Research ArticleAdomian Decomposition Method with ModifiedBernstein Polynomials for Solving Ordinary and PartialDifferential Equations

Ahmed Farooq Qasim and Ekhlass S AL-Rawi

College of Computer Sciences and Mathematics University of Mosul Iraq

Correspondence should be addressed to Ahmed Farooq Qasim ahmednumericalyahoocom

Received 7 July 2018 Accepted 24 September 2018 Published 11 October 2018

Academic Editor Jafar Biazar

Copyright copy 2018 Ahmed Farooq Qasim and Ekhlass S AL-Rawi This is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

In this paper we used Bernstein polynomials to modify the Adomian decomposition method which can be used to solve linearand nonlinear equations This scheme is tested for four examples from ordinary and partial differential equations furthermorethe obtained results demonstrate reliability and activity of the proposed technique This strategy gives a precise and productivesystem in comparison with other traditional techniques and the arrangements methodology is extremely straightforward and fewemphasis prompts high exact solution The numerical outcomes showed that the acquired estimated solutions were in appropriateconcurrence with the correct solution

1 Introduction

Adomian decomposition technique was established byGeorge Adomian and has as of late turned into an extremelyrecognized strategy in connected sciences The techniquedoes not require any diminutiveness presumptions or lin-earization to solve the ordinary and partial differentialequations and this produces the strategy extremely effectiveamong alternate strategies Recently many iteration tech-niques have been used for solving nonlinear equations fromordinary partial and fractional equations [1] like variationaliteration method and differential transform method [2]homotopy perturbation and analysismethods [3] Numerousworks have been tested in various different regions forexample warmth or mass exchange incompressible fluidnonlinear optics and gas elements wonders [4 5] frac-tional Maxwell fluid [6 7] and the Oldroyd-B fluid model[8]

The approximation used polynomials extremely impor-tant in scientific experiments where many rely on topicssuch as the study of statistics different population and thetemperatures and others on the approximation theory Inaddition many experiments rely mainly on the approximatemeasurements and observations to be studied and processed

by the appropriate scientific methods in order to reach theresults expected from the study

The Adomian decomposition technique is improved viaChebyshev polynomials in [9 10] with Legendre polynomials[11] and with Laguerre polynomials [12]

This paper is organized as follows In Section 2 the basicideas of the modified Bernstein polynomials are describedSection 3 is devoted to solving a nonlinear differentialequations using Adomian decomposition method based onmodified Bernstein polynomials the results and comparisonsof the numerical solutions are presented in Section 4 andconcluding remarks are given in Section 5

2 The Modified Bernstein Polynomials

Polynomials are the mathematical technique as these canbe characterized figured separated and incorporated effort-lessly The Bernstein premise polynomials are trying toinexact the capacities Bernstein polynomials are the betterguess to a capacity with a couple of terms These polynomialsare utilized as a part of the fields of connected arithmetic andmaterial science and PC helped geometric outlines and arelikewise joined with different techniques like Galerkin and

HindawiJournal of Applied MathematicsVolume 2018 Article ID 1803107 9 pageshttpsdoiorg10115520181803107

2 Journal of Applied Mathematics

collocation technique to solve some differential and integralequations [13]

Definition 1 (Bernstein basis polynomials) The Bernsteinbasis polynomials of degree m over the interval [0 1] aredefined by

119861119894119898 (119909) = (119898119894 )119909119894 (1 minus 119909)119898minus119894 (1)

where the binomial coefficient is

(119898119894 ) = 119898119894 (119898 minus 119894) (2)

For example when m=5 then the Bernstein terms are

11986105 (119909) = (1 minus 119909)511986115 (119909) = 5119909 (1 minus 119909)411986125 (119909) = 101199092 (1 minus 119909)311986135 (119909) = 101199093 (1 minus 119909)211986145 (119909) = 51199094 (1 minus 119909)11986155 (119909) = 1199095

(3)

Definition 2 (Bernstein polynomials) A linear combinationof Bernstein basis polynomials

119861119898 (119909) = 119898sum119894=0

119861119894119898 (119909) 120573119894 (4)

is called the Bernstein polynomials of degree m where 120573119894 arethe Bernstein coefficients

Definition 3 Let 119891 be a real valued function defined andbounded on [0 1] let 119861119898(119891) be the polynomial on [0 1]defined by

119861119898 (119891) = 119898sum119894=0

(119898119894 )119909119894 (1 minus 119909)119898minus119894 119891( 119894119898) (5)

where 119861119898(119891) is the m-th Bernstein polynomials for 119891(119909)For each function 119891 [0 1] 997888rarr 119877 we have

lim119898997888rarrinfin

119861119891119898 (119909) = 119891 (119909) (6)

Example If 119891(119909) = 119890119909 119909 isin [0 1] then the Bernsteinexpanded for the function 119891(119909) when m=5 is

119861119898 (119891) = 119891 (0) (1 minus 119909)5 + 119891(15) 5119909 (1 minus 119909)4+ 119891(25) 101199092 (1 minus 119909)3+ 119891(35) 101199093 (1 minus 119909)2+ 119891(45) 51199094 (1 minus 119909) + 119891 (1) 1199095

119861119898 (119891) = 1198900 (1 minus 119909)5 + 511989015119909 (1 minus 119909)4+ 10119890251199092 (1 minus 119909)3 + 10119890351199093 (1 minus 119909)2+ 5119890451199094 (1 minus 119909) + 11989011199095

(7)

In (1986) [14] Lorentz prove that if the 2k-th order derivative1198912119896(119909) is bounded in the interval (01) then for each 119909 isin [0 1]119861119891119898 (119909) = 119891 (119909) + 2119896minus1sum

119886=2

119891(119886) (119909)119886119898119886 119879119898119886 (119909) + 119874( 1119898119896 ) (8)

where

119879119898119886 (119909) = sum119896

(119896 minus 119898119909)119886 (119898119896)119909119896 (1 minus 119909)119898minus119896 (9)

Remark (see [15]) Notice that 119879119898119886(119909) is the a-th centralmoment of a random variable with a binomial appropriationwith parameters 119898 and 119909 Clearly 1198791198980 = 1 1198791198981 = 0 It iswell known that the sequence 119879119898119886(119909) satisfies the followingrecurrence

119879119898119886+1 (119909) = 119909 (1 minus 119909) (1198791015840119898119886 (119909) + 119898119886119879119898119886minus1 (119909)) (10)

If we apply (8) to k = 1 2 3 then we obtain

119861119891119898 (119909) = 119891 (119909) + 119874( 1119898)119861119891119898 (119909) = 119891 (119909) + 119909 (1 minus 119909) 11989110158401015840 (119909)2119898 + 119874( 11198982)119861119891119898 (119909) = 119891 (119909) + 119909 (1 minus 119909) 11989110158401015840 (119909)2119898

+ 119909 (1 minus 119909) (4 (1 minus 2119909) 119891(3) (119909) + 3119909 (1 minus 119909) 119891(4) (119909))241198982

+ 119874( 11198983)

(11)

and higher level approximations can be computed

3 ADM Based on ModifiedBernstein Polynomials

Let us consider the following equation

119871119906 + 119873119906 + 119877119906 = 119892 (119909) (12)

where 119871 is an invertible linear term 119873 represents thenonlinear term and 119877 is the remaining linear part from (12)we have

119871119906 = 119892 (119909) minus 119873119906 minus 119877119906 (13)

Now applying the inverse factor119871minus1 to both sides of (13) thenvia the initial conditions we find

119906 = 119891 (119909) minus 119871minus1119873119906 minus 119871minus1119877119906 (14)

Journal of Applied Mathematics 3

where 119871minus1 = int1199090() ds and 119891(119909) are the terms having from

integrating the rest of the term g (x) and from utilizing thegiven initial or boundary conditions The ADM assumes thatN(u) (nonlinear term) can be decomposed by an infiniteseries of polynomials which is expressed in form

119873(119906) = infinsum119899=0

119860119899 (119906119900 1199061 119906119899) (15)

where 119860n are the Adomianrsquos polynomials [16] defined as

119860119899 = 1119899 119889119899119889120582n [119873(infinsum119894=0

120582iui)]120582=0

n = 0 1 2 (16)

We expand the function 119892(119909) by Bernstein series

119892 (119909) = 119898sum119894=0

119886119894119861119894 (119909) (17)

where 119861119894(119909) is the Bernstein polynomialsNow using (14) and (17) we have

1199060 = 119871minus1 (11988601198610 (119909) + 11988611198611 (119909) + 11988621198612 (119909)+ sdot sdot sdot 119886119898119861119898 (119909)) + 120579 (119909)

1199061 = minus119871minus1 (1198771199060) minus 119871minus1 (1198731199060) 1199062 = minus119871minus1 (1198771199061) minus 119871minus1 (1198731199061)

(18)

and so onThese formulas are easy to compute by usingMaple13 software

In this paper we improve the function 119892(119909) using modi-fied Bernstein series

119892 (119909) = 119898sum119894=0

(119898119894 )119909119894 (1 minus 119909)119898minus119894 119891( 119894119898)

minus 2119896minus1sum119886=2

119891(119886) (119909)119886119898119886 119879119898119886 (119909)(19)

And we can approach the derivatives using the Bernsteinpolynomials

119889119889119909119861119894119898 (119909) = 119898 (119861119894minus1119898minus1 (119909) minus 119861119894119898minus1 (119909)) (20)

Then (19) becomes

119892 (119909) = 119898sum119894=0

(119898119894 )119909119894 (1 minus 119909)119898minus119894 119891( 119894119898)

minus 2119896minus1sum119886=2

(119889(119886)119889119909(119886)) 119861119894119898 (119909)119886119898119886 119879119898119886 (119909)(21)

Now using (18) and (21) we have

1199060 = 119871minus1 (119861119894119898 (119909)) + 120579 (119909) 1199061 = minus119871minus1 (1198771199060) minus 119871minus1 (1198731199060) 1199062 = minus119871minus1 (1198771199061) minus 119871minus1 (1198731199061)

(22)

The above equation is governing equation of ADM usingmodified Bernstein polynomials The obtained approximatesolution 120596119881(119909) = sum119881119894=0 119906119894 by (22) has a comparison with theclassic approximation solution and the correct solution

4 Numerical Results

In this section we solve ordinary and partial differentialequations by ADM based on Bernstein polynomials and wecompare with ADM based on classical Bernstein polynomial

Example 1 Consider the ordinary equation

11988921199101198891199052 + 119905119889119910119889119905 + 11990521199103 = (2 + 61199052) 1198901199052 + 119905211989031199052

119910 (0) = 1119889119910119889119905 (0) = 0

(23)

with the exact solution (119905) = 1198901199052 Using (12) we have119871119910 + 119873119910 + 119877119910 = 119892 (119909) (24)

where 119871 = 11988921198891199052 119877119910 = 119905(119889119889119905) 119873119910 = 11990521199103 and 119892(119905) = (2 +61199052)1198901199052 + 119905211989031199052

The Adomian polynomials for representing the nonlinearterm Ny are

1198600 = 119905211991030 1198601 = 1199052 (3119910201199101)

1198602 = 1199052 (3119910201199102 + 3119910011991021)

(25)

Now 119871minus1 = int1199050int1199050()119889119905 119889119905 then using (5) the classical Bernstein

polynomials of 119892(119905) when v=m=6 are

119892119887 (119905) = 2 + 1547324119905 + 92901641199052 + 7832891199053+ 97518871199054 + 76596681199055 + 37498641199056 (26)

Andmodified Bernstein polynomials (21) of 119892(119905)with k=2 are119892119898119887 (119905) = 2 minus 0001037119905 + 69220821199052 + 19974411199053

+ 67376621199054 + 110511211199055 + 131245231199056 (27)

4 Journal of Applied Mathematics

Table 1 Comparison of absolute errors using 1199106 when m=v=6 and k=2

119905 1003816100381610038161003816119910119890119909119886119888119905 minus 1199101198871003816100381610038161003816 1003816100381610038161003816119910119890119909119886119888119905 minus 11991011989811988710038161003816100381610038160 0 0001 2580000 E-7 2000000 E-9002 2068000 E-6 2900000 E-8003 6992000 E-6 1420000 E-7004 1660300 E-5 4450000 E-7005 3249600 E-5 1074000 E-6006 5629000 E-5 2207000 E-6007 8962800 E-5 4057000 E-6008 1341850 E-4 6872000 E-6009 1916690 E-4 1093000 E-501 2638350 E-4 1654900 E-5MSE 1369383957 E-8 4632475500 E-11

00100080060040020

000025

000020

000015

000010

000005

Ybern

t(a)

0000016

0000014

0000012

0000010

0000008

0000006

0000004

0000002

0

Ymbern

0100080060040020t

(b)

Figure 1 The absolute error between ADM with modified Bernstein polynomials and the exact solution when m=v=6 and k=2

By (22) we have

1199100 = 119871minus1 (119892119898119887 (119905)) + 119910 (0) + 119889119910119889119905 (0) 119905 = 1 + 1199052minus 00001731199053 + 0576841199054 + 00998721199055+ 02245891199056 + 02631221199057 + 02343671199058

1199101 = minus119871minus1 (119905 1198891198891199051199100) minus 119871minus1 (1198600) = minus0251199054+ 00000261199055 minus 01769121199056 minus 00118771199057 + sdot sdot sdot

1199102 = minus119871minus1 (119905 1198891198891199051199101) minus 119871minus1 (1198601) = 00333331199056

minus 00000031199057 + 00323481199058 + 00115361199059 + sdot sdot sdot

(28)

And we obtain

119910119898119887 (119905) = 6sum119894=0

119910119894= 1 + 1199052 minus 00001731199053 + 0326841199054 + sdot sdot sdot

(29)

The absolute error of 119910119898119887(119905) and 119910119887(119905) is presented in Table 1and Figure 1

Figure 1 presents the absolute error of ADM with Bern-stein polynomial in (a) and ADM with modified Bernstein

Journal of Applied Mathematics 5

Table 2 Comparison of absolute errors using 11991010 when m=16 v=4 and k=2

119905 1003816100381610038161003816119910119890119909119886119888119905 minus 1199101198871003816100381610038161003816 1003816100381610038161003816119910119890119909119886119888119905 minus 11991011989811988710038161003816100381610038160 0 001 2061126 E-6 3074560 E-702 1145239 E-5 1058636 E-503 1315732 E-5 5114716 E-504 2165603 E-4 1331415 E-405 8646500 E-4 2420463 E-406 2311555 E-3 3299021 E-407 4905204 E-3 3231831 E-408 8829753 E-3 1540876 E-409 1391114 E-2 1870564 E-41 1946226 E-2 6088701 E-4MSE 6804632 E-5 7217798344 E-8

polynomial in (b) at m=v=6 and k=2 The absolute errorsgenerated using the ADM with Bernstein polynomial areof 10minus4 while the errors yielded from ADM with modifiedBernstein polynomial are of 10minus5Example 2 Consider the ordinary equation

11988921199101198891199052 + 119910119889119910119889119905 = 119905 sin (21199052) minus 41199052 sin (1199052) + 2 cos (1199052)

0 le 119905 le 1119910 (0) = 0

119889119910119889119905 (0) = 0

(30)

with the exact solution (119905) = sin(1199052)Here 119871 = 11988921198891199052119873119910 = 119910(119889119910119889119905) and 119892(119905) = 119905 sin(21199052) minus41199052 sin(1199052) + 2 cos(1199052)The Adomian polynomials for represent the nonlinear

term Nu are

1198600 = 1199100 11988911988911990511991001198601 = 1199101 1198891198891199051199100 + 1199100 1198891198891199051199101

1198602 = 1199102 1198891198891199051199100 + 1199101 1198891198891199051199101 + 1199100 1198891198891199051199101

(31)

Then using (5) the classical Bernstein polynomials of 119892(119905)when v=4 and m=16 is

119892119887 (119905) = 2000659171119905 + 022331901199052 + 00980851199053minus 3395401199054 + sdot sdot sdot (32)

Andmodified Bernstein polynomials (21) of g(t) with k=2 are

119892119898119887 (119905) = 2 minus 0007366119905 + 021888551199052 + 13897511199053minus 4502131199054 + sdot sdot sdot (33)

By (22) we have

119910119898119887 (119905) = 4sum119894=0

119910119894= 1199052 minus 00012281199053 + 00182411199054 minus 00305131199055

+ sdot sdot sdot (34)

The absolute error of 119910119898119887(119905) and 119910119887(119905) is presented in Table 2and Figure 2

Figure 2 presents the absolute error of ADM with Bern-stein polynomial in (a) and ADM with modified Bernsteinpolynomial in (b) at m=v=10 and k=3 The absolute errorsgenerated using the ADM with Bernstein polynomial areof 10minus2 while the errors yielded from ADM with modifiedBernstein polynomial are of 10minus4Example 3 Consider the ordinary equation

119889119910119889119905 minus 119905119910 + 1199102 = 1198901199052 119910 (0) = 1

(35)

with the exact solution (119905) = 11989011990522Here 119871 = 119889119889119905 119877119906 = minus119905119910119873119910 = 1199102 and 119892(119905) = 1198901199052 Then using (5) the classical Bernstein polynomials of 119892(119905)

when v=8 and m=12 is

119892119887 (119905) = 1 + 0083623119905 + 09391771199052 + 01977291199053+ 03315721199054 + sdot sdot sdot (36)

6 Journal of Applied Mathematics

Table 3 Comparison of absolute errors using 11991010 when m=12 v=8 and k=3

119905 1003816100381610038161003816119910119890119909119886119888119905 minus 1199101198871003816100381610038161003816 1003816100381610038161003816119910119890119909119886119888119905 minus 11991011989811988710038161003816100381610038160 0 0001 4134000 E-6 1750000 E-7002 1635300 E-5 6400000 E-7003 3639700 E-5 1314000 E-6004 6402400 E-5 2123000 E-6005 9900900 E-5 2999000 E-6006 1411460 E-4 3883000 E-6007 1902410 E-4 4720000 E-6008 2461180 E-4 5463000 E-6009 3086100 E-4 6069000 E-601 3775690 E-4 6501000 E-6MSE 3699974901 E-8 1619641710 E-11

Ybern

10806040200

001

002

003

t(a)

1080604020t

00016

00014

00012

00010

00008

00006

00004

00002

0

Ymbern

(b)

Figure 2The absolute error between ADM with modified Bernstein polynomials and the exact solution when m=v=10 and k=3

andmodified Bernstein polynomials (21) of 119892(119905) with k=3 are119892119898119887 (119905) = 1 + 0003794119905 + 09570941199052 + 01035091199053

+ 04102891199054 + sdot sdot sdot (37)

By (22) we have

119906119898119887 (119905) = 8sum119894=0

119906119894= 1 + 05018971199052 minus 00155671199053 + 01591351199054

+ sdot sdot sdot (38)

The absolute error of 119906119898119887(119905) and 119906119887(119905) is presented in Table 3and Figure 3

Figure 3 presents the absolute error of ADM with Bern-stein polynomial in (a) and ADM with modified Bernstein

polynomial in (b) at m=12 v=8 and k=3 The absolute errorsgenerated using the ADM with Bernstein polynomial areof 10minus4 while the errors yielded from ADM with modifiedBernstein polynomial are of 10minus6Example 4 Consider the partial differential equation

120597119910120597119905 + 119910120597119910120597119909 minus V12059721199101205971199092 = 119909 (2119905 cos (1199052) + sin2 (1199052))

119910 (119909 0) = 0(39)

Using (12) we have

119871119910 + 119873119910 + 119877119910 = 119892 (119909 119905) (40)

where 119871 = 119889119889119905 119877119910 = minusV(12059721199101205971199092) 119873119910 = 119910(120597119910120597119909) and119892(119909 119905) = 119909(2119905 cos(1199052) + sin2(1199052))

Journal of Applied Mathematics 7

0

00001

00002

00003

Ybern

0100080060040020t

(a)

0

1 times 10minus6

2 times 10minus6

3 times 10minus6

4 times 10minus6

5 times 10minus6

6 times 10minus6

Ymbern

0100080060040020t

(b)

Figure 3 The absolute error between ADM with modified Bernstein polynomials and the exact solution when m=12 v=8 and k=3

The Adomian polynomials for Ny are

1198600 = 119910119900 120597119910119900120597119909 1198601 = 119910119900 1205971199101120597119909 + 1199101 120597119910119900120597119909

1198602 = 119910119900 1205971199102120597119909 + 1199102 120597119910119900120597119909 + 1199101 1205971199101120597119909

(41)

Then using (5) the classical Bernstein polynomials of 119892(119909 119905)when v=m=6 is119892119887 (119909 119905) = 2003859119909119905 + 01034751199091199052 + 01499541199091199053

minus 0279351199091199054 + sdot sdot sdot (42)

and modified Bernstein polynomials (21) of 119892(119909 119905) with k=2are

119892119898119887 (119909 119905) = 1986611119909119905 + 00457441199091199052+ 05042811199091199053 minus 0253751199091199054 + sdot sdot sdot (43)

By (22) with V = 1 we have1199100 = 119871minus1 (119892119898119887 (119909 119905)) + 119910 (119909 0) = 09933061199091199052

+ 00152481199091199053 + 0126071199091199054 minus 00507501199091199055 1199101 = minus119871minus1 (minusV120597211991001205971199092 ) minus 119871minus1 (1198600) = minus01973311199091199055

minus 00050491199091199056 minus 00358121199091199057 + 00121221199091199058+ sdot sdot sdot

1199102 = minus119871minus1(minusV120597211991011205971199092 ) minus 119871minus1 (1198601) = 00490031199091199058+ 00017831199091199059 + 001210511990911990510 minus 000379511990911990511+ sdot sdot sdot

(44)

And we obtain

119910119898119887 (119909 119905) = 6sum119894=0

119910119894= 09933061199091199052 + 00152481199091199053

+ 01260701199091199054 minus 02480811199091199055minus 00837491199091199056 + sdot sdot sdot

(45)

The absolute error of 119910119898119887(119909 119905) and 119910119887(119909 119905) is presented inTable 4 and Figure 4 with the exact solution 119910(119909 119905) =119909 sin(1199052)

Also Figure 4 presents the absolute error of ADM withBernstein polynomial in (a) and ADM with modified Bern-stein polynomial in (c) atm=v=6 and k=2The absolute errorsgenerated using the ADM with Bernstein polynomial areof 10minus3 while the errors yielded from ADM with modifiedBernstein polynomial are of 10minus45 Conclusions

In this paper we show that utilizing modified Bernsteinpolynomials is smartly thought to modify the performance

8 Journal of Applied Mathematics

Table 4 Comparison of absolute errors using 1199106 when m=v=6 k=2 and x=01

119905 1003816100381610038161003816119910119890119909119886119888119905 minus 1199101198871003816100381610038161003816 1003816100381610038161003816119910119890119909119886119888119905 minus 11991011989811988710038161003816100381610038160 0 001 5512102 E-6 4149093 E-602 3401160 E-5 1849504 E-603 8851342 E-5 2838128 E-504 1404426 E-5 9121566 E-505 1256008 E-4 1620426 E-406 4250973 E-5 1880417 E-407 4360788 E-4 1086168 E-408 1050197 E-3 1020235 E-409 1718854 E-3 3752218 E-41 2024768 E-3 4863297 E-4MSE 8393551358 E-7 4702782622 E-8

108

0604

020

t

010008

006004

0020

00005

00010

00015Error

x

(a) The absolute error for classical Bernstein polynomials

108

0604

020

t

010008

006004

002

008

006005

007

003004

002001

0

x

Ybern

(b) The numerical solution for classical Bernstein polynomials

108

0604

020

t

010008

006004

0020

00001

00002

00003

00004

Error

x

(c) The absolute error for modified Bernstein polynomials

1

08

06

04

02

0

t

010008

006

004

002

008

006005

007

003004

002001

0

x

Ymbern

(d) The numerical solution for modified Bernstein polynomials

Figure 4The absolute error between ADMwith classical andmodified Bernstein polynomials and the exact solution using 1199106 whenm=v=6k=2 and x=01

Journal of Applied Mathematics 9

of the Adomian decomposition technique The fundamentalpreferred standpoint of this strategy is that it can be usedspecifically for all sort of differential and integral equations

We utilize modified Bernstein extensions of the nonlinearterm to get more exact outcomes Figures empower usto consider the difference between utilizing two strategiesgraphically Tables are additionally given to demonstrate thevariety of the outright mistakes for bigger estimation tobe specific for bigger m We observed from the numericaloutcomes in Tables 1ndash4 and Figures 1ndash4 that the ADMwith modification Bernstein polynomials gives more exactand robust numerical solution than the classical Bernsteinpolynomials Every one of the calculations was done with theguide of Maple 13 programming

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The research is supported by College of Computer Sciencesand Mathematics University of Mosul Republic of Iraqunder Project no 11251357

References

[1] S Muhammad and I Muhammad ldquoSome multi-step iterativemethods for solving nonlinear equationsrdquo Open Journal ofMathematical Sciences vol 1 no 1 pp 25ndash33 2017

[2] Y Yang and L Hua ldquoVariational iteration transformmethod forfractional differential equationswith local fractional derivativerdquoAbstract and Applied Analysis vol 2014 Article ID 760957 9pages 2014

[3] U R Hamood S S Muhammad and A Ayesha ldquoCombinationof homotopy perturbationmethod (HPM) and double sumudutransform to solve fractional KDV equationsrdquo Open Journal ofMathematical Sciences vol 2 no 1 pp 29ndash38 2018

[4] A M Wazwaz ldquoConstruction of solitary wave solutions andrational solutions for the KdV equation by Adomian decom-position methodrdquo Chaos Solitons amp Fractals vol 12 no 12 pp2283ndash2293 2001

[5] N Bildik and A Konuralp ldquoThe use of variational itera-tion method differential transform method and Adomiandecomposition method for solving different types of nonlinearpartial differential equationsrdquo International Journal ofNonlinearSciences andNumerical Simulation vol 7 no 1 pp 65ndash70 2006

[6] T Madeeha N N Muhammad S Rabia V Dumitru IMuhammad and S Naeem ldquoOn unsteady flow of a viscoelasticfluid through rotating cylindersrdquo Open Physics vol 1 no 1 pp1ndash15 2017

[7] A Sannia Q Haitao A Muhammad J Maria and I Muham-mad ldquoExact solutions of fractional Maxwell fluid between two

cylindersrdquo Open Journal of Mathematical Sciences vol 1 no 1pp 52ndash61 2017

[8] QHaitao FNidaWHassan and S Junaid ldquoAnalytical solutionfor the flow of a generalized Oldroyd-B fluid in a circularcylinderrdquo Open Journal of Mathematical Sciences vol 1 no 1pp 85ndash96 2017

[9] MMHosseini ldquoAdomian decompositionmethodwith Cheby-shev polynomialsrdquo Applied Mathematics and Computation vol175 no 2 pp 1685ndash1693 2006

[10] N Bildik and Deniz ldquoModified Adomian decompositionmethod for solving Riccati differential equationsrdquo Review of theAir Force Academy vol 3 no 30 pp 21ndash26 2015

[11] Y Liu ldquoAdomian decompositionmethod with orthogonal poly-nomials Legendre polynomialsrdquo Mathematical and ComputerModelling vol 49 no 5-6 pp 1268ndash1273 2009

[12] Y Mahmoudi et al ldquoAdomian decomposition method withLaguerre polynomials for solving ordinary differential equa-tionrdquo Journal of Basic and Applied Scientific Research vol 2 no12 pp 12236ndash12241 2012

[13] D Rani and V Mishra ldquoApproximate solution of boundaryvalue problem with bernstein polynomial laplace decompo-sition methodrdquo International Journal of Pure and AppliedMathematics vol 114 no 4 pp 823ndash833 2017

[14] GG LorentzBernstein Polynomials Chelsea Publishing Series1986

[15] J Cicho and Z Goi ldquoOn Bernoulli sums and Bernsteinpolynomialsrdquo Discrete Mathematics and eoretical ComputerScience pp 1ndash12 2009

[16] G Adomian Nonlinear Stochastic Operator Equations Aca-demic Press San Diego CA USA 1986

Hindawiwwwhindawicom Volume 2018

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Mathematical Problems in Engineering

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Hindawiwwwhindawicom Volume 2018

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

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Hindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

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Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Stochastic AnalysisInternational Journal of

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Journal of Applied Mathematics 3

where 119871minus1 = int1199090() ds and 119891(119909) are the terms having from

integrating the rest of the term g (x) and from utilizing thegiven initial or boundary conditions The ADM assumes thatN(u) (nonlinear term) can be decomposed by an infiniteseries of polynomials which is expressed in form

119873(119906) = infinsum119899=0

119860119899 (119906119900 1199061 119906119899) (15)

where 119860n are the Adomianrsquos polynomials [16] defined as

119860119899 = 1119899 119889119899119889120582n [119873(infinsum119894=0

120582iui)]120582=0

n = 0 1 2 (16)

We expand the function 119892(119909) by Bernstein series

119892 (119909) = 119898sum119894=0

119886119894119861119894 (119909) (17)

where 119861119894(119909) is the Bernstein polynomialsNow using (14) and (17) we have

1199060 = 119871minus1 (11988601198610 (119909) + 11988611198611 (119909) + 11988621198612 (119909)+ sdot sdot sdot 119886119898119861119898 (119909)) + 120579 (119909)

1199061 = minus119871minus1 (1198771199060) minus 119871minus1 (1198731199060) 1199062 = minus119871minus1 (1198771199061) minus 119871minus1 (1198731199061)

(18)

and so onThese formulas are easy to compute by usingMaple13 software

In this paper we improve the function 119892(119909) using modi-fied Bernstein series

119892 (119909) = 119898sum119894=0

(119898119894 )119909119894 (1 minus 119909)119898minus119894 119891( 119894119898)

minus 2119896minus1sum119886=2

119891(119886) (119909)119886119898119886 119879119898119886 (119909)(19)

And we can approach the derivatives using the Bernsteinpolynomials

119889119889119909119861119894119898 (119909) = 119898 (119861119894minus1119898minus1 (119909) minus 119861119894119898minus1 (119909)) (20)

Then (19) becomes

119892 (119909) = 119898sum119894=0

(119898119894 )119909119894 (1 minus 119909)119898minus119894 119891( 119894119898)

minus 2119896minus1sum119886=2

(119889(119886)119889119909(119886)) 119861119894119898 (119909)119886119898119886 119879119898119886 (119909)(21)

Now using (18) and (21) we have

1199060 = 119871minus1 (119861119894119898 (119909)) + 120579 (119909) 1199061 = minus119871minus1 (1198771199060) minus 119871minus1 (1198731199060) 1199062 = minus119871minus1 (1198771199061) minus 119871minus1 (1198731199061)

(22)

The above equation is governing equation of ADM usingmodified Bernstein polynomials The obtained approximatesolution 120596119881(119909) = sum119881119894=0 119906119894 by (22) has a comparison with theclassic approximation solution and the correct solution

4 Numerical Results

In this section we solve ordinary and partial differentialequations by ADM based on Bernstein polynomials and wecompare with ADM based on classical Bernstein polynomial

Example 1 Consider the ordinary equation

11988921199101198891199052 + 119905119889119910119889119905 + 11990521199103 = (2 + 61199052) 1198901199052 + 119905211989031199052

119910 (0) = 1119889119910119889119905 (0) = 0

(23)

with the exact solution (119905) = 1198901199052 Using (12) we have119871119910 + 119873119910 + 119877119910 = 119892 (119909) (24)

where 119871 = 11988921198891199052 119877119910 = 119905(119889119889119905) 119873119910 = 11990521199103 and 119892(119905) = (2 +61199052)1198901199052 + 119905211989031199052

The Adomian polynomials for representing the nonlinearterm Ny are

1198600 = 119905211991030 1198601 = 1199052 (3119910201199101)

1198602 = 1199052 (3119910201199102 + 3119910011991021)

(25)

Now 119871minus1 = int1199050int1199050()119889119905 119889119905 then using (5) the classical Bernstein

polynomials of 119892(119905) when v=m=6 are

119892119887 (119905) = 2 + 1547324119905 + 92901641199052 + 7832891199053+ 97518871199054 + 76596681199055 + 37498641199056 (26)

Andmodified Bernstein polynomials (21) of 119892(119905)with k=2 are119892119898119887 (119905) = 2 minus 0001037119905 + 69220821199052 + 19974411199053

+ 67376621199054 + 110511211199055 + 131245231199056 (27)

4 Journal of Applied Mathematics

Table 1 Comparison of absolute errors using 1199106 when m=v=6 and k=2

119905 1003816100381610038161003816119910119890119909119886119888119905 minus 1199101198871003816100381610038161003816 1003816100381610038161003816119910119890119909119886119888119905 minus 11991011989811988710038161003816100381610038160 0 0001 2580000 E-7 2000000 E-9002 2068000 E-6 2900000 E-8003 6992000 E-6 1420000 E-7004 1660300 E-5 4450000 E-7005 3249600 E-5 1074000 E-6006 5629000 E-5 2207000 E-6007 8962800 E-5 4057000 E-6008 1341850 E-4 6872000 E-6009 1916690 E-4 1093000 E-501 2638350 E-4 1654900 E-5MSE 1369383957 E-8 4632475500 E-11

00100080060040020

000025

000020

000015

000010

000005

Ybern

t(a)

0000016

0000014

0000012

0000010

0000008

0000006

0000004

0000002

0

Ymbern

0100080060040020t

(b)

Figure 1 The absolute error between ADM with modified Bernstein polynomials and the exact solution when m=v=6 and k=2

By (22) we have

1199100 = 119871minus1 (119892119898119887 (119905)) + 119910 (0) + 119889119910119889119905 (0) 119905 = 1 + 1199052minus 00001731199053 + 0576841199054 + 00998721199055+ 02245891199056 + 02631221199057 + 02343671199058

1199101 = minus119871minus1 (119905 1198891198891199051199100) minus 119871minus1 (1198600) = minus0251199054+ 00000261199055 minus 01769121199056 minus 00118771199057 + sdot sdot sdot

1199102 = minus119871minus1 (119905 1198891198891199051199101) minus 119871minus1 (1198601) = 00333331199056

minus 00000031199057 + 00323481199058 + 00115361199059 + sdot sdot sdot

(28)

And we obtain

119910119898119887 (119905) = 6sum119894=0

119910119894= 1 + 1199052 minus 00001731199053 + 0326841199054 + sdot sdot sdot

(29)

The absolute error of 119910119898119887(119905) and 119910119887(119905) is presented in Table 1and Figure 1

Figure 1 presents the absolute error of ADM with Bern-stein polynomial in (a) and ADM with modified Bernstein

Journal of Applied Mathematics 5

Table 2 Comparison of absolute errors using 11991010 when m=16 v=4 and k=2

119905 1003816100381610038161003816119910119890119909119886119888119905 minus 1199101198871003816100381610038161003816 1003816100381610038161003816119910119890119909119886119888119905 minus 11991011989811988710038161003816100381610038160 0 001 2061126 E-6 3074560 E-702 1145239 E-5 1058636 E-503 1315732 E-5 5114716 E-504 2165603 E-4 1331415 E-405 8646500 E-4 2420463 E-406 2311555 E-3 3299021 E-407 4905204 E-3 3231831 E-408 8829753 E-3 1540876 E-409 1391114 E-2 1870564 E-41 1946226 E-2 6088701 E-4MSE 6804632 E-5 7217798344 E-8

polynomial in (b) at m=v=6 and k=2 The absolute errorsgenerated using the ADM with Bernstein polynomial areof 10minus4 while the errors yielded from ADM with modifiedBernstein polynomial are of 10minus5Example 2 Consider the ordinary equation

11988921199101198891199052 + 119910119889119910119889119905 = 119905 sin (21199052) minus 41199052 sin (1199052) + 2 cos (1199052)

0 le 119905 le 1119910 (0) = 0

119889119910119889119905 (0) = 0

(30)

with the exact solution (119905) = sin(1199052)Here 119871 = 11988921198891199052119873119910 = 119910(119889119910119889119905) and 119892(119905) = 119905 sin(21199052) minus41199052 sin(1199052) + 2 cos(1199052)The Adomian polynomials for represent the nonlinear

term Nu are

1198600 = 1199100 11988911988911990511991001198601 = 1199101 1198891198891199051199100 + 1199100 1198891198891199051199101

1198602 = 1199102 1198891198891199051199100 + 1199101 1198891198891199051199101 + 1199100 1198891198891199051199101

(31)

Then using (5) the classical Bernstein polynomials of 119892(119905)when v=4 and m=16 is

119892119887 (119905) = 2000659171119905 + 022331901199052 + 00980851199053minus 3395401199054 + sdot sdot sdot (32)

Andmodified Bernstein polynomials (21) of g(t) with k=2 are

119892119898119887 (119905) = 2 minus 0007366119905 + 021888551199052 + 13897511199053minus 4502131199054 + sdot sdot sdot (33)

By (22) we have

119910119898119887 (119905) = 4sum119894=0

119910119894= 1199052 minus 00012281199053 + 00182411199054 minus 00305131199055

+ sdot sdot sdot (34)

The absolute error of 119910119898119887(119905) and 119910119887(119905) is presented in Table 2and Figure 2

Figure 2 presents the absolute error of ADM with Bern-stein polynomial in (a) and ADM with modified Bernsteinpolynomial in (b) at m=v=10 and k=3 The absolute errorsgenerated using the ADM with Bernstein polynomial areof 10minus2 while the errors yielded from ADM with modifiedBernstein polynomial are of 10minus4Example 3 Consider the ordinary equation

119889119910119889119905 minus 119905119910 + 1199102 = 1198901199052 119910 (0) = 1

(35)

with the exact solution (119905) = 11989011990522Here 119871 = 119889119889119905 119877119906 = minus119905119910119873119910 = 1199102 and 119892(119905) = 1198901199052 Then using (5) the classical Bernstein polynomials of 119892(119905)

when v=8 and m=12 is

119892119887 (119905) = 1 + 0083623119905 + 09391771199052 + 01977291199053+ 03315721199054 + sdot sdot sdot (36)

6 Journal of Applied Mathematics

Table 3 Comparison of absolute errors using 11991010 when m=12 v=8 and k=3

119905 1003816100381610038161003816119910119890119909119886119888119905 minus 1199101198871003816100381610038161003816 1003816100381610038161003816119910119890119909119886119888119905 minus 11991011989811988710038161003816100381610038160 0 0001 4134000 E-6 1750000 E-7002 1635300 E-5 6400000 E-7003 3639700 E-5 1314000 E-6004 6402400 E-5 2123000 E-6005 9900900 E-5 2999000 E-6006 1411460 E-4 3883000 E-6007 1902410 E-4 4720000 E-6008 2461180 E-4 5463000 E-6009 3086100 E-4 6069000 E-601 3775690 E-4 6501000 E-6MSE 3699974901 E-8 1619641710 E-11

Ybern

10806040200

001

002

003

t(a)

1080604020t

00016

00014

00012

00010

00008

00006

00004

00002

0

Ymbern

(b)

Figure 2The absolute error between ADM with modified Bernstein polynomials and the exact solution when m=v=10 and k=3

andmodified Bernstein polynomials (21) of 119892(119905) with k=3 are119892119898119887 (119905) = 1 + 0003794119905 + 09570941199052 + 01035091199053

+ 04102891199054 + sdot sdot sdot (37)

By (22) we have

119906119898119887 (119905) = 8sum119894=0

119906119894= 1 + 05018971199052 minus 00155671199053 + 01591351199054

+ sdot sdot sdot (38)

The absolute error of 119906119898119887(119905) and 119906119887(119905) is presented in Table 3and Figure 3

Figure 3 presents the absolute error of ADM with Bern-stein polynomial in (a) and ADM with modified Bernstein

polynomial in (b) at m=12 v=8 and k=3 The absolute errorsgenerated using the ADM with Bernstein polynomial areof 10minus4 while the errors yielded from ADM with modifiedBernstein polynomial are of 10minus6Example 4 Consider the partial differential equation

120597119910120597119905 + 119910120597119910120597119909 minus V12059721199101205971199092 = 119909 (2119905 cos (1199052) + sin2 (1199052))

119910 (119909 0) = 0(39)

Using (12) we have

119871119910 + 119873119910 + 119877119910 = 119892 (119909 119905) (40)

where 119871 = 119889119889119905 119877119910 = minusV(12059721199101205971199092) 119873119910 = 119910(120597119910120597119909) and119892(119909 119905) = 119909(2119905 cos(1199052) + sin2(1199052))

Journal of Applied Mathematics 7

0

00001

00002

00003

Ybern

0100080060040020t

(a)

0

1 times 10minus6

2 times 10minus6

3 times 10minus6

4 times 10minus6

5 times 10minus6

6 times 10minus6

Ymbern

0100080060040020t

(b)

Figure 3 The absolute error between ADM with modified Bernstein polynomials and the exact solution when m=12 v=8 and k=3

The Adomian polynomials for Ny are

1198600 = 119910119900 120597119910119900120597119909 1198601 = 119910119900 1205971199101120597119909 + 1199101 120597119910119900120597119909

1198602 = 119910119900 1205971199102120597119909 + 1199102 120597119910119900120597119909 + 1199101 1205971199101120597119909

(41)

Then using (5) the classical Bernstein polynomials of 119892(119909 119905)when v=m=6 is119892119887 (119909 119905) = 2003859119909119905 + 01034751199091199052 + 01499541199091199053

minus 0279351199091199054 + sdot sdot sdot (42)

and modified Bernstein polynomials (21) of 119892(119909 119905) with k=2are

119892119898119887 (119909 119905) = 1986611119909119905 + 00457441199091199052+ 05042811199091199053 minus 0253751199091199054 + sdot sdot sdot (43)

By (22) with V = 1 we have1199100 = 119871minus1 (119892119898119887 (119909 119905)) + 119910 (119909 0) = 09933061199091199052

+ 00152481199091199053 + 0126071199091199054 minus 00507501199091199055 1199101 = minus119871minus1 (minusV120597211991001205971199092 ) minus 119871minus1 (1198600) = minus01973311199091199055

minus 00050491199091199056 minus 00358121199091199057 + 00121221199091199058+ sdot sdot sdot

1199102 = minus119871minus1(minusV120597211991011205971199092 ) minus 119871minus1 (1198601) = 00490031199091199058+ 00017831199091199059 + 001210511990911990510 minus 000379511990911990511+ sdot sdot sdot

(44)

And we obtain

119910119898119887 (119909 119905) = 6sum119894=0

119910119894= 09933061199091199052 + 00152481199091199053

+ 01260701199091199054 minus 02480811199091199055minus 00837491199091199056 + sdot sdot sdot

(45)

The absolute error of 119910119898119887(119909 119905) and 119910119887(119909 119905) is presented inTable 4 and Figure 4 with the exact solution 119910(119909 119905) =119909 sin(1199052)

Also Figure 4 presents the absolute error of ADM withBernstein polynomial in (a) and ADM with modified Bern-stein polynomial in (c) atm=v=6 and k=2The absolute errorsgenerated using the ADM with Bernstein polynomial areof 10minus3 while the errors yielded from ADM with modifiedBernstein polynomial are of 10minus45 Conclusions

In this paper we show that utilizing modified Bernsteinpolynomials is smartly thought to modify the performance

8 Journal of Applied Mathematics

Table 4 Comparison of absolute errors using 1199106 when m=v=6 k=2 and x=01

119905 1003816100381610038161003816119910119890119909119886119888119905 minus 1199101198871003816100381610038161003816 1003816100381610038161003816119910119890119909119886119888119905 minus 11991011989811988710038161003816100381610038160 0 001 5512102 E-6 4149093 E-602 3401160 E-5 1849504 E-603 8851342 E-5 2838128 E-504 1404426 E-5 9121566 E-505 1256008 E-4 1620426 E-406 4250973 E-5 1880417 E-407 4360788 E-4 1086168 E-408 1050197 E-3 1020235 E-409 1718854 E-3 3752218 E-41 2024768 E-3 4863297 E-4MSE 8393551358 E-7 4702782622 E-8

108

0604

020

t

010008

006004

0020

00005

00010

00015Error

x

(a) The absolute error for classical Bernstein polynomials

108

0604

020

t

010008

006004

002

008

006005

007

003004

002001

0

x

Ybern

(b) The numerical solution for classical Bernstein polynomials

108

0604

020

t

010008

006004

0020

00001

00002

00003

00004

Error

x

(c) The absolute error for modified Bernstein polynomials

1

08

06

04

02

0

t

010008

006

004

002

008

006005

007

003004

002001

0

x

Ymbern

(d) The numerical solution for modified Bernstein polynomials

Figure 4The absolute error between ADMwith classical andmodified Bernstein polynomials and the exact solution using 1199106 whenm=v=6k=2 and x=01

Journal of Applied Mathematics 9

of the Adomian decomposition technique The fundamentalpreferred standpoint of this strategy is that it can be usedspecifically for all sort of differential and integral equations

We utilize modified Bernstein extensions of the nonlinearterm to get more exact outcomes Figures empower usto consider the difference between utilizing two strategiesgraphically Tables are additionally given to demonstrate thevariety of the outright mistakes for bigger estimation tobe specific for bigger m We observed from the numericaloutcomes in Tables 1ndash4 and Figures 1ndash4 that the ADMwith modification Bernstein polynomials gives more exactand robust numerical solution than the classical Bernsteinpolynomials Every one of the calculations was done with theguide of Maple 13 programming

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The research is supported by College of Computer Sciencesand Mathematics University of Mosul Republic of Iraqunder Project no 11251357

References

[1] S Muhammad and I Muhammad ldquoSome multi-step iterativemethods for solving nonlinear equationsrdquo Open Journal ofMathematical Sciences vol 1 no 1 pp 25ndash33 2017

[2] Y Yang and L Hua ldquoVariational iteration transformmethod forfractional differential equationswith local fractional derivativerdquoAbstract and Applied Analysis vol 2014 Article ID 760957 9pages 2014

[3] U R Hamood S S Muhammad and A Ayesha ldquoCombinationof homotopy perturbationmethod (HPM) and double sumudutransform to solve fractional KDV equationsrdquo Open Journal ofMathematical Sciences vol 2 no 1 pp 29ndash38 2018

[4] A M Wazwaz ldquoConstruction of solitary wave solutions andrational solutions for the KdV equation by Adomian decom-position methodrdquo Chaos Solitons amp Fractals vol 12 no 12 pp2283ndash2293 2001

[5] N Bildik and A Konuralp ldquoThe use of variational itera-tion method differential transform method and Adomiandecomposition method for solving different types of nonlinearpartial differential equationsrdquo International Journal ofNonlinearSciences andNumerical Simulation vol 7 no 1 pp 65ndash70 2006

[6] T Madeeha N N Muhammad S Rabia V Dumitru IMuhammad and S Naeem ldquoOn unsteady flow of a viscoelasticfluid through rotating cylindersrdquo Open Physics vol 1 no 1 pp1ndash15 2017

[7] A Sannia Q Haitao A Muhammad J Maria and I Muham-mad ldquoExact solutions of fractional Maxwell fluid between two

cylindersrdquo Open Journal of Mathematical Sciences vol 1 no 1pp 52ndash61 2017

[8] QHaitao FNidaWHassan and S Junaid ldquoAnalytical solutionfor the flow of a generalized Oldroyd-B fluid in a circularcylinderrdquo Open Journal of Mathematical Sciences vol 1 no 1pp 85ndash96 2017

[9] MMHosseini ldquoAdomian decompositionmethodwith Cheby-shev polynomialsrdquo Applied Mathematics and Computation vol175 no 2 pp 1685ndash1693 2006

[10] N Bildik and Deniz ldquoModified Adomian decompositionmethod for solving Riccati differential equationsrdquo Review of theAir Force Academy vol 3 no 30 pp 21ndash26 2015

[11] Y Liu ldquoAdomian decompositionmethod with orthogonal poly-nomials Legendre polynomialsrdquo Mathematical and ComputerModelling vol 49 no 5-6 pp 1268ndash1273 2009

[12] Y Mahmoudi et al ldquoAdomian decomposition method withLaguerre polynomials for solving ordinary differential equa-tionrdquo Journal of Basic and Applied Scientific Research vol 2 no12 pp 12236ndash12241 2012

[13] D Rani and V Mishra ldquoApproximate solution of boundaryvalue problem with bernstein polynomial laplace decompo-sition methodrdquo International Journal of Pure and AppliedMathematics vol 114 no 4 pp 823ndash833 2017

[14] GG LorentzBernstein Polynomials Chelsea Publishing Series1986

[15] J Cicho and Z Goi ldquoOn Bernoulli sums and Bernsteinpolynomialsrdquo Discrete Mathematics and eoretical ComputerScience pp 1ndash12 2009

[16] G Adomian Nonlinear Stochastic Operator Equations Aca-demic Press San Diego CA USA 1986

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

4 Journal of Applied Mathematics

Table 1 Comparison of absolute errors using 1199106 when m=v=6 and k=2

119905 1003816100381610038161003816119910119890119909119886119888119905 minus 1199101198871003816100381610038161003816 1003816100381610038161003816119910119890119909119886119888119905 minus 11991011989811988710038161003816100381610038160 0 0001 2580000 E-7 2000000 E-9002 2068000 E-6 2900000 E-8003 6992000 E-6 1420000 E-7004 1660300 E-5 4450000 E-7005 3249600 E-5 1074000 E-6006 5629000 E-5 2207000 E-6007 8962800 E-5 4057000 E-6008 1341850 E-4 6872000 E-6009 1916690 E-4 1093000 E-501 2638350 E-4 1654900 E-5MSE 1369383957 E-8 4632475500 E-11

00100080060040020

000025

000020

000015

000010

000005

Ybern

t(a)

0000016

0000014

0000012

0000010

0000008

0000006

0000004

0000002

0

Ymbern

0100080060040020t

(b)

Figure 1 The absolute error between ADM with modified Bernstein polynomials and the exact solution when m=v=6 and k=2

By (22) we have

1199100 = 119871minus1 (119892119898119887 (119905)) + 119910 (0) + 119889119910119889119905 (0) 119905 = 1 + 1199052minus 00001731199053 + 0576841199054 + 00998721199055+ 02245891199056 + 02631221199057 + 02343671199058

1199101 = minus119871minus1 (119905 1198891198891199051199100) minus 119871minus1 (1198600) = minus0251199054+ 00000261199055 minus 01769121199056 minus 00118771199057 + sdot sdot sdot

1199102 = minus119871minus1 (119905 1198891198891199051199101) minus 119871minus1 (1198601) = 00333331199056

minus 00000031199057 + 00323481199058 + 00115361199059 + sdot sdot sdot

(28)

And we obtain

119910119898119887 (119905) = 6sum119894=0

119910119894= 1 + 1199052 minus 00001731199053 + 0326841199054 + sdot sdot sdot

(29)

The absolute error of 119910119898119887(119905) and 119910119887(119905) is presented in Table 1and Figure 1

Figure 1 presents the absolute error of ADM with Bern-stein polynomial in (a) and ADM with modified Bernstein

Journal of Applied Mathematics 5

Table 2 Comparison of absolute errors using 11991010 when m=16 v=4 and k=2

119905 1003816100381610038161003816119910119890119909119886119888119905 minus 1199101198871003816100381610038161003816 1003816100381610038161003816119910119890119909119886119888119905 minus 11991011989811988710038161003816100381610038160 0 001 2061126 E-6 3074560 E-702 1145239 E-5 1058636 E-503 1315732 E-5 5114716 E-504 2165603 E-4 1331415 E-405 8646500 E-4 2420463 E-406 2311555 E-3 3299021 E-407 4905204 E-3 3231831 E-408 8829753 E-3 1540876 E-409 1391114 E-2 1870564 E-41 1946226 E-2 6088701 E-4MSE 6804632 E-5 7217798344 E-8

polynomial in (b) at m=v=6 and k=2 The absolute errorsgenerated using the ADM with Bernstein polynomial areof 10minus4 while the errors yielded from ADM with modifiedBernstein polynomial are of 10minus5Example 2 Consider the ordinary equation

11988921199101198891199052 + 119910119889119910119889119905 = 119905 sin (21199052) minus 41199052 sin (1199052) + 2 cos (1199052)

0 le 119905 le 1119910 (0) = 0

119889119910119889119905 (0) = 0

(30)

with the exact solution (119905) = sin(1199052)Here 119871 = 11988921198891199052119873119910 = 119910(119889119910119889119905) and 119892(119905) = 119905 sin(21199052) minus41199052 sin(1199052) + 2 cos(1199052)The Adomian polynomials for represent the nonlinear

term Nu are

1198600 = 1199100 11988911988911990511991001198601 = 1199101 1198891198891199051199100 + 1199100 1198891198891199051199101

1198602 = 1199102 1198891198891199051199100 + 1199101 1198891198891199051199101 + 1199100 1198891198891199051199101

(31)

Then using (5) the classical Bernstein polynomials of 119892(119905)when v=4 and m=16 is

119892119887 (119905) = 2000659171119905 + 022331901199052 + 00980851199053minus 3395401199054 + sdot sdot sdot (32)

Andmodified Bernstein polynomials (21) of g(t) with k=2 are

119892119898119887 (119905) = 2 minus 0007366119905 + 021888551199052 + 13897511199053minus 4502131199054 + sdot sdot sdot (33)

By (22) we have

119910119898119887 (119905) = 4sum119894=0

119910119894= 1199052 minus 00012281199053 + 00182411199054 minus 00305131199055

+ sdot sdot sdot (34)

The absolute error of 119910119898119887(119905) and 119910119887(119905) is presented in Table 2and Figure 2

Figure 2 presents the absolute error of ADM with Bern-stein polynomial in (a) and ADM with modified Bernsteinpolynomial in (b) at m=v=10 and k=3 The absolute errorsgenerated using the ADM with Bernstein polynomial areof 10minus2 while the errors yielded from ADM with modifiedBernstein polynomial are of 10minus4Example 3 Consider the ordinary equation

119889119910119889119905 minus 119905119910 + 1199102 = 1198901199052 119910 (0) = 1

(35)

with the exact solution (119905) = 11989011990522Here 119871 = 119889119889119905 119877119906 = minus119905119910119873119910 = 1199102 and 119892(119905) = 1198901199052 Then using (5) the classical Bernstein polynomials of 119892(119905)

when v=8 and m=12 is

119892119887 (119905) = 1 + 0083623119905 + 09391771199052 + 01977291199053+ 03315721199054 + sdot sdot sdot (36)

6 Journal of Applied Mathematics

Table 3 Comparison of absolute errors using 11991010 when m=12 v=8 and k=3

119905 1003816100381610038161003816119910119890119909119886119888119905 minus 1199101198871003816100381610038161003816 1003816100381610038161003816119910119890119909119886119888119905 minus 11991011989811988710038161003816100381610038160 0 0001 4134000 E-6 1750000 E-7002 1635300 E-5 6400000 E-7003 3639700 E-5 1314000 E-6004 6402400 E-5 2123000 E-6005 9900900 E-5 2999000 E-6006 1411460 E-4 3883000 E-6007 1902410 E-4 4720000 E-6008 2461180 E-4 5463000 E-6009 3086100 E-4 6069000 E-601 3775690 E-4 6501000 E-6MSE 3699974901 E-8 1619641710 E-11

Ybern

10806040200

001

002

003

t(a)

1080604020t

00016

00014

00012

00010

00008

00006

00004

00002

0

Ymbern

(b)

Figure 2The absolute error between ADM with modified Bernstein polynomials and the exact solution when m=v=10 and k=3

andmodified Bernstein polynomials (21) of 119892(119905) with k=3 are119892119898119887 (119905) = 1 + 0003794119905 + 09570941199052 + 01035091199053

+ 04102891199054 + sdot sdot sdot (37)

By (22) we have

119906119898119887 (119905) = 8sum119894=0

119906119894= 1 + 05018971199052 minus 00155671199053 + 01591351199054

+ sdot sdot sdot (38)

The absolute error of 119906119898119887(119905) and 119906119887(119905) is presented in Table 3and Figure 3

Figure 3 presents the absolute error of ADM with Bern-stein polynomial in (a) and ADM with modified Bernstein

polynomial in (b) at m=12 v=8 and k=3 The absolute errorsgenerated using the ADM with Bernstein polynomial areof 10minus4 while the errors yielded from ADM with modifiedBernstein polynomial are of 10minus6Example 4 Consider the partial differential equation

120597119910120597119905 + 119910120597119910120597119909 minus V12059721199101205971199092 = 119909 (2119905 cos (1199052) + sin2 (1199052))

119910 (119909 0) = 0(39)

Using (12) we have

119871119910 + 119873119910 + 119877119910 = 119892 (119909 119905) (40)

where 119871 = 119889119889119905 119877119910 = minusV(12059721199101205971199092) 119873119910 = 119910(120597119910120597119909) and119892(119909 119905) = 119909(2119905 cos(1199052) + sin2(1199052))

Journal of Applied Mathematics 7

0

00001

00002

00003

Ybern

0100080060040020t

(a)

0

1 times 10minus6

2 times 10minus6

3 times 10minus6

4 times 10minus6

5 times 10minus6

6 times 10minus6

Ymbern

0100080060040020t

(b)

Figure 3 The absolute error between ADM with modified Bernstein polynomials and the exact solution when m=12 v=8 and k=3

The Adomian polynomials for Ny are

1198600 = 119910119900 120597119910119900120597119909 1198601 = 119910119900 1205971199101120597119909 + 1199101 120597119910119900120597119909

1198602 = 119910119900 1205971199102120597119909 + 1199102 120597119910119900120597119909 + 1199101 1205971199101120597119909

(41)

Then using (5) the classical Bernstein polynomials of 119892(119909 119905)when v=m=6 is119892119887 (119909 119905) = 2003859119909119905 + 01034751199091199052 + 01499541199091199053

minus 0279351199091199054 + sdot sdot sdot (42)

and modified Bernstein polynomials (21) of 119892(119909 119905) with k=2are

119892119898119887 (119909 119905) = 1986611119909119905 + 00457441199091199052+ 05042811199091199053 minus 0253751199091199054 + sdot sdot sdot (43)

By (22) with V = 1 we have1199100 = 119871minus1 (119892119898119887 (119909 119905)) + 119910 (119909 0) = 09933061199091199052

+ 00152481199091199053 + 0126071199091199054 minus 00507501199091199055 1199101 = minus119871minus1 (minusV120597211991001205971199092 ) minus 119871minus1 (1198600) = minus01973311199091199055

minus 00050491199091199056 minus 00358121199091199057 + 00121221199091199058+ sdot sdot sdot

1199102 = minus119871minus1(minusV120597211991011205971199092 ) minus 119871minus1 (1198601) = 00490031199091199058+ 00017831199091199059 + 001210511990911990510 minus 000379511990911990511+ sdot sdot sdot

(44)

And we obtain

119910119898119887 (119909 119905) = 6sum119894=0

119910119894= 09933061199091199052 + 00152481199091199053

+ 01260701199091199054 minus 02480811199091199055minus 00837491199091199056 + sdot sdot sdot

(45)

The absolute error of 119910119898119887(119909 119905) and 119910119887(119909 119905) is presented inTable 4 and Figure 4 with the exact solution 119910(119909 119905) =119909 sin(1199052)

Also Figure 4 presents the absolute error of ADM withBernstein polynomial in (a) and ADM with modified Bern-stein polynomial in (c) atm=v=6 and k=2The absolute errorsgenerated using the ADM with Bernstein polynomial areof 10minus3 while the errors yielded from ADM with modifiedBernstein polynomial are of 10minus45 Conclusions

In this paper we show that utilizing modified Bernsteinpolynomials is smartly thought to modify the performance

8 Journal of Applied Mathematics

Table 4 Comparison of absolute errors using 1199106 when m=v=6 k=2 and x=01

119905 1003816100381610038161003816119910119890119909119886119888119905 minus 1199101198871003816100381610038161003816 1003816100381610038161003816119910119890119909119886119888119905 minus 11991011989811988710038161003816100381610038160 0 001 5512102 E-6 4149093 E-602 3401160 E-5 1849504 E-603 8851342 E-5 2838128 E-504 1404426 E-5 9121566 E-505 1256008 E-4 1620426 E-406 4250973 E-5 1880417 E-407 4360788 E-4 1086168 E-408 1050197 E-3 1020235 E-409 1718854 E-3 3752218 E-41 2024768 E-3 4863297 E-4MSE 8393551358 E-7 4702782622 E-8

108

0604

020

t

010008

006004

0020

00005

00010

00015Error

x

(a) The absolute error for classical Bernstein polynomials

108

0604

020

t

010008

006004

002

008

006005

007

003004

002001

0

x

Ybern

(b) The numerical solution for classical Bernstein polynomials

108

0604

020

t

010008

006004

0020

00001

00002

00003

00004

Error

x

(c) The absolute error for modified Bernstein polynomials

1

08

06

04

02

0

t

010008

006

004

002

008

006005

007

003004

002001

0

x

Ymbern

(d) The numerical solution for modified Bernstein polynomials

Figure 4The absolute error between ADMwith classical andmodified Bernstein polynomials and the exact solution using 1199106 whenm=v=6k=2 and x=01

Journal of Applied Mathematics 9

of the Adomian decomposition technique The fundamentalpreferred standpoint of this strategy is that it can be usedspecifically for all sort of differential and integral equations

We utilize modified Bernstein extensions of the nonlinearterm to get more exact outcomes Figures empower usto consider the difference between utilizing two strategiesgraphically Tables are additionally given to demonstrate thevariety of the outright mistakes for bigger estimation tobe specific for bigger m We observed from the numericaloutcomes in Tables 1ndash4 and Figures 1ndash4 that the ADMwith modification Bernstein polynomials gives more exactand robust numerical solution than the classical Bernsteinpolynomials Every one of the calculations was done with theguide of Maple 13 programming

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The research is supported by College of Computer Sciencesand Mathematics University of Mosul Republic of Iraqunder Project no 11251357

References

[1] S Muhammad and I Muhammad ldquoSome multi-step iterativemethods for solving nonlinear equationsrdquo Open Journal ofMathematical Sciences vol 1 no 1 pp 25ndash33 2017

[2] Y Yang and L Hua ldquoVariational iteration transformmethod forfractional differential equationswith local fractional derivativerdquoAbstract and Applied Analysis vol 2014 Article ID 760957 9pages 2014

[3] U R Hamood S S Muhammad and A Ayesha ldquoCombinationof homotopy perturbationmethod (HPM) and double sumudutransform to solve fractional KDV equationsrdquo Open Journal ofMathematical Sciences vol 2 no 1 pp 29ndash38 2018

[4] A M Wazwaz ldquoConstruction of solitary wave solutions andrational solutions for the KdV equation by Adomian decom-position methodrdquo Chaos Solitons amp Fractals vol 12 no 12 pp2283ndash2293 2001

[5] N Bildik and A Konuralp ldquoThe use of variational itera-tion method differential transform method and Adomiandecomposition method for solving different types of nonlinearpartial differential equationsrdquo International Journal ofNonlinearSciences andNumerical Simulation vol 7 no 1 pp 65ndash70 2006

[6] T Madeeha N N Muhammad S Rabia V Dumitru IMuhammad and S Naeem ldquoOn unsteady flow of a viscoelasticfluid through rotating cylindersrdquo Open Physics vol 1 no 1 pp1ndash15 2017

[7] A Sannia Q Haitao A Muhammad J Maria and I Muham-mad ldquoExact solutions of fractional Maxwell fluid between two

cylindersrdquo Open Journal of Mathematical Sciences vol 1 no 1pp 52ndash61 2017

[8] QHaitao FNidaWHassan and S Junaid ldquoAnalytical solutionfor the flow of a generalized Oldroyd-B fluid in a circularcylinderrdquo Open Journal of Mathematical Sciences vol 1 no 1pp 85ndash96 2017

[9] MMHosseini ldquoAdomian decompositionmethodwith Cheby-shev polynomialsrdquo Applied Mathematics and Computation vol175 no 2 pp 1685ndash1693 2006

[10] N Bildik and Deniz ldquoModified Adomian decompositionmethod for solving Riccati differential equationsrdquo Review of theAir Force Academy vol 3 no 30 pp 21ndash26 2015

[11] Y Liu ldquoAdomian decompositionmethod with orthogonal poly-nomials Legendre polynomialsrdquo Mathematical and ComputerModelling vol 49 no 5-6 pp 1268ndash1273 2009

[12] Y Mahmoudi et al ldquoAdomian decomposition method withLaguerre polynomials for solving ordinary differential equa-tionrdquo Journal of Basic and Applied Scientific Research vol 2 no12 pp 12236ndash12241 2012

[13] D Rani and V Mishra ldquoApproximate solution of boundaryvalue problem with bernstein polynomial laplace decompo-sition methodrdquo International Journal of Pure and AppliedMathematics vol 114 no 4 pp 823ndash833 2017

[14] GG LorentzBernstein Polynomials Chelsea Publishing Series1986

[15] J Cicho and Z Goi ldquoOn Bernoulli sums and Bernsteinpolynomialsrdquo Discrete Mathematics and eoretical ComputerScience pp 1ndash12 2009

[16] G Adomian Nonlinear Stochastic Operator Equations Aca-demic Press San Diego CA USA 1986

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Stochastic AnalysisInternational Journal of

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Journal of Applied Mathematics 5

Table 2 Comparison of absolute errors using 11991010 when m=16 v=4 and k=2

119905 1003816100381610038161003816119910119890119909119886119888119905 minus 1199101198871003816100381610038161003816 1003816100381610038161003816119910119890119909119886119888119905 minus 11991011989811988710038161003816100381610038160 0 001 2061126 E-6 3074560 E-702 1145239 E-5 1058636 E-503 1315732 E-5 5114716 E-504 2165603 E-4 1331415 E-405 8646500 E-4 2420463 E-406 2311555 E-3 3299021 E-407 4905204 E-3 3231831 E-408 8829753 E-3 1540876 E-409 1391114 E-2 1870564 E-41 1946226 E-2 6088701 E-4MSE 6804632 E-5 7217798344 E-8

polynomial in (b) at m=v=6 and k=2 The absolute errorsgenerated using the ADM with Bernstein polynomial areof 10minus4 while the errors yielded from ADM with modifiedBernstein polynomial are of 10minus5Example 2 Consider the ordinary equation

11988921199101198891199052 + 119910119889119910119889119905 = 119905 sin (21199052) minus 41199052 sin (1199052) + 2 cos (1199052)

0 le 119905 le 1119910 (0) = 0

119889119910119889119905 (0) = 0

(30)

with the exact solution (119905) = sin(1199052)Here 119871 = 11988921198891199052119873119910 = 119910(119889119910119889119905) and 119892(119905) = 119905 sin(21199052) minus41199052 sin(1199052) + 2 cos(1199052)The Adomian polynomials for represent the nonlinear

term Nu are

1198600 = 1199100 11988911988911990511991001198601 = 1199101 1198891198891199051199100 + 1199100 1198891198891199051199101

1198602 = 1199102 1198891198891199051199100 + 1199101 1198891198891199051199101 + 1199100 1198891198891199051199101

(31)

Then using (5) the classical Bernstein polynomials of 119892(119905)when v=4 and m=16 is

119892119887 (119905) = 2000659171119905 + 022331901199052 + 00980851199053minus 3395401199054 + sdot sdot sdot (32)

Andmodified Bernstein polynomials (21) of g(t) with k=2 are

119892119898119887 (119905) = 2 minus 0007366119905 + 021888551199052 + 13897511199053minus 4502131199054 + sdot sdot sdot (33)

By (22) we have

119910119898119887 (119905) = 4sum119894=0

119910119894= 1199052 minus 00012281199053 + 00182411199054 minus 00305131199055

+ sdot sdot sdot (34)

The absolute error of 119910119898119887(119905) and 119910119887(119905) is presented in Table 2and Figure 2

Figure 2 presents the absolute error of ADM with Bern-stein polynomial in (a) and ADM with modified Bernsteinpolynomial in (b) at m=v=10 and k=3 The absolute errorsgenerated using the ADM with Bernstein polynomial areof 10minus2 while the errors yielded from ADM with modifiedBernstein polynomial are of 10minus4Example 3 Consider the ordinary equation

119889119910119889119905 minus 119905119910 + 1199102 = 1198901199052 119910 (0) = 1

(35)

with the exact solution (119905) = 11989011990522Here 119871 = 119889119889119905 119877119906 = minus119905119910119873119910 = 1199102 and 119892(119905) = 1198901199052 Then using (5) the classical Bernstein polynomials of 119892(119905)

when v=8 and m=12 is

119892119887 (119905) = 1 + 0083623119905 + 09391771199052 + 01977291199053+ 03315721199054 + sdot sdot sdot (36)

6 Journal of Applied Mathematics

Table 3 Comparison of absolute errors using 11991010 when m=12 v=8 and k=3

119905 1003816100381610038161003816119910119890119909119886119888119905 minus 1199101198871003816100381610038161003816 1003816100381610038161003816119910119890119909119886119888119905 minus 11991011989811988710038161003816100381610038160 0 0001 4134000 E-6 1750000 E-7002 1635300 E-5 6400000 E-7003 3639700 E-5 1314000 E-6004 6402400 E-5 2123000 E-6005 9900900 E-5 2999000 E-6006 1411460 E-4 3883000 E-6007 1902410 E-4 4720000 E-6008 2461180 E-4 5463000 E-6009 3086100 E-4 6069000 E-601 3775690 E-4 6501000 E-6MSE 3699974901 E-8 1619641710 E-11

Ybern

10806040200

001

002

003

t(a)

1080604020t

00016

00014

00012

00010

00008

00006

00004

00002

0

Ymbern

(b)

Figure 2The absolute error between ADM with modified Bernstein polynomials and the exact solution when m=v=10 and k=3

andmodified Bernstein polynomials (21) of 119892(119905) with k=3 are119892119898119887 (119905) = 1 + 0003794119905 + 09570941199052 + 01035091199053

+ 04102891199054 + sdot sdot sdot (37)

By (22) we have

119906119898119887 (119905) = 8sum119894=0

119906119894= 1 + 05018971199052 minus 00155671199053 + 01591351199054

+ sdot sdot sdot (38)

The absolute error of 119906119898119887(119905) and 119906119887(119905) is presented in Table 3and Figure 3

Figure 3 presents the absolute error of ADM with Bern-stein polynomial in (a) and ADM with modified Bernstein

polynomial in (b) at m=12 v=8 and k=3 The absolute errorsgenerated using the ADM with Bernstein polynomial areof 10minus4 while the errors yielded from ADM with modifiedBernstein polynomial are of 10minus6Example 4 Consider the partial differential equation

120597119910120597119905 + 119910120597119910120597119909 minus V12059721199101205971199092 = 119909 (2119905 cos (1199052) + sin2 (1199052))

119910 (119909 0) = 0(39)

Using (12) we have

119871119910 + 119873119910 + 119877119910 = 119892 (119909 119905) (40)

where 119871 = 119889119889119905 119877119910 = minusV(12059721199101205971199092) 119873119910 = 119910(120597119910120597119909) and119892(119909 119905) = 119909(2119905 cos(1199052) + sin2(1199052))

Journal of Applied Mathematics 7

0

00001

00002

00003

Ybern

0100080060040020t

(a)

0

1 times 10minus6

2 times 10minus6

3 times 10minus6

4 times 10minus6

5 times 10minus6

6 times 10minus6

Ymbern

0100080060040020t

(b)

Figure 3 The absolute error between ADM with modified Bernstein polynomials and the exact solution when m=12 v=8 and k=3

The Adomian polynomials for Ny are

1198600 = 119910119900 120597119910119900120597119909 1198601 = 119910119900 1205971199101120597119909 + 1199101 120597119910119900120597119909

1198602 = 119910119900 1205971199102120597119909 + 1199102 120597119910119900120597119909 + 1199101 1205971199101120597119909

(41)

Then using (5) the classical Bernstein polynomials of 119892(119909 119905)when v=m=6 is119892119887 (119909 119905) = 2003859119909119905 + 01034751199091199052 + 01499541199091199053

minus 0279351199091199054 + sdot sdot sdot (42)

and modified Bernstein polynomials (21) of 119892(119909 119905) with k=2are

119892119898119887 (119909 119905) = 1986611119909119905 + 00457441199091199052+ 05042811199091199053 minus 0253751199091199054 + sdot sdot sdot (43)

By (22) with V = 1 we have1199100 = 119871minus1 (119892119898119887 (119909 119905)) + 119910 (119909 0) = 09933061199091199052

+ 00152481199091199053 + 0126071199091199054 minus 00507501199091199055 1199101 = minus119871minus1 (minusV120597211991001205971199092 ) minus 119871minus1 (1198600) = minus01973311199091199055

minus 00050491199091199056 minus 00358121199091199057 + 00121221199091199058+ sdot sdot sdot

1199102 = minus119871minus1(minusV120597211991011205971199092 ) minus 119871minus1 (1198601) = 00490031199091199058+ 00017831199091199059 + 001210511990911990510 minus 000379511990911990511+ sdot sdot sdot

(44)

And we obtain

119910119898119887 (119909 119905) = 6sum119894=0

119910119894= 09933061199091199052 + 00152481199091199053

+ 01260701199091199054 minus 02480811199091199055minus 00837491199091199056 + sdot sdot sdot

(45)

The absolute error of 119910119898119887(119909 119905) and 119910119887(119909 119905) is presented inTable 4 and Figure 4 with the exact solution 119910(119909 119905) =119909 sin(1199052)

Also Figure 4 presents the absolute error of ADM withBernstein polynomial in (a) and ADM with modified Bern-stein polynomial in (c) atm=v=6 and k=2The absolute errorsgenerated using the ADM with Bernstein polynomial areof 10minus3 while the errors yielded from ADM with modifiedBernstein polynomial are of 10minus45 Conclusions

In this paper we show that utilizing modified Bernsteinpolynomials is smartly thought to modify the performance

8 Journal of Applied Mathematics

Table 4 Comparison of absolute errors using 1199106 when m=v=6 k=2 and x=01

119905 1003816100381610038161003816119910119890119909119886119888119905 minus 1199101198871003816100381610038161003816 1003816100381610038161003816119910119890119909119886119888119905 minus 11991011989811988710038161003816100381610038160 0 001 5512102 E-6 4149093 E-602 3401160 E-5 1849504 E-603 8851342 E-5 2838128 E-504 1404426 E-5 9121566 E-505 1256008 E-4 1620426 E-406 4250973 E-5 1880417 E-407 4360788 E-4 1086168 E-408 1050197 E-3 1020235 E-409 1718854 E-3 3752218 E-41 2024768 E-3 4863297 E-4MSE 8393551358 E-7 4702782622 E-8

108

0604

020

t

010008

006004

0020

00005

00010

00015Error

x

(a) The absolute error for classical Bernstein polynomials

108

0604

020

t

010008

006004

002

008

006005

007

003004

002001

0

x

Ybern

(b) The numerical solution for classical Bernstein polynomials

108

0604

020

t

010008

006004

0020

00001

00002

00003

00004

Error

x

(c) The absolute error for modified Bernstein polynomials

1

08

06

04

02

0

t

010008

006

004

002

008

006005

007

003004

002001

0

x

Ymbern

(d) The numerical solution for modified Bernstein polynomials

Figure 4The absolute error between ADMwith classical andmodified Bernstein polynomials and the exact solution using 1199106 whenm=v=6k=2 and x=01

Journal of Applied Mathematics 9

of the Adomian decomposition technique The fundamentalpreferred standpoint of this strategy is that it can be usedspecifically for all sort of differential and integral equations

We utilize modified Bernstein extensions of the nonlinearterm to get more exact outcomes Figures empower usto consider the difference between utilizing two strategiesgraphically Tables are additionally given to demonstrate thevariety of the outright mistakes for bigger estimation tobe specific for bigger m We observed from the numericaloutcomes in Tables 1ndash4 and Figures 1ndash4 that the ADMwith modification Bernstein polynomials gives more exactand robust numerical solution than the classical Bernsteinpolynomials Every one of the calculations was done with theguide of Maple 13 programming

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The research is supported by College of Computer Sciencesand Mathematics University of Mosul Republic of Iraqunder Project no 11251357

References

[1] S Muhammad and I Muhammad ldquoSome multi-step iterativemethods for solving nonlinear equationsrdquo Open Journal ofMathematical Sciences vol 1 no 1 pp 25ndash33 2017

[2] Y Yang and L Hua ldquoVariational iteration transformmethod forfractional differential equationswith local fractional derivativerdquoAbstract and Applied Analysis vol 2014 Article ID 760957 9pages 2014

[3] U R Hamood S S Muhammad and A Ayesha ldquoCombinationof homotopy perturbationmethod (HPM) and double sumudutransform to solve fractional KDV equationsrdquo Open Journal ofMathematical Sciences vol 2 no 1 pp 29ndash38 2018

[4] A M Wazwaz ldquoConstruction of solitary wave solutions andrational solutions for the KdV equation by Adomian decom-position methodrdquo Chaos Solitons amp Fractals vol 12 no 12 pp2283ndash2293 2001

[5] N Bildik and A Konuralp ldquoThe use of variational itera-tion method differential transform method and Adomiandecomposition method for solving different types of nonlinearpartial differential equationsrdquo International Journal ofNonlinearSciences andNumerical Simulation vol 7 no 1 pp 65ndash70 2006

[6] T Madeeha N N Muhammad S Rabia V Dumitru IMuhammad and S Naeem ldquoOn unsteady flow of a viscoelasticfluid through rotating cylindersrdquo Open Physics vol 1 no 1 pp1ndash15 2017

[7] A Sannia Q Haitao A Muhammad J Maria and I Muham-mad ldquoExact solutions of fractional Maxwell fluid between two

cylindersrdquo Open Journal of Mathematical Sciences vol 1 no 1pp 52ndash61 2017

[8] QHaitao FNidaWHassan and S Junaid ldquoAnalytical solutionfor the flow of a generalized Oldroyd-B fluid in a circularcylinderrdquo Open Journal of Mathematical Sciences vol 1 no 1pp 85ndash96 2017

[9] MMHosseini ldquoAdomian decompositionmethodwith Cheby-shev polynomialsrdquo Applied Mathematics and Computation vol175 no 2 pp 1685ndash1693 2006

[10] N Bildik and Deniz ldquoModified Adomian decompositionmethod for solving Riccati differential equationsrdquo Review of theAir Force Academy vol 3 no 30 pp 21ndash26 2015

[11] Y Liu ldquoAdomian decompositionmethod with orthogonal poly-nomials Legendre polynomialsrdquo Mathematical and ComputerModelling vol 49 no 5-6 pp 1268ndash1273 2009

[12] Y Mahmoudi et al ldquoAdomian decomposition method withLaguerre polynomials for solving ordinary differential equa-tionrdquo Journal of Basic and Applied Scientific Research vol 2 no12 pp 12236ndash12241 2012

[13] D Rani and V Mishra ldquoApproximate solution of boundaryvalue problem with bernstein polynomial laplace decompo-sition methodrdquo International Journal of Pure and AppliedMathematics vol 114 no 4 pp 823ndash833 2017

[14] GG LorentzBernstein Polynomials Chelsea Publishing Series1986

[15] J Cicho and Z Goi ldquoOn Bernoulli sums and Bernsteinpolynomialsrdquo Discrete Mathematics and eoretical ComputerScience pp 1ndash12 2009

[16] G Adomian Nonlinear Stochastic Operator Equations Aca-demic Press San Diego CA USA 1986

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 Journal of Applied Mathematics

Table 3 Comparison of absolute errors using 11991010 when m=12 v=8 and k=3

119905 1003816100381610038161003816119910119890119909119886119888119905 minus 1199101198871003816100381610038161003816 1003816100381610038161003816119910119890119909119886119888119905 minus 11991011989811988710038161003816100381610038160 0 0001 4134000 E-6 1750000 E-7002 1635300 E-5 6400000 E-7003 3639700 E-5 1314000 E-6004 6402400 E-5 2123000 E-6005 9900900 E-5 2999000 E-6006 1411460 E-4 3883000 E-6007 1902410 E-4 4720000 E-6008 2461180 E-4 5463000 E-6009 3086100 E-4 6069000 E-601 3775690 E-4 6501000 E-6MSE 3699974901 E-8 1619641710 E-11

Ybern

10806040200

001

002

003

t(a)

1080604020t

00016

00014

00012

00010

00008

00006

00004

00002

0

Ymbern

(b)

Figure 2The absolute error between ADM with modified Bernstein polynomials and the exact solution when m=v=10 and k=3

andmodified Bernstein polynomials (21) of 119892(119905) with k=3 are119892119898119887 (119905) = 1 + 0003794119905 + 09570941199052 + 01035091199053

+ 04102891199054 + sdot sdot sdot (37)

By (22) we have

119906119898119887 (119905) = 8sum119894=0

119906119894= 1 + 05018971199052 minus 00155671199053 + 01591351199054

+ sdot sdot sdot (38)

The absolute error of 119906119898119887(119905) and 119906119887(119905) is presented in Table 3and Figure 3

Figure 3 presents the absolute error of ADM with Bern-stein polynomial in (a) and ADM with modified Bernstein

polynomial in (b) at m=12 v=8 and k=3 The absolute errorsgenerated using the ADM with Bernstein polynomial areof 10minus4 while the errors yielded from ADM with modifiedBernstein polynomial are of 10minus6Example 4 Consider the partial differential equation

120597119910120597119905 + 119910120597119910120597119909 minus V12059721199101205971199092 = 119909 (2119905 cos (1199052) + sin2 (1199052))

119910 (119909 0) = 0(39)

Using (12) we have

119871119910 + 119873119910 + 119877119910 = 119892 (119909 119905) (40)

where 119871 = 119889119889119905 119877119910 = minusV(12059721199101205971199092) 119873119910 = 119910(120597119910120597119909) and119892(119909 119905) = 119909(2119905 cos(1199052) + sin2(1199052))

Journal of Applied Mathematics 7

0

00001

00002

00003

Ybern

0100080060040020t

(a)

0

1 times 10minus6

2 times 10minus6

3 times 10minus6

4 times 10minus6

5 times 10minus6

6 times 10minus6

Ymbern

0100080060040020t

(b)

Figure 3 The absolute error between ADM with modified Bernstein polynomials and the exact solution when m=12 v=8 and k=3

The Adomian polynomials for Ny are

1198600 = 119910119900 120597119910119900120597119909 1198601 = 119910119900 1205971199101120597119909 + 1199101 120597119910119900120597119909

1198602 = 119910119900 1205971199102120597119909 + 1199102 120597119910119900120597119909 + 1199101 1205971199101120597119909

(41)

Then using (5) the classical Bernstein polynomials of 119892(119909 119905)when v=m=6 is119892119887 (119909 119905) = 2003859119909119905 + 01034751199091199052 + 01499541199091199053

minus 0279351199091199054 + sdot sdot sdot (42)

and modified Bernstein polynomials (21) of 119892(119909 119905) with k=2are

119892119898119887 (119909 119905) = 1986611119909119905 + 00457441199091199052+ 05042811199091199053 minus 0253751199091199054 + sdot sdot sdot (43)

By (22) with V = 1 we have1199100 = 119871minus1 (119892119898119887 (119909 119905)) + 119910 (119909 0) = 09933061199091199052

+ 00152481199091199053 + 0126071199091199054 minus 00507501199091199055 1199101 = minus119871minus1 (minusV120597211991001205971199092 ) minus 119871minus1 (1198600) = minus01973311199091199055

minus 00050491199091199056 minus 00358121199091199057 + 00121221199091199058+ sdot sdot sdot

1199102 = minus119871minus1(minusV120597211991011205971199092 ) minus 119871minus1 (1198601) = 00490031199091199058+ 00017831199091199059 + 001210511990911990510 minus 000379511990911990511+ sdot sdot sdot

(44)

And we obtain

119910119898119887 (119909 119905) = 6sum119894=0

119910119894= 09933061199091199052 + 00152481199091199053

+ 01260701199091199054 minus 02480811199091199055minus 00837491199091199056 + sdot sdot sdot

(45)

The absolute error of 119910119898119887(119909 119905) and 119910119887(119909 119905) is presented inTable 4 and Figure 4 with the exact solution 119910(119909 119905) =119909 sin(1199052)

Also Figure 4 presents the absolute error of ADM withBernstein polynomial in (a) and ADM with modified Bern-stein polynomial in (c) atm=v=6 and k=2The absolute errorsgenerated using the ADM with Bernstein polynomial areof 10minus3 while the errors yielded from ADM with modifiedBernstein polynomial are of 10minus45 Conclusions

In this paper we show that utilizing modified Bernsteinpolynomials is smartly thought to modify the performance

8 Journal of Applied Mathematics

Table 4 Comparison of absolute errors using 1199106 when m=v=6 k=2 and x=01

119905 1003816100381610038161003816119910119890119909119886119888119905 minus 1199101198871003816100381610038161003816 1003816100381610038161003816119910119890119909119886119888119905 minus 11991011989811988710038161003816100381610038160 0 001 5512102 E-6 4149093 E-602 3401160 E-5 1849504 E-603 8851342 E-5 2838128 E-504 1404426 E-5 9121566 E-505 1256008 E-4 1620426 E-406 4250973 E-5 1880417 E-407 4360788 E-4 1086168 E-408 1050197 E-3 1020235 E-409 1718854 E-3 3752218 E-41 2024768 E-3 4863297 E-4MSE 8393551358 E-7 4702782622 E-8

108

0604

020

t

010008

006004

0020

00005

00010

00015Error

x

(a) The absolute error for classical Bernstein polynomials

108

0604

020

t

010008

006004

002

008

006005

007

003004

002001

0

x

Ybern

(b) The numerical solution for classical Bernstein polynomials

108

0604

020

t

010008

006004

0020

00001

00002

00003

00004

Error

x

(c) The absolute error for modified Bernstein polynomials

1

08

06

04

02

0

t

010008

006

004

002

008

006005

007

003004

002001

0

x

Ymbern

(d) The numerical solution for modified Bernstein polynomials

Figure 4The absolute error between ADMwith classical andmodified Bernstein polynomials and the exact solution using 1199106 whenm=v=6k=2 and x=01

Journal of Applied Mathematics 9

of the Adomian decomposition technique The fundamentalpreferred standpoint of this strategy is that it can be usedspecifically for all sort of differential and integral equations

We utilize modified Bernstein extensions of the nonlinearterm to get more exact outcomes Figures empower usto consider the difference between utilizing two strategiesgraphically Tables are additionally given to demonstrate thevariety of the outright mistakes for bigger estimation tobe specific for bigger m We observed from the numericaloutcomes in Tables 1ndash4 and Figures 1ndash4 that the ADMwith modification Bernstein polynomials gives more exactand robust numerical solution than the classical Bernsteinpolynomials Every one of the calculations was done with theguide of Maple 13 programming

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The research is supported by College of Computer Sciencesand Mathematics University of Mosul Republic of Iraqunder Project no 11251357

References

[1] S Muhammad and I Muhammad ldquoSome multi-step iterativemethods for solving nonlinear equationsrdquo Open Journal ofMathematical Sciences vol 1 no 1 pp 25ndash33 2017

[2] Y Yang and L Hua ldquoVariational iteration transformmethod forfractional differential equationswith local fractional derivativerdquoAbstract and Applied Analysis vol 2014 Article ID 760957 9pages 2014

[3] U R Hamood S S Muhammad and A Ayesha ldquoCombinationof homotopy perturbationmethod (HPM) and double sumudutransform to solve fractional KDV equationsrdquo Open Journal ofMathematical Sciences vol 2 no 1 pp 29ndash38 2018

[4] A M Wazwaz ldquoConstruction of solitary wave solutions andrational solutions for the KdV equation by Adomian decom-position methodrdquo Chaos Solitons amp Fractals vol 12 no 12 pp2283ndash2293 2001

[5] N Bildik and A Konuralp ldquoThe use of variational itera-tion method differential transform method and Adomiandecomposition method for solving different types of nonlinearpartial differential equationsrdquo International Journal ofNonlinearSciences andNumerical Simulation vol 7 no 1 pp 65ndash70 2006

[6] T Madeeha N N Muhammad S Rabia V Dumitru IMuhammad and S Naeem ldquoOn unsteady flow of a viscoelasticfluid through rotating cylindersrdquo Open Physics vol 1 no 1 pp1ndash15 2017

[7] A Sannia Q Haitao A Muhammad J Maria and I Muham-mad ldquoExact solutions of fractional Maxwell fluid between two

cylindersrdquo Open Journal of Mathematical Sciences vol 1 no 1pp 52ndash61 2017

[8] QHaitao FNidaWHassan and S Junaid ldquoAnalytical solutionfor the flow of a generalized Oldroyd-B fluid in a circularcylinderrdquo Open Journal of Mathematical Sciences vol 1 no 1pp 85ndash96 2017

[9] MMHosseini ldquoAdomian decompositionmethodwith Cheby-shev polynomialsrdquo Applied Mathematics and Computation vol175 no 2 pp 1685ndash1693 2006

[10] N Bildik and Deniz ldquoModified Adomian decompositionmethod for solving Riccati differential equationsrdquo Review of theAir Force Academy vol 3 no 30 pp 21ndash26 2015

[11] Y Liu ldquoAdomian decompositionmethod with orthogonal poly-nomials Legendre polynomialsrdquo Mathematical and ComputerModelling vol 49 no 5-6 pp 1268ndash1273 2009

[12] Y Mahmoudi et al ldquoAdomian decomposition method withLaguerre polynomials for solving ordinary differential equa-tionrdquo Journal of Basic and Applied Scientific Research vol 2 no12 pp 12236ndash12241 2012

[13] D Rani and V Mishra ldquoApproximate solution of boundaryvalue problem with bernstein polynomial laplace decompo-sition methodrdquo International Journal of Pure and AppliedMathematics vol 114 no 4 pp 823ndash833 2017

[14] GG LorentzBernstein Polynomials Chelsea Publishing Series1986

[15] J Cicho and Z Goi ldquoOn Bernoulli sums and Bernsteinpolynomialsrdquo Discrete Mathematics and eoretical ComputerScience pp 1ndash12 2009

[16] G Adomian Nonlinear Stochastic Operator Equations Aca-demic Press San Diego CA USA 1986

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Journal of Applied Mathematics 7

0

00001

00002

00003

Ybern

0100080060040020t

(a)

0

1 times 10minus6

2 times 10minus6

3 times 10minus6

4 times 10minus6

5 times 10minus6

6 times 10minus6

Ymbern

0100080060040020t

(b)

Figure 3 The absolute error between ADM with modified Bernstein polynomials and the exact solution when m=12 v=8 and k=3

The Adomian polynomials for Ny are

1198600 = 119910119900 120597119910119900120597119909 1198601 = 119910119900 1205971199101120597119909 + 1199101 120597119910119900120597119909

1198602 = 119910119900 1205971199102120597119909 + 1199102 120597119910119900120597119909 + 1199101 1205971199101120597119909

(41)

Then using (5) the classical Bernstein polynomials of 119892(119909 119905)when v=m=6 is119892119887 (119909 119905) = 2003859119909119905 + 01034751199091199052 + 01499541199091199053

minus 0279351199091199054 + sdot sdot sdot (42)

and modified Bernstein polynomials (21) of 119892(119909 119905) with k=2are

119892119898119887 (119909 119905) = 1986611119909119905 + 00457441199091199052+ 05042811199091199053 minus 0253751199091199054 + sdot sdot sdot (43)

By (22) with V = 1 we have1199100 = 119871minus1 (119892119898119887 (119909 119905)) + 119910 (119909 0) = 09933061199091199052

+ 00152481199091199053 + 0126071199091199054 minus 00507501199091199055 1199101 = minus119871minus1 (minusV120597211991001205971199092 ) minus 119871minus1 (1198600) = minus01973311199091199055

minus 00050491199091199056 minus 00358121199091199057 + 00121221199091199058+ sdot sdot sdot

1199102 = minus119871minus1(minusV120597211991011205971199092 ) minus 119871minus1 (1198601) = 00490031199091199058+ 00017831199091199059 + 001210511990911990510 minus 000379511990911990511+ sdot sdot sdot

(44)

And we obtain

119910119898119887 (119909 119905) = 6sum119894=0

119910119894= 09933061199091199052 + 00152481199091199053

+ 01260701199091199054 minus 02480811199091199055minus 00837491199091199056 + sdot sdot sdot

(45)

The absolute error of 119910119898119887(119909 119905) and 119910119887(119909 119905) is presented inTable 4 and Figure 4 with the exact solution 119910(119909 119905) =119909 sin(1199052)

Also Figure 4 presents the absolute error of ADM withBernstein polynomial in (a) and ADM with modified Bern-stein polynomial in (c) atm=v=6 and k=2The absolute errorsgenerated using the ADM with Bernstein polynomial areof 10minus3 while the errors yielded from ADM with modifiedBernstein polynomial are of 10minus45 Conclusions

In this paper we show that utilizing modified Bernsteinpolynomials is smartly thought to modify the performance

8 Journal of Applied Mathematics

Table 4 Comparison of absolute errors using 1199106 when m=v=6 k=2 and x=01

119905 1003816100381610038161003816119910119890119909119886119888119905 minus 1199101198871003816100381610038161003816 1003816100381610038161003816119910119890119909119886119888119905 minus 11991011989811988710038161003816100381610038160 0 001 5512102 E-6 4149093 E-602 3401160 E-5 1849504 E-603 8851342 E-5 2838128 E-504 1404426 E-5 9121566 E-505 1256008 E-4 1620426 E-406 4250973 E-5 1880417 E-407 4360788 E-4 1086168 E-408 1050197 E-3 1020235 E-409 1718854 E-3 3752218 E-41 2024768 E-3 4863297 E-4MSE 8393551358 E-7 4702782622 E-8

108

0604

020

t

010008

006004

0020

00005

00010

00015Error

x

(a) The absolute error for classical Bernstein polynomials

108

0604

020

t

010008

006004

002

008

006005

007

003004

002001

0

x

Ybern

(b) The numerical solution for classical Bernstein polynomials

108

0604

020

t

010008

006004

0020

00001

00002

00003

00004

Error

x

(c) The absolute error for modified Bernstein polynomials

1

08

06

04

02

0

t

010008

006

004

002

008

006005

007

003004

002001

0

x

Ymbern

(d) The numerical solution for modified Bernstein polynomials

Figure 4The absolute error between ADMwith classical andmodified Bernstein polynomials and the exact solution using 1199106 whenm=v=6k=2 and x=01

Journal of Applied Mathematics 9

of the Adomian decomposition technique The fundamentalpreferred standpoint of this strategy is that it can be usedspecifically for all sort of differential and integral equations

We utilize modified Bernstein extensions of the nonlinearterm to get more exact outcomes Figures empower usto consider the difference between utilizing two strategiesgraphically Tables are additionally given to demonstrate thevariety of the outright mistakes for bigger estimation tobe specific for bigger m We observed from the numericaloutcomes in Tables 1ndash4 and Figures 1ndash4 that the ADMwith modification Bernstein polynomials gives more exactand robust numerical solution than the classical Bernsteinpolynomials Every one of the calculations was done with theguide of Maple 13 programming

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The research is supported by College of Computer Sciencesand Mathematics University of Mosul Republic of Iraqunder Project no 11251357

References

[1] S Muhammad and I Muhammad ldquoSome multi-step iterativemethods for solving nonlinear equationsrdquo Open Journal ofMathematical Sciences vol 1 no 1 pp 25ndash33 2017

[2] Y Yang and L Hua ldquoVariational iteration transformmethod forfractional differential equationswith local fractional derivativerdquoAbstract and Applied Analysis vol 2014 Article ID 760957 9pages 2014

[3] U R Hamood S S Muhammad and A Ayesha ldquoCombinationof homotopy perturbationmethod (HPM) and double sumudutransform to solve fractional KDV equationsrdquo Open Journal ofMathematical Sciences vol 2 no 1 pp 29ndash38 2018

[4] A M Wazwaz ldquoConstruction of solitary wave solutions andrational solutions for the KdV equation by Adomian decom-position methodrdquo Chaos Solitons amp Fractals vol 12 no 12 pp2283ndash2293 2001

[5] N Bildik and A Konuralp ldquoThe use of variational itera-tion method differential transform method and Adomiandecomposition method for solving different types of nonlinearpartial differential equationsrdquo International Journal ofNonlinearSciences andNumerical Simulation vol 7 no 1 pp 65ndash70 2006

[6] T Madeeha N N Muhammad S Rabia V Dumitru IMuhammad and S Naeem ldquoOn unsteady flow of a viscoelasticfluid through rotating cylindersrdquo Open Physics vol 1 no 1 pp1ndash15 2017

[7] A Sannia Q Haitao A Muhammad J Maria and I Muham-mad ldquoExact solutions of fractional Maxwell fluid between two

cylindersrdquo Open Journal of Mathematical Sciences vol 1 no 1pp 52ndash61 2017

[8] QHaitao FNidaWHassan and S Junaid ldquoAnalytical solutionfor the flow of a generalized Oldroyd-B fluid in a circularcylinderrdquo Open Journal of Mathematical Sciences vol 1 no 1pp 85ndash96 2017

[9] MMHosseini ldquoAdomian decompositionmethodwith Cheby-shev polynomialsrdquo Applied Mathematics and Computation vol175 no 2 pp 1685ndash1693 2006

[10] N Bildik and Deniz ldquoModified Adomian decompositionmethod for solving Riccati differential equationsrdquo Review of theAir Force Academy vol 3 no 30 pp 21ndash26 2015

[11] Y Liu ldquoAdomian decompositionmethod with orthogonal poly-nomials Legendre polynomialsrdquo Mathematical and ComputerModelling vol 49 no 5-6 pp 1268ndash1273 2009

[12] Y Mahmoudi et al ldquoAdomian decomposition method withLaguerre polynomials for solving ordinary differential equa-tionrdquo Journal of Basic and Applied Scientific Research vol 2 no12 pp 12236ndash12241 2012

[13] D Rani and V Mishra ldquoApproximate solution of boundaryvalue problem with bernstein polynomial laplace decompo-sition methodrdquo International Journal of Pure and AppliedMathematics vol 114 no 4 pp 823ndash833 2017

[14] GG LorentzBernstein Polynomials Chelsea Publishing Series1986

[15] J Cicho and Z Goi ldquoOn Bernoulli sums and Bernsteinpolynomialsrdquo Discrete Mathematics and eoretical ComputerScience pp 1ndash12 2009

[16] G Adomian Nonlinear Stochastic Operator Equations Aca-demic Press San Diego CA USA 1986

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Journal of Applied Mathematics

Table 4 Comparison of absolute errors using 1199106 when m=v=6 k=2 and x=01

119905 1003816100381610038161003816119910119890119909119886119888119905 minus 1199101198871003816100381610038161003816 1003816100381610038161003816119910119890119909119886119888119905 minus 11991011989811988710038161003816100381610038160 0 001 5512102 E-6 4149093 E-602 3401160 E-5 1849504 E-603 8851342 E-5 2838128 E-504 1404426 E-5 9121566 E-505 1256008 E-4 1620426 E-406 4250973 E-5 1880417 E-407 4360788 E-4 1086168 E-408 1050197 E-3 1020235 E-409 1718854 E-3 3752218 E-41 2024768 E-3 4863297 E-4MSE 8393551358 E-7 4702782622 E-8

108

0604

020

t

010008

006004

0020

00005

00010

00015Error

x

(a) The absolute error for classical Bernstein polynomials

108

0604

020

t

010008

006004

002

008

006005

007

003004

002001

0

x

Ybern

(b) The numerical solution for classical Bernstein polynomials

108

0604

020

t

010008

006004

0020

00001

00002

00003

00004

Error

x

(c) The absolute error for modified Bernstein polynomials

1

08

06

04

02

0

t

010008

006

004

002

008

006005

007

003004

002001

0

x

Ymbern

(d) The numerical solution for modified Bernstein polynomials

Figure 4The absolute error between ADMwith classical andmodified Bernstein polynomials and the exact solution using 1199106 whenm=v=6k=2 and x=01

Journal of Applied Mathematics 9

of the Adomian decomposition technique The fundamentalpreferred standpoint of this strategy is that it can be usedspecifically for all sort of differential and integral equations

We utilize modified Bernstein extensions of the nonlinearterm to get more exact outcomes Figures empower usto consider the difference between utilizing two strategiesgraphically Tables are additionally given to demonstrate thevariety of the outright mistakes for bigger estimation tobe specific for bigger m We observed from the numericaloutcomes in Tables 1ndash4 and Figures 1ndash4 that the ADMwith modification Bernstein polynomials gives more exactand robust numerical solution than the classical Bernsteinpolynomials Every one of the calculations was done with theguide of Maple 13 programming

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The research is supported by College of Computer Sciencesand Mathematics University of Mosul Republic of Iraqunder Project no 11251357

References

[1] S Muhammad and I Muhammad ldquoSome multi-step iterativemethods for solving nonlinear equationsrdquo Open Journal ofMathematical Sciences vol 1 no 1 pp 25ndash33 2017

[2] Y Yang and L Hua ldquoVariational iteration transformmethod forfractional differential equationswith local fractional derivativerdquoAbstract and Applied Analysis vol 2014 Article ID 760957 9pages 2014

[3] U R Hamood S S Muhammad and A Ayesha ldquoCombinationof homotopy perturbationmethod (HPM) and double sumudutransform to solve fractional KDV equationsrdquo Open Journal ofMathematical Sciences vol 2 no 1 pp 29ndash38 2018

[4] A M Wazwaz ldquoConstruction of solitary wave solutions andrational solutions for the KdV equation by Adomian decom-position methodrdquo Chaos Solitons amp Fractals vol 12 no 12 pp2283ndash2293 2001

[5] N Bildik and A Konuralp ldquoThe use of variational itera-tion method differential transform method and Adomiandecomposition method for solving different types of nonlinearpartial differential equationsrdquo International Journal ofNonlinearSciences andNumerical Simulation vol 7 no 1 pp 65ndash70 2006

[6] T Madeeha N N Muhammad S Rabia V Dumitru IMuhammad and S Naeem ldquoOn unsteady flow of a viscoelasticfluid through rotating cylindersrdquo Open Physics vol 1 no 1 pp1ndash15 2017

[7] A Sannia Q Haitao A Muhammad J Maria and I Muham-mad ldquoExact solutions of fractional Maxwell fluid between two

cylindersrdquo Open Journal of Mathematical Sciences vol 1 no 1pp 52ndash61 2017

[8] QHaitao FNidaWHassan and S Junaid ldquoAnalytical solutionfor the flow of a generalized Oldroyd-B fluid in a circularcylinderrdquo Open Journal of Mathematical Sciences vol 1 no 1pp 85ndash96 2017

[9] MMHosseini ldquoAdomian decompositionmethodwith Cheby-shev polynomialsrdquo Applied Mathematics and Computation vol175 no 2 pp 1685ndash1693 2006

[10] N Bildik and Deniz ldquoModified Adomian decompositionmethod for solving Riccati differential equationsrdquo Review of theAir Force Academy vol 3 no 30 pp 21ndash26 2015

[11] Y Liu ldquoAdomian decompositionmethod with orthogonal poly-nomials Legendre polynomialsrdquo Mathematical and ComputerModelling vol 49 no 5-6 pp 1268ndash1273 2009

[12] Y Mahmoudi et al ldquoAdomian decomposition method withLaguerre polynomials for solving ordinary differential equa-tionrdquo Journal of Basic and Applied Scientific Research vol 2 no12 pp 12236ndash12241 2012

[13] D Rani and V Mishra ldquoApproximate solution of boundaryvalue problem with bernstein polynomial laplace decompo-sition methodrdquo International Journal of Pure and AppliedMathematics vol 114 no 4 pp 823ndash833 2017

[14] GG LorentzBernstein Polynomials Chelsea Publishing Series1986

[15] J Cicho and Z Goi ldquoOn Bernoulli sums and Bernsteinpolynomialsrdquo Discrete Mathematics and eoretical ComputerScience pp 1ndash12 2009

[16] G Adomian Nonlinear Stochastic Operator Equations Aca-demic Press San Diego CA USA 1986

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Journal of Applied Mathematics 9

of the Adomian decomposition technique The fundamentalpreferred standpoint of this strategy is that it can be usedspecifically for all sort of differential and integral equations

We utilize modified Bernstein extensions of the nonlinearterm to get more exact outcomes Figures empower usto consider the difference between utilizing two strategiesgraphically Tables are additionally given to demonstrate thevariety of the outright mistakes for bigger estimation tobe specific for bigger m We observed from the numericaloutcomes in Tables 1ndash4 and Figures 1ndash4 that the ADMwith modification Bernstein polynomials gives more exactand robust numerical solution than the classical Bernsteinpolynomials Every one of the calculations was done with theguide of Maple 13 programming

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The research is supported by College of Computer Sciencesand Mathematics University of Mosul Republic of Iraqunder Project no 11251357

References

[1] S Muhammad and I Muhammad ldquoSome multi-step iterativemethods for solving nonlinear equationsrdquo Open Journal ofMathematical Sciences vol 1 no 1 pp 25ndash33 2017

[2] Y Yang and L Hua ldquoVariational iteration transformmethod forfractional differential equationswith local fractional derivativerdquoAbstract and Applied Analysis vol 2014 Article ID 760957 9pages 2014

[3] U R Hamood S S Muhammad and A Ayesha ldquoCombinationof homotopy perturbationmethod (HPM) and double sumudutransform to solve fractional KDV equationsrdquo Open Journal ofMathematical Sciences vol 2 no 1 pp 29ndash38 2018

[4] A M Wazwaz ldquoConstruction of solitary wave solutions andrational solutions for the KdV equation by Adomian decom-position methodrdquo Chaos Solitons amp Fractals vol 12 no 12 pp2283ndash2293 2001

[5] N Bildik and A Konuralp ldquoThe use of variational itera-tion method differential transform method and Adomiandecomposition method for solving different types of nonlinearpartial differential equationsrdquo International Journal ofNonlinearSciences andNumerical Simulation vol 7 no 1 pp 65ndash70 2006

[6] T Madeeha N N Muhammad S Rabia V Dumitru IMuhammad and S Naeem ldquoOn unsteady flow of a viscoelasticfluid through rotating cylindersrdquo Open Physics vol 1 no 1 pp1ndash15 2017

[7] A Sannia Q Haitao A Muhammad J Maria and I Muham-mad ldquoExact solutions of fractional Maxwell fluid between two

cylindersrdquo Open Journal of Mathematical Sciences vol 1 no 1pp 52ndash61 2017

[8] QHaitao FNidaWHassan and S Junaid ldquoAnalytical solutionfor the flow of a generalized Oldroyd-B fluid in a circularcylinderrdquo Open Journal of Mathematical Sciences vol 1 no 1pp 85ndash96 2017

[9] MMHosseini ldquoAdomian decompositionmethodwith Cheby-shev polynomialsrdquo Applied Mathematics and Computation vol175 no 2 pp 1685ndash1693 2006

[10] N Bildik and Deniz ldquoModified Adomian decompositionmethod for solving Riccati differential equationsrdquo Review of theAir Force Academy vol 3 no 30 pp 21ndash26 2015

[11] Y Liu ldquoAdomian decompositionmethod with orthogonal poly-nomials Legendre polynomialsrdquo Mathematical and ComputerModelling vol 49 no 5-6 pp 1268ndash1273 2009

[12] Y Mahmoudi et al ldquoAdomian decomposition method withLaguerre polynomials for solving ordinary differential equa-tionrdquo Journal of Basic and Applied Scientific Research vol 2 no12 pp 12236ndash12241 2012

[13] D Rani and V Mishra ldquoApproximate solution of boundaryvalue problem with bernstein polynomial laplace decompo-sition methodrdquo International Journal of Pure and AppliedMathematics vol 114 no 4 pp 823ndash833 2017

[14] GG LorentzBernstein Polynomials Chelsea Publishing Series1986

[15] J Cicho and Z Goi ldquoOn Bernoulli sums and Bernsteinpolynomialsrdquo Discrete Mathematics and eoretical ComputerScience pp 1ndash12 2009

[16] G Adomian Nonlinear Stochastic Operator Equations Aca-demic Press San Diego CA USA 1986

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom