243

1199054

+

17

27

1199053+

8

3

1199052+

16

3

119905

(B2A)

41=

1

162

1199053+

1

9

1199052+ 119905 + 4

(B2A)

42=

1

486

1199054+

4

81

1199053+

5

9

1199052+

10

3

119905 + 8

(B2A)

43=

1

2916

1199055+

5

486

1199054+

4

27

1199053+

11

9

1199052+

17

3

119905 + 12

(B2A)

44=

1

26244

1199056+

1

729

1199055+

2

81

1199054+

22

81

1199053+

17

9

1199052

+ 8119905 + 16

BA2

=(

(

(BA2)11(BA2

)12(BA2

)13(BA2

)14

(BA2)21(BA2

)22(BA2

)23(BA2

)24

(BA2)31(BA2

)32(BA2

)33(BA2

)34

(BA2)41(BA2

)42(BA2

)43(BA2

)44

)

)

(79)

where

(BA2)11=

1

162

1199053

(BA2)12=

1

243

1199054+

1

81

1199053

(BA2)13=

1

729

1199055+

2

243

1199054+

1

54

1199053

(BA2)14=

2

6561

1199056+

2

729

1199055+

1

81

1199054+

2

81

1199053

(BA2)21=

1

18

1199052

(BA2)22=

1

27

1199053+

1

9

1199052

(BA2)23=

1

81

1199054+

2

27

1199053+

1

6

1199052

(BA2)24=

2

729

1199055+

2

81

1199054+

1

9

1199053+

2

9

1199052

(BA2)31=

1

3

119905

(BA2)32=

2

9

1199052+

2

3

119905

(BA2)33=

2

27

1199053+

4

9

1199052+ 119905

(BA2)34=

4

243

1199054+

4

27

1199053+

2

3

1199052+

4

3

119905

(BA2)41= 1

(BA2)42=

2

3

119905 + 2

(BA2)43=

2

9

1199052+

4

3

119905 + 3

(BA2)44=

4

81

1199053+

4

9

1199052+ 2119905 + 4


BAB

= (


)

(80)

where

(BAB)11 =2

6561

1199056+

2

729

1199055+

1

81

1199054+

2

81

1199053

(BAB)11 =4

6561

1199056+

4

729

1199055+

2

81

1199054+

4

81

1199053

(BAB)11 =2

2187

1199056+

2

243

1199055+

1

27

1199054+

2

27

1199053

(BAB)11 =8

6561

1199056+

8

729

1199055+

4

81

1199054+

8

81

1199053

(BAB)11 =2

729

1199055+

2

81

1199054+

1

9

1199053+

2

9

1199052

(BAB)11 =4

729

1199055+

4

81

1199054+

2

9

1199053+

4

9

1199052

(BAB)11 =2

243

1199055+

2

27

1199054+

1

3

1199053+

2

3

1199052

(BAB)11 =8

729

1199055+

8

81

1199054+

4

9

1199053+

8

9

1199052

(BAB)11 =4

243

1199054+

4

27

1199053+

2

3

1199052+

4

3

119905

(BAB)11 =8

243

1199054+

8

27

1199053+

4

3

1199052+

8

3

119905

(BAB)11 =4

81

1199054+

4

9

1199053+ 2119905

2+ 4119905

(BAB)11 =16

243

1199054+

16

27

1199053+

8

3

1199052+

16

3

119905

(BAB)11 =4

81

1199053+

4

9

1199052+ 2119905 + 4

(BAB)11 =8

81

1199053+

8

9

1199052+ 4119905 + 8

(BAB)11 =4

27

1199053+

4

3

1199052+ 6119905 + 12

(BAB)11 =16

81

1199053+

16

9

1199052+ 8119905 + 16

(81)


(AB)3

0= A

3 (AB)

3

3=B

3

(AB)3

1= (A

2B +BA

2+ABA)

= (

12057211120572121205721312057214

12057221120572221205722312057224

12057231120572321205723312057234

12057241120572421205724312057244

)

(82)

where

12057211=

2

9

1199053

12057212=

4

9

1199053+

5

243

1199054

12057213=

2

3

1199053+

1

243

1199055+

10

243

1199054

12057214=

8

9

1199053+

4

6561

1199056+

2

243

1199055+

5

81

1199054

12057221=

7

9

1199052

12057222=

14

9

1199052+

1

9

1199053

12057223=

7

3

1199052+

2

81

1199054+

2

9

1199053

12057224=

28

9

1199052+

1

243

1199055+

4

81

1199054+

1

3

1199053

12057231= 2119905

12057232= 4119905 +

4

9

1199052

12057233= 6119905 +

1

9

1199053+

8

9

1199052

12057234= 8119905 +

5

243

1199054+

2

9

1199053+

4

3

1199052

12057241= 3

12057242= 6 + 119905

12057243= 9 +

5

18

1199052+ 2119905

12057244= 12 +

1

18

1199053+

5

9

1199052+ 3119905

(AB)3

2= (AB

2+B

2A +BAB)

= (

12057311120573121205731312057314

12057321120573221205732312057324

12057331120573321205733312057334

12057341120573421205734312057344

)

(83)

being

12057311=

17

26244

1199056+

13

1458

1199055+

11

162

1199054+

20

81

1199053

12057312=

10

6561

1199056+

29

1458

1199055+

35

243

1199054+

40

81

1199053+

1

78732

1199057

12057313=

2

729

1199056+

47

1458

1199055+

107

486

1199054+

20

27

1199053

+

1

472392

1199058+

5

78732

1199057


12057314=

1

243

1199056+

65

1458

1199055+

8

27

1199054+

80

81

1199053+

1

4251528

1199059

+

1

118098

1199058+

1

6561

1199057

12057321=

13

2916

1199055+

1

18

1199054+

7

18

1199053+

4

3

1199052

12057322=

8

729

1199055+

7

54

1199054+

23

27

1199053+

8

3

1199052+

1

8748

1199056

12057323=

5

243

1199055+

35

162

1199054+

71

54

1199053+ 4119905

2+

1

52488

1199057+

5

8748

1199056

12057324=

23

729

1199055+

49

162

1199054+

16

9

1199053+

16

3

1199052+

1

472392

1199058

+

1

13122

1199057+

1

729

1199056

12057331=

11

486

1199054+

7

27

1199053+

5

3

1199052+

16

3

119905

12057332=

14

243

1199054+

17

27

1199053+

34

9

1199052+

32

3

119905 +

1

1458

1199055

12057333=

1

9

1199054+

29

27

1199053+

53

9

1199052+ 16119905 +

1

8748

1199056+

5

1458

1199055

12057334=

14

81

1199054+

41

27

1199053+ 8119905

2+

64

3

119905 +

1

78732

1199057

+

1

2187

1199056+

2

243

1199055

12057341=

5

81

1199053+

2

3

1199052+ 4119905 + 12

12057342=

13

81

1199053+

5

3

1199052+

28

3

119905 + 24 +

1

486

1199054

12057343=

17

54

1199053+

26

9

1199052+

44

3

119905 + 36 +

1

2916

1199055+

5

486

1199054

12057344=

40

81

1199053+

37

9

1199052+ 20119905 + 48 +

1

26244

1199056

+

1

729

1199055+

2

81

1199054

(84)



+ + 119909 = sin 119905 (85)

Matrix A

0 1

minus1 minus1


Matrix B

0 1

0 0



119909 (0) = 0 (0) = 1 (86)



2

119905) + radic3119890minus1199052 sin(

radic3

2

119905)

(87)





exp (Bm)

0

((tm)0)(10) ((tm)1)(11)

((tm)0)(10)


Matrix C

0 0

minus1 minus1








exp (Cm)

01




Automatic procedure


Precision

Digits

t

Value of m

18

13

1

1000



exp (A)

06596974858612 05335045275926

minus05335071951242 01261956258061




0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15


minus15

minus1

minus05

0

05

1

15

0 2 4 6 8 10


0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15



for119898 = 1000

Step 119905 119909(119905)

1 01 009516653368102 02 018133052249203 03 025948012999554 04 033058564791315 05 039559200286866 06 045541222101157 07 051092181897118 08 056295408367889 09 0612296198681210 10 06596861707192



119905 |119909119878(119905) minus 119909(119905)| |119909

119877(119905) minus 119909(119905)|

01 00000047853239 0000000167203104 00000712453497 0000035875126707 00002074165163 0000354460586810 00003710415717 0001430787361713 00004547161744 0003825311783616 00003284030320 0008043689956219 00000622406880 0014419715717422 00006216534547 0023032955185125 00011277539714 0033667545654128 00013572764337 0045813471294231 00012347982225 0058707643120834 00008953272672 0071407807759437 00006002231245 0082889689449740 00005492334199 0092155687292443 00007154938930 0098343146001446 00008300921248 0100820433067849 00005487250652 0099260855115152 00002963137375 0093686827986155 00015298326576 0084479859363158 00026838757810 0072355393614361 00032571351779 0058304626062764 00030289419613 0043509187995267 00022189180928 0029236498885170 00013691340100 0016725709670673 00010043530807 0007075055562276 00012821439144 0001140917647379 00018689726696 0000541733952482 00021436005756 0002189733032385 00016148602301 0009108229122588 00002869277587 0019614707845191 00012796703448 0032789720352394 00022771057968 0047474001777797 00021518593816 00623710756305100 00009803998394 00761625070468


0 2 4 6 8 100

002

004

006

008

01

012



5 Conclusions


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

(B2A)

11=

1

26244

1199056+

1

1458

1199055+

1

162

1199054+

2

81

1199053

(B2A)

12=

1

78732

1199057+

2

6561

1199056+

5

1458

1199055+

5

243

1199054+

4

81

1199053

(B2A)

13=

1

472392

1199058+

5

78732

1199057+

2

2187

1199056+

11

1458

1199055

+

17

486

1199054+

2

27

1199053

(B2A)

14=

1

4251528

1199059+

1

118098

1199058+

1

6561

1199057+

11

6561

1199056

+

17

1458

1199055+

4

81

1199054+

8

81

1199053

(B2A)

21=

1

2916

1199055+

1

162

1199054+

1

18

1199053+

2

9

1199052

(B2A)

22=

1

8748

1199056+

2

729

1199055+

5

162

1199054+

5

27

1199053+

4

9

1199052

(B2A)

23=

1

52488

1199057+

5

8748

1199056+

2

243

1199055+

11

162

1199054

+

17

54

1199053+

2

3

1199052

(B2A)

24=

1

472392

1199058+

1

13122

1199057+

1

729

1199056+

11

729

1199055

+

17

162

1199054+

4

9

1199053+

8

9

1199052

(B2A)

31=

1

486

1199054+

1

27

1199053+

1

3

1199052+

4

3

119905

(B2A)

32=

1

1458

1199055+

4

243

1199054+

5

27

1199053+

10

9

1199052+

8

3

119905

(B2A)

33=

1

8748

1199056+

5

1458

1199055+

4

81

1199054+

11

27

1199053+

17

9

1199052+ 4119905

(B2A)

34=

1

78732

1199057+

1

2187

1199056+

2

243

1199055+

22

243

1199054

+

17

27

1199053+

8

3

1199052+

16

3

119905

(B2A)

41=

1

162

1199053+

1

9

1199052+ 119905 + 4

(B2A)

42=

1

486

1199054+

4

81

1199053+

5

9

1199052+

10

3

119905 + 8

(B2A)

43=

1

2916

1199055+

5

486

1199054+

4

27

1199053+

11

9

1199052+

17

3

119905 + 12

(B2A)

44=

1

26244

1199056+

1

729

1199055+

2

81

1199054+

22

81

1199053+

17

9

1199052

+ 8119905 + 16

BA2

=(

(

(BA2)11(BA2

)12(BA2

)13(BA2

)14

(BA2)21(BA2

)22(BA2

)23(BA2

)24

(BA2)31(BA2

)32(BA2

)33(BA2

)34

(BA2)41(BA2

)42(BA2

)43(BA2

)44

)

)

(79)

where

(BA2)11=

1

162

1199053

(BA2)12=

1

243

1199054+

1

81

1199053

(BA2)13=

1

729

1199055+

2

243

1199054+

1

54

1199053

(BA2)14=

2

6561

1199056+

2

729

1199055+

1

81

1199054+

2

81

1199053

(BA2)21=

1

18

1199052

(BA2)22=

1

27

1199053+

1

9

1199052

(BA2)23=

1

81

1199054+

2

27

1199053+

1

6

1199052

(BA2)24=

2

729

1199055+

2

81

1199054+

1

9

1199053+

2

9

1199052

(BA2)31=

1

3

119905

(BA2)32=

2

9

1199052+

2

3

119905

(BA2)33=

2

27

1199053+

4

9

1199052+ 119905

(BA2)34=

4

243

1199054+

4

27

1199053+

2

3

1199052+

4

3

119905

(BA2)41= 1

(BA2)42=

2

3

119905 + 2

(BA2)43=

2

9

1199052+

4

3

119905 + 3

(BA2)44=

4

81

1199053+

4

9

1199052+ 2119905 + 4


BAB

= (


)

(80)

where

(BAB)11 =2

6561

1199056+

2

729

1199055+

1

81

1199054+

2

81

1199053

(BAB)11 =4

6561

1199056+

4

729

1199055+

2

81

1199054+

4

81

1199053

(BAB)11 =2

2187

1199056+

2

243

1199055+

1

27

1199054+

2

27

1199053

(BAB)11 =8

6561

1199056+

8

729

1199055+

4

81

1199054+

8

81

1199053

(BAB)11 =2

729

1199055+

2

81

1199054+

1

9

1199053+

2

9

1199052

(BAB)11 =4

729

1199055+

4

81

1199054+

2

9

1199053+

4

9

1199052

(BAB)11 =2

243

1199055+

2

27

1199054+

1

3

1199053+

2

3

1199052

(BAB)11 =8

729

1199055+

8

81

1199054+

4

9

1199053+

8

9

1199052

(BAB)11 =4

243

1199054+

4

27

1199053+

2

3

1199052+

4

3

119905

(BAB)11 =8

243

1199054+

8

27

1199053+

4

3

1199052+

8

3

119905

(BAB)11 =4

81

1199054+

4

9

1199053+ 2119905

2+ 4119905

(BAB)11 =16

243

1199054+

16

27

1199053+

8

3

1199052+

16

3

119905

(BAB)11 =4

81

1199053+

4

9

1199052+ 2119905 + 4

(BAB)11 =8

81

1199053+

8

9

1199052+ 4119905 + 8

(BAB)11 =4

27

1199053+

4

3

1199052+ 6119905 + 12

(BAB)11 =16

81

1199053+

16

9

1199052+ 8119905 + 16

(81)


(AB)3

0= A

3 (AB)

3

3=B

3

(AB)3

1= (A

2B +BA

2+ABA)

= (

12057211120572121205721312057214

12057221120572221205722312057224

12057231120572321205723312057234

12057241120572421205724312057244

)

(82)

where

12057211=

2

9

1199053

12057212=

4

9

1199053+

5

243

1199054

12057213=

2

3

1199053+

1

243

1199055+

10

243

1199054

12057214=

8

9

1199053+

4

6561

1199056+

2

243

1199055+

5

81

1199054

12057221=

7

9

1199052

12057222=

14

9

1199052+

1

9

1199053

12057223=

7

3

1199052+

2

81

1199054+

2

9

1199053

12057224=

28

9

1199052+

1

243

1199055+

4

81

1199054+

1

3

1199053

12057231= 2119905

12057232= 4119905 +

4

9

1199052

12057233= 6119905 +

1

9

1199053+

8

9

1199052

12057234= 8119905 +

5

243

1199054+

2

9

1199053+

4

3

1199052

12057241= 3

12057242= 6 + 119905

12057243= 9 +

5

18

1199052+ 2119905

12057244= 12 +

1

18

1199053+

5

9

1199052+ 3119905

(AB)3

2= (AB

2+B

2A +BAB)

= (

12057311120573121205731312057314

12057321120573221205732312057324

12057331120573321205733312057334

12057341120573421205734312057344

)

(83)

being

12057311=

17

26244

1199056+

13

1458

1199055+

11

162

1199054+

20

81

1199053

12057312=

10

6561

1199056+

29

1458

1199055+

35

243

1199054+

40

81

1199053+

1

78732

1199057

12057313=

2

729

1199056+

47

1458

1199055+

107

486

1199054+

20

27

1199053

+

1

472392

1199058+

5

78732

1199057


12057314=

1

243

1199056+

65

1458

1199055+

8

27

1199054+

80

81

1199053+

1

4251528

1199059

+

1

118098

1199058+

1

6561

1199057

12057321=

13

2916

1199055+

1

18

1199054+

7

18

1199053+

4

3

1199052

12057322=

8

729

1199055+

7

54

1199054+

23

27

1199053+

8

3

1199052+

1

8748

1199056

12057323=

5

243

1199055+

35

162

1199054+

71

54

1199053+ 4119905

2+

1

52488

1199057+

5

8748

1199056

12057324=

23

729

1199055+

49

162

1199054+

16

9

1199053+

16

3

1199052+

1

472392

1199058

+

1

13122

1199057+

1

729

1199056

12057331=

11

486

1199054+

7

27

1199053+

5

3

1199052+

16

3

119905

12057332=

14

243

1199054+

17

27

1199053+

34

9

1199052+

32

3

119905 +

1

1458

1199055

12057333=

1

9

1199054+

29

27

1199053+

53

9

1199052+ 16119905 +

1

8748

1199056+

5

1458

1199055

12057334=

14

81

1199054+

41

27

1199053+ 8119905

2+

64

3

119905 +

1

78732

1199057

+

1

2187

1199056+

2

243

1199055

12057341=

5

81

1199053+

2

3

1199052+ 4119905 + 12

12057342=

13

81

1199053+

5

3

1199052+

28

3

119905 + 24 +

1

486

1199054

12057343=

17

54

1199053+

26

9

1199052+

44

3

119905 + 36 +

1

2916

1199055+

5

486

1199054

12057344=

40

81

1199053+

37

9

1199052+ 20119905 + 48 +

1

26244

1199056

+

1

729

1199055+

2

81

1199054

(84)



+ + 119909 = sin 119905 (85)

Matrix A

0 1

minus1 minus1


Matrix B

0 1

0 0



119909 (0) = 0 (0) = 1 (86)



2

119905) + radic3119890minus1199052 sin(

radic3

2

119905)

(87)





exp (Bm)

0

((tm)0)(10) ((tm)1)(11)

((tm)0)(10)


Matrix C

0 0

minus1 minus1








exp (Cm)

01




Automatic procedure


Precision

Digits

t

Value of m

18

13

1

1000



exp (A)

06596974858612 05335045275926

minus05335071951242 01261956258061




0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15


minus15

minus1

minus05

0

05

1

15

0 2 4 6 8 10


0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15



for119898 = 1000

Step 119905 119909(119905)

1 01 009516653368102 02 018133052249203 03 025948012999554 04 033058564791315 05 039559200286866 06 045541222101157 07 051092181897118 08 056295408367889 09 0612296198681210 10 06596861707192



119905 |119909119878(119905) minus 119909(119905)| |119909

119877(119905) minus 119909(119905)|

01 00000047853239 0000000167203104 00000712453497 0000035875126707 00002074165163 0000354460586810 00003710415717 0001430787361713 00004547161744 0003825311783616 00003284030320 0008043689956219 00000622406880 0014419715717422 00006216534547 0023032955185125 00011277539714 0033667545654128 00013572764337 0045813471294231 00012347982225 0058707643120834 00008953272672 0071407807759437 00006002231245 0082889689449740 00005492334199 0092155687292443 00007154938930 0098343146001446 00008300921248 0100820433067849 00005487250652 0099260855115152 00002963137375 0093686827986155 00015298326576 0084479859363158 00026838757810 0072355393614361 00032571351779 0058304626062764 00030289419613 0043509187995267 00022189180928 0029236498885170 00013691340100 0016725709670673 00010043530807 0007075055562276 00012821439144 0001140917647379 00018689726696 0000541733952482 00021436005756 0002189733032385 00016148602301 0009108229122588 00002869277587 0019614707845191 00012796703448 0032789720352394 00022771057968 0047474001777797 00021518593816 00623710756305100 00009803998394 00761625070468


0 2 4 6 8 100

002

004

006

008

01

012



5 Conclusions


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










BAB

= (


)

(80)

where

(BAB)11 =2

6561

1199056+

2

729

1199055+

1

81

1199054+

2

81

1199053

(BAB)11 =4

6561

1199056+

4

729

1199055+

2

81

1199054+

4

81

1199053

(BAB)11 =2

2187

1199056+

2

243

1199055+

1

27

1199054+

2

27

1199053

(BAB)11 =8

6561

1199056+

8

729

1199055+

4

81

1199054+

8

81

1199053

(BAB)11 =2

729

1199055+

2

81

1199054+

1

9

1199053+

2

9

1199052

(BAB)11 =4

729

1199055+

4

81

1199054+

2

9

1199053+

4

9

1199052

(BAB)11 =2

243

1199055+

2

27

1199054+

1

3

1199053+

2

3

1199052

(BAB)11 =8

729

1199055+

8

81

1199054+

4

9

1199053+

8

9

1199052

(BAB)11 =4

243

1199054+

4

27

1199053+

2

3

1199052+

4

3

119905

(BAB)11 =8

243

1199054+

8

27

1199053+

4

3

1199052+

8

3

119905

(BAB)11 =4

81

1199054+

4

9

1199053+ 2119905

2+ 4119905

(BAB)11 =16

243

1199054+

16

27

1199053+

8

3

1199052+

16

3

119905

(BAB)11 =4

81

1199053+

4

9

1199052+ 2119905 + 4

(BAB)11 =8

81

1199053+

8

9

1199052+ 4119905 + 8

(BAB)11 =4

27

1199053+

4

3

1199052+ 6119905 + 12

(BAB)11 =16

81

1199053+

16

9

1199052+ 8119905 + 16

(81)


(AB)3

0= A

3 (AB)

3

3=B

3

(AB)3

1= (A

2B +BA

2+ABA)

= (

12057211120572121205721312057214

12057221120572221205722312057224

12057231120572321205723312057234

12057241120572421205724312057244

)

(82)

where

12057211=

2

9

1199053

12057212=

4

9

1199053+

5

243

1199054

12057213=

2

3

1199053+

1

243

1199055+

10

243

1199054

12057214=

8

9

1199053+

4

6561

1199056+

2

243

1199055+

5

81

1199054

12057221=

7

9

1199052

12057222=

14

9

1199052+

1

9

1199053

12057223=

7

3

1199052+

2

81

1199054+

2

9

1199053

12057224=

28

9

1199052+

1

243

1199055+

4

81

1199054+

1

3

1199053

12057231= 2119905

12057232= 4119905 +

4

9

1199052

12057233= 6119905 +

1

9

1199053+

8

9

1199052

12057234= 8119905 +

5

243

1199054+

2

9

1199053+

4

3

1199052

12057241= 3

12057242= 6 + 119905

12057243= 9 +

5

18

1199052+ 2119905

12057244= 12 +

1

18

1199053+

5

9

1199052+ 3119905

(AB)3

2= (AB

2+B

2A +BAB)

= (

12057311120573121205731312057314

12057321120573221205732312057324

12057331120573321205733312057334

12057341120573421205734312057344

)

(83)

being

12057311=

17

26244

1199056+

13

1458

1199055+

11

162

1199054+

20

81

1199053

12057312=

10

6561

1199056+

29

1458

1199055+

35

243

1199054+

40

81

1199053+

1

78732

1199057

12057313=

2

729

1199056+

47

1458

1199055+

107

486

1199054+

20

27

1199053

+

1

472392

1199058+

5

78732

1199057


12057314=

1

243

1199056+

65

1458

1199055+

8

27

1199054+

80

81

1199053+

1

4251528

1199059

+

1

118098

1199058+

1

6561

1199057

12057321=

13

2916

1199055+

1

18

1199054+

7

18

1199053+

4

3

1199052

12057322=

8

729

1199055+

7

54

1199054+

23

27

1199053+

8

3

1199052+

1

8748

1199056

12057323=

5

243

1199055+

35

162

1199054+

71

54

1199053+ 4119905

2+

1

52488

1199057+

5

8748

1199056

12057324=

23

729

1199055+

49

162

1199054+

16

9

1199053+

16

3

1199052+

1

472392

1199058

+

1

13122

1199057+

1

729

1199056

12057331=

11

486

1199054+

7

27

1199053+

5

3

1199052+

16

3

119905

12057332=

14

243

1199054+

17

27

1199053+

34

9

1199052+

32

3

119905 +

1

1458

1199055

12057333=

1

9

1199054+

29

27

1199053+

53

9

1199052+ 16119905 +

1

8748

1199056+

5

1458

1199055

12057334=

14

81

1199054+

41

27

1199053+ 8119905

2+

64

3

119905 +

1

78732

1199057

+

1

2187

1199056+

2

243

1199055

12057341=

5

81

1199053+

2

3

1199052+ 4119905 + 12

12057342=

13

81

1199053+

5

3

1199052+

28

3

119905 + 24 +

1

486

1199054

12057343=

17

54

1199053+

26

9

1199052+

44

3

119905 + 36 +

1

2916

1199055+

5

486

1199054

12057344=

40

81

1199053+

37

9

1199052+ 20119905 + 48 +

1

26244

1199056

+

1

729

1199055+

2

81

1199054

(84)



+ + 119909 = sin 119905 (85)

Matrix A

0 1

minus1 minus1


Matrix B

0 1

0 0



119909 (0) = 0 (0) = 1 (86)



2

119905) + radic3119890minus1199052 sin(

radic3

2

119905)

(87)





exp (Bm)

0

((tm)0)(10) ((tm)1)(11)

((tm)0)(10)


Matrix C

0 0

minus1 minus1








exp (Cm)

01




Automatic procedure


Precision

Digits

t

Value of m

18

13

1

1000



exp (A)

06596974858612 05335045275926

minus05335071951242 01261956258061




0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15


minus15

minus1

minus05

0

05

1

15

0 2 4 6 8 10


0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15



for119898 = 1000

Step 119905 119909(119905)

1 01 009516653368102 02 018133052249203 03 025948012999554 04 033058564791315 05 039559200286866 06 045541222101157 07 051092181897118 08 056295408367889 09 0612296198681210 10 06596861707192



119905 |119909119878(119905) minus 119909(119905)| |119909

119877(119905) minus 119909(119905)|

01 00000047853239 0000000167203104 00000712453497 0000035875126707 00002074165163 0000354460586810 00003710415717 0001430787361713 00004547161744 0003825311783616 00003284030320 0008043689956219 00000622406880 0014419715717422 00006216534547 0023032955185125 00011277539714 0033667545654128 00013572764337 0045813471294231 00012347982225 0058707643120834 00008953272672 0071407807759437 00006002231245 0082889689449740 00005492334199 0092155687292443 00007154938930 0098343146001446 00008300921248 0100820433067849 00005487250652 0099260855115152 00002963137375 0093686827986155 00015298326576 0084479859363158 00026838757810 0072355393614361 00032571351779 0058304626062764 00030289419613 0043509187995267 00022189180928 0029236498885170 00013691340100 0016725709670673 00010043530807 0007075055562276 00012821439144 0001140917647379 00018689726696 0000541733952482 00021436005756 0002189733032385 00016148602301 0009108229122588 00002869277587 0019614707845191 00012796703448 0032789720352394 00022771057968 0047474001777797 00021518593816 00623710756305100 00009803998394 00761625070468


0 2 4 6 8 100

002

004

006

008

01

012



5 Conclusions


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










12057314=

1

243

1199056+

65

1458

1199055+

8

27

1199054+

80

81

1199053+

1

4251528

1199059

+

1

118098

1199058+

1

6561

1199057

12057321=

13

2916

1199055+

1

18

1199054+

7

18

1199053+

4

3

1199052

12057322=

8

729

1199055+

7

54

1199054+

23

27

1199053+

8

3

1199052+

1

8748

1199056

12057323=

5

243

1199055+

35

162

1199054+

71

54

1199053+ 4119905

2+

1

52488

1199057+

5

8748

1199056

12057324=

23

729

1199055+

49

162

1199054+

16

9

1199053+

16

3

1199052+

1

472392

1199058

+

1

13122

1199057+

1

729

1199056

12057331=

11

486

1199054+

7

27

1199053+

5

3

1199052+

16

3

119905

12057332=

14

243

1199054+

17

27

1199053+

34

9

1199052+

32

3

119905 +

1

1458

1199055

12057333=

1

9

1199054+

29

27

1199053+

53

9

1199052+ 16119905 +

1

8748

1199056+

5

1458

1199055

12057334=

14

81

1199054+

41

27

1199053+ 8119905

2+

64

3

119905 +

1

78732

1199057

+

1

2187

1199056+

2

243

1199055

12057341=

5

81

1199053+

2

3

1199052+ 4119905 + 12

12057342=

13

81

1199053+

5

3

1199052+

28

3

119905 + 24 +

1

486

1199054

12057343=

17

54

1199053+

26

9

1199052+

44

3

119905 + 36 +

1

2916

1199055+

5

486

1199054

12057344=

40

81

1199053+

37

9

1199052+ 20119905 + 48 +

1

26244

1199056

+

1

729

1199055+

2

81

1199054

(84)



+ + 119909 = sin 119905 (85)

Matrix A

0 1

minus1 minus1


Matrix B

0 1

0 0



119909 (0) = 0 (0) = 1 (86)



2

119905) + radic3119890minus1199052 sin(

radic3

2

119905)

(87)





exp (Bm)

0

((tm)0)(10) ((tm)1)(11)

((tm)0)(10)


Matrix C

0 0

minus1 minus1








exp (Cm)

01




Automatic procedure


Precision

Digits

t

Value of m

18

13

1

1000



exp (A)

06596974858612 05335045275926

minus05335071951242 01261956258061




0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15


minus15

minus1

minus05

0

05

1

15

0 2 4 6 8 10


0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15



for119898 = 1000

Step 119905 119909(119905)

1 01 009516653368102 02 018133052249203 03 025948012999554 04 033058564791315 05 039559200286866 06 045541222101157 07 051092181897118 08 056295408367889 09 0612296198681210 10 06596861707192



119905 |119909119878(119905) minus 119909(119905)| |119909

119877(119905) minus 119909(119905)|

01 00000047853239 0000000167203104 00000712453497 0000035875126707 00002074165163 0000354460586810 00003710415717 0001430787361713 00004547161744 0003825311783616 00003284030320 0008043689956219 00000622406880 0014419715717422 00006216534547 0023032955185125 00011277539714 0033667545654128 00013572764337 0045813471294231 00012347982225 0058707643120834 00008953272672 0071407807759437 00006002231245 0082889689449740 00005492334199 0092155687292443 00007154938930 0098343146001446 00008300921248 0100820433067849 00005487250652 0099260855115152 00002963137375 0093686827986155 00015298326576 0084479859363158 00026838757810 0072355393614361 00032571351779 0058304626062764 00030289419613 0043509187995267 00022189180928 0029236498885170 00013691340100 0016725709670673 00010043530807 0007075055562276 00012821439144 0001140917647379 00018689726696 0000541733952482 00021436005756 0002189733032385 00016148602301 0009108229122588 00002869277587 0019614707845191 00012796703448 0032789720352394 00022771057968 0047474001777797 00021518593816 00623710756305100 00009803998394 00761625070468


0 2 4 6 8 100

002

004

006

008

01

012



5 Conclusions


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










exp (Bm)

0

((tm)0)(10) ((tm)1)(11)

((tm)0)(10)


Matrix C

0 0

minus1 minus1








exp (Cm)

01




Automatic procedure


Precision

Digits

t

Value of m

18

13

1

1000



exp (A)

06596974858612 05335045275926

minus05335071951242 01261956258061




0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15


minus15

minus1

minus05

0

05

1

15

0 2 4 6 8 10


0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15



for119898 = 1000

Step 119905 119909(119905)

1 01 009516653368102 02 018133052249203 03 025948012999554 04 033058564791315 05 039559200286866 06 045541222101157 07 051092181897118 08 056295408367889 09 0612296198681210 10 06596861707192



119905 |119909119878(119905) minus 119909(119905)| |119909

119877(119905) minus 119909(119905)|

01 00000047853239 0000000167203104 00000712453497 0000035875126707 00002074165163 0000354460586810 00003710415717 0001430787361713 00004547161744 0003825311783616 00003284030320 0008043689956219 00000622406880 0014419715717422 00006216534547 0023032955185125 00011277539714 0033667545654128 00013572764337 0045813471294231 00012347982225 0058707643120834 00008953272672 0071407807759437 00006002231245 0082889689449740 00005492334199 0092155687292443 00007154938930 0098343146001446 00008300921248 0100820433067849 00005487250652 0099260855115152 00002963137375 0093686827986155 00015298326576 0084479859363158 00026838757810 0072355393614361 00032571351779 0058304626062764 00030289419613 0043509187995267 00022189180928 0029236498885170 00013691340100 0016725709670673 00010043530807 0007075055562276 00012821439144 0001140917647379 00018689726696 0000541733952482 00021436005756 0002189733032385 00016148602301 0009108229122588 00002869277587 0019614707845191 00012796703448 0032789720352394 00022771057968 0047474001777797 00021518593816 00623710756305100 00009803998394 00761625070468


0 2 4 6 8 100

002

004

006

008

01

012



5 Conclusions


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15


minus15

minus1

minus05

0

05

1

15

0 2 4 6 8 10


0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15



for119898 = 1000

Step 119905 119909(119905)

1 01 009516653368102 02 018133052249203 03 025948012999554 04 033058564791315 05 039559200286866 06 045541222101157 07 051092181897118 08 056295408367889 09 0612296198681210 10 06596861707192



119905 |119909119878(119905) minus 119909(119905)| |119909

119877(119905) minus 119909(119905)|

01 00000047853239 0000000167203104 00000712453497 0000035875126707 00002074165163 0000354460586810 00003710415717 0001430787361713 00004547161744 0003825311783616 00003284030320 0008043689956219 00000622406880 0014419715717422 00006216534547 0023032955185125 00011277539714 0033667545654128 00013572764337 0045813471294231 00012347982225 0058707643120834 00008953272672 0071407807759437 00006002231245 0082889689449740 00005492334199 0092155687292443 00007154938930 0098343146001446 00008300921248 0100820433067849 00005487250652 0099260855115152 00002963137375 0093686827986155 00015298326576 0084479859363158 00026838757810 0072355393614361 00032571351779 0058304626062764 00030289419613 0043509187995267 00022189180928 0029236498885170 00013691340100 0016725709670673 00010043530807 0007075055562276 00012821439144 0001140917647379 00018689726696 0000541733952482 00021436005756 0002189733032385 00016148602301 0009108229122588 00002869277587 0019614707845191 00012796703448 0032789720352394 00022771057968 0047474001777797 00021518593816 00623710756305100 00009803998394 00761625070468


0 2 4 6 8 100

002

004

006

008

01

012



5 Conclusions


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 100

002

004

006

008

01

012



5 Conclusions


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article symbolic solution to complete ordinary...

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