odibat (2007) generalized taylor's formula

5/21/2018 Odibat (2007) Generalized Taylor's Formula

1/8

Generalized Taylors formula

Zaid M. Odibat a,*, Nabil T. Shawagfeh b

a Prince Abdullah Bin Ghazi Faculty of Science & IT, Al-Balqa Applied University, Salt, Jordanb Department of Mathematics, University of Jordan, Amman, Jordan

Abstract

In this paper, a new generalized Taylors formula of the kind

fx Xnj0

ajax aja Ranx;

whereaj 2 R,x >a, 0


2/8

with m< a, x> x0P a and

Rn;m Jana D

_ana fx

1

Ca m

Z x00

x tam1D

_am1a ftdt; 1:3

whereD_

ana is the RiemannLiouville fractional derivative of ordera + n. This fractional derivative operator is

defined for a> 0, a 2 R and x> a as follows:

D_

aafx

dm

dxm1

Cm a

Z xa

ft

x ta1m

dt

" # 1:4

for m 1 0) is said to be in the space Ca

(a 2 R) if it can be written asf(x) = xpf1(x) forsome p> a where f1(x) is continuous in [0, 1), and it is said to be in the spaceC

ma iff

(m) 2 Ca, m 2 N.

Definition 2. The RiemannLiouville integral operator of ordera> 0 with aP 0 is defined as

Jaafx 1

Ca

Z xa

x sa1

fsds; x >a; 2:1

J0afx fx: 2:2

Properties of the operator can be found in[12]. We only need here the following:

For f2 Ca, a,b> 0, aP 0, c 2 R and c> 1, we have

JaaJbafx J

baJ

aafx J

aba fx; 2:3

Ja

axc

xca

CaBxa

x

a; c 1; 2:4

Z.M. Odibat, N.T. Shawagfeh / Applied Mathematics and Computation 186 (2007) 286293 287


3/8

where Bs(a,c+ 1) is the incomplete beta function which is defined as

Bsa; c 1

Z s0

ta11 tcdt; 2:5

Jaaecx eacx aaX

1

k0

cx ak

Ca

k

1

: 2:6

The RiemannLiouville derivative has certain disadvantages when trying to model real-world phenomena

with fractional differential equations. Therefore, we shall introduce a modified fractional differential operator

Daa proposed by Caputo in his work on the theory of viscoelasticity.

Definition 3. The Caputo fractional derivative off(x) of order a> 0 with aP 0 is defined as

Daafx Jmaa f

mx 1

Cm a

Z xa

fmt

x ta1m

dt

" # 2:7

for m 1 < a 6 m, m 2 N, xP a, fx 2Cm1.The Caputo fractional derivative was investigated by many authors, for m 1 < a 6 m, fx 2Cma and

aP 1, we have

JaaDaafx J

mDmfx fx Xm1k0

fkax a

k

k! : 2:8

3. Generalized Taylors formula

In this section we will introduce a new generalization of Taylors formula that involving Caputo fractional

derivatives. We will begin with the generalized mean value theorem.

Theorem 1 (Generalized mean value theorem). Suppose that f(x) 2 C[a,b] and Daafx 2Ca; b, for

0< a 6 1, then we have

fx fa 1

CaDaafn x a

a 3:1

with a 6 n 6 x, "x 2 (a,b].

Proof. Form(2.1) and (2.7), we have

JaaDaafx

1

Ca

Z xa

x ta1Daaftdt: 3:2

Using the integral mean value theorem, we get

JaaDaafx

1

CaDaafn

Z xa

x ta1dt; 3:3

1

CaDaafn x a

a 3:4

for 0 6 n 6 x.

On the other hand, form(2.8), we have

JaaDaafx fx fa: 3:5

So, from(3.4) and (3.5), (3.1)is obtained. In case ofa = 1, the generalized mean value theorem reduces to the

classical mean value theorem. Before we present the generalized Taylors formula in the Caputo sense, we need

the following relation. h

288 Z.M. Odibat, N.T. Shawagfeh / Applied Mathematics and Computation 186 (2007) 286293
http://-/?-http://-/?-


4/8

Theorem 2. Suppose that Dnaa fx;Dn1aa fx 2 Ca; b, for 0< a 6 1, then we have

Jnaa Dnaa fx J

n1aa D

n1aa fx

x ana

Cna 1Dnaa fa; 3:6

where

Dnaa Daa D

aa D

aa n-times: 3:7

Proof. We have, using(2.3)

Jnaa Dnaa fx J

n1aa D

n1aa fx J

naa D

naa fx J

aaD

n1aa fx; 3:8

Jnaa Dnaa fx J

aaD

aaD

naa fx; 3:9

Jnaa Dnaa fa using 2:8; 3:10

x a

na

Cna 1Dnaa fa using 4:4: 3:11

Theorem 3 (Generalized Taylors formula). Suppose that Dkaa fx 2Ca; b for k = 0,1, . . ., n + 1, where0< a 6 1, then we have

fx Xni0

x aia

Cia 1Diaafa

Dn1aa fn

Cn 1a 1 x an1a 3:12

with a 6 n 6 x, "x 2 (a,b],

where

Dnaa Daa D

aa D

aa n-times:

Proof. Form(3.6), we have

Xni0

JiaaDiaafx J

i1aa D

i1aa fx

Xni0

x aia

Cia 1Diaafa; 3:13

that is

fx Jn1aa Dn1aa fx

Xni0

x aia

Cia 1Diaafa: 3:14

Applying the integral mean value theorem yields

Jn1aa Dn1aa fx 1Cn 1a 1

Z xa

x tn1aDn1aa ftdt; 3:15

Dn1aa fn

Cn 1a 1

Z xa

x tn1a

dt; 3:16

Dn1aa fn

Cn 1a 1 x a

n1a: 3:17

From (3.14) and (3.17), (3.12) is obtained. In case ofa= 1, the Caputo generalized Taylors formula (3.12)

reduces to the classical Taylors formula. The radius of convergence, R, for the generalized Taylors series

X1

i0

x aia

Cia 1

Diaafa 3:18



5/8

depends on f(x) and a, and is given by

R jx aja limn!1

Cna 1

Cn 1a 1

Dn1aa fa

Dnaa fa

: 3:19

Observe that the advantage of the presented Generalized Taylors formula if compared with the RiemannLiouville Generalized Taylors formulas(1.2) and (1.5)that the second requires more conditions on the func-

tionf(x), see[5], and the coefficients on the first appears more simple and easy to compute. Furthermore, the

Caputo fractional derivative is considered recently in many physical and engineering problems [8] more than

the RiemannLiouville fractional derivatives, because it allows traditional initial and boundary conditions to

be included in the formulation of the problem. h

4. Applications: approximation of functions

In this section, we use the generalized Taylors formula(3.12)to approximate functions at a given points.

The method of approximations is described in the following theorem.

Theorem 4. Suppose that Dkaa fx 2Ca; b for k = 0,1, . . .,n + 1, where 0 < a 6 1. If x 2 [a,b], then

fx ffiPaNx XNi0

x aia

Cia 1Diaafa: 4:1

Furthermore, there is a value n with a 6 n 6 x so that the error term RaNx has the form

RaNx DN1a fn

CN 1a 1 x a

N1a: 4:2

The proof follows directly form Theorem 3. The accuracy of PaNx increases when we choose large N anddecreases as the value of x moves away from the center a. Hence, we must choose N large enough so that the error

does not exceed a specified bound.

Example 1. The MittagLeffler function[13]with parameter a> 0 is defined as

Eax X1k0

xk

Cak 1: 4:3

This function plays a very important role in the solution of linear fractional differential equations [3,8], the

solutions of such equations are obtained in terms ofEa(xa). Note thatDkaa Eax

a 2C0;1, for k2 N anda> 0. Using(4.1) and (4.2), Ea(x

a) can be approximated as

Eaxa ffi PaNx 1 xa

Ca 1 x

2a

C2a 1 x

Na

CNa 1; 4:4

where the error term RaNx has the form

RaNx Ean

CN 1a 1 x aN1a; a 6 n 6 x: 4:5

Table 1shows approximate vales ofEa(xa) for different vales ofx and a, where a= 0 and N= 10.

Example 2. The Wright function[9] with parametersa and b is defined as

Wx; a;b X1

k0

xk

k!Cak b: 4:6



6/8

This function appears in the solutions of fractional partial differential equations[3], the solutions of such equa-

tions are obtained in terms of W(xa, a,1). Note that Dkaa Wxa;a; 1 2 C0; 1, for k2 N and a 2 R.

Using(4.1) and (4.2), W(xa, a,1) can be approximated as

Wxa;a; 1 ffi PaNx 1 xa

Ca 1

x2a

2!C2a 1 1N

xNa

N!CNa 1; 4:7

where the error term RaNx has the form

RaNx 1NWna;a; 1

CN 1a 1 x a

N1a; a 6 n 6 x: 4:8

5. Applications: series solutions of fractional differential equations

Fractional differential equations have been the focus of many studies due to their frequent appearance in

various fields such as physics, chemistry and engineering [1,4,614]. In this section, we use the generalized

Taylors formula to solve fractional differential equations. This method is very useful and can be applied to

solve many important fractional differential equations with non constant coefficients.

The idea behind solving a fractional differential equation using the generalized Taylors formula is simple.First, assume that the solution can be written as a fractional power series of the form:X1

n0

cnxna

Cna 1: 5:1

Second, write each term in the differential equation as a fractional power series. Third, equate the coefficients

of the resulting series on both sides of the equation. Finally, solve for the unknown coefficients in the series

representation of the assumed solution.

To demonstrate the applicability of the method as a computational tool for solving fractional differential

equations, we present the following examples.

Example 3. Consider the initial value problemDa0yx kyx; y0 y0; 5:2

where 0 0. Using the generalized Taylors formula, assuming that the solu-

tion y(x) can be written as

yx X1n0

cnxna

Cna 1: 5:3

From the definition of Caputo fractional derivative(2.7), we obtain

Da0yx X1

n1

cnxn1a

Cn 1a 1: 5:4

Table 1

Approximate values ofEa(xa)

x a= 0.2 a= 0.4 a= 0.6 a= 0.8 a= 1.0

0.0 1 1 1 1 1

0.5 5.88448295 3.33855319 2.39749137 1.92807992 1.64872127

1.0 9.91764643 6.10160337 4.24801113 3.29456410 2.71828180

1.5 14.51647891 10.30526704 7.21639172 5.51534629 4.481686602.0 19.71112855 16.57373140 12.02222332 9.15829024 7.38899471

2.5 25.49628706 25.60773056 15.73453592 15.14051286 12.18174319

3.0 31.86088315 38.19723789 31.90425034 24.93644082 20.07966518



7/8

Substituting(5.3) and (5.4)into(5.2)yieldsX1n0

cn1xna

Cna 1 k

X1n0

cnxna

Cna 1 0: 5:5

Equating the coefficient ofxna to zero in(5.5), we get

cn1 kcn c0 y0; 5:6

that is

cn kny0: 5:7

Substituting(5.7)into(5.3), we obtain the solution

yx y0X1n0

kn x

na

Cna 1;

y0Eakxa; 5:8

where Ea(x) is the MittagLeffler function.

Example 4. Consider the fractional differential equation

D2a0 yx Da0yx 2 yx 0; 5:9

whereD2a0 Da0 D

a0 and x > 0. Using the generalized Taylors formula, assuming that the solutiony(x) can be

written as

yx X1n0

cnxa

Cna 1: 5:10

From the definition of Caputo fractional derivative(2.7), we obtain

Da0yx X1n1

cnxn1a

Cn 1a 1 5:11

and

D2a0 yx X1n2

cnxn2a

Cn 2a 1: 5:12

Substituting(5.10), (5.11) and (5.12)into(5.8), yields

X1n0

cn2 cn1 cn

xna

Cna 1 0: 5:13

Equating the coefficient ofxna to zero in(5.13)and identifying the coefficients, we obtain recursively

cn2 2cn cn1 0; n2 N: 5:14

This gives

c2 2c0 c1;

c3 2c0 3c1;

c4 6c0 5c1;

c5 10c0 11c1;

c6 22c0 21c1:

5:15



8/8

Therefore, we obtain the following solutions:

y1x c0 1 2

C2a 1x2a

2

C3a 1x3a

6

C4a 1x4a

10

C5a 1x5a

5:16

and

y2x c11

Ca 1xa 1

C2a 1x2a 3

C3a 1x3a 5

C4a 1x4a

: 5:17

References

[1] F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, in: A. Carpinteri, F. Mainardi (Eds.),

Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wien, 1997, pp. 291348.

[2] G. Hardy, Riemanns form of Taylor series, J. London Math. 20 (1945) 4857.

[3] I. Podlubny, The Laplace transform method for linear differential equations of fractional order, Slovac Academy of Science, Slovak

Republic, 1994.

[4] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, 1999.

[5] J. Truilljo, M. Rivero, B. Bonilla, On a RiemannLiouville generalized Taylors formula, J. Math. Anal. 231 (1999) 255265.

[6] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.[7] K. Diethelm, J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002) 229248.

[8] R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order, Fractals and Fractional

Calculus, Carpinteri & Mainardi, New York, 1997.

[9] R. Gorenflo, Yu. Luchko, F. Mainardi, Analytical properties and applications of the Wrigth function, Fract. Calc. Appl. Anal. 2 (4)

(1999) 383414.

[10] R.L. Bagley, On the fractional order initial value problem and its engineering applications, in: K. Nishimoto (Ed.), Fractional

Calculus and Its Applications, Nihon University, Tokyo, 1990, pp. 1220.

[11] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach,

London, 1993.

[12] S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, USA, 1993,

p. 2.

[13] W.R. Schneider, Completely monotone generalized MittagLeffler functions, Exposition. Math. 14 (1) (1996) 316.

[14] Y. Luchko, H.M. Srivastava, The exact solution of certain differential equations of fractional order by using operational calculus,

Comput. Math. Appl. 29 (1995) 7385.

[15] Y. Wantanable, Notes on the generalized derivatives of RiemannLiouville and its application to Leibntzs formula, Thoku Math. J.

34, 2841.


odibat (2007) generalized taylor's formula

Documents