Research ArticleThe Variational Homotopy Perturbation Method for Solving((119899 times 119899) + 1) Dimensional Burgersrsquo Equations

F A Hendi1 B S Kashkari1 and A A Alderremy2

1Department of Mathematics King Abdulaziz University Jeddah 21442 Saudi Arabia2Department of Mathematics King Khalid University Abha 61321 Saudi Arabia

Correspondence should be addressed to F A Hendi falhendikauedusa

Received 2 February 2016 Accepted 30 March 2016

Academic Editor Jisheng Kou

Copyright copy 2016 F A Hendi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The variational homotopy perturbation method VHPM is used for solving 119899-dimensional Burgersrsquo system Some examples areexamined to validate that the method reduced the calculation size treating the difficulty of nonlinear term and the accuracy

1 Introduction

The variational iteration method VIM and the homotopyperturbation method HPM were proposed by He in [1ndash6] Many researchers used these methods in a variety ofscientific fields of partial differential equations PDEs includ-ing Burgersrsquo equation which arises in many of physicallyimportant phenomena [7ndash9] It was shown that the methodsare stronger than other techniques such as the Adomiandecomposition method [10ndash18] In our work 119899-dimensionalBurgersrsquo equation is solved by the variational homotopyperturbation method VHPM which is combination of VIMand HPM The VHPM was proposed in [19ndash21] VectorBurgersrsquo system is given by [22]

119880119905+ (119880 sdot nabla)119880 = 120583Δ119880 (1)

where 1199061 1199062 119906

119899are the velocity components and 120583 is the

kinematic viscosity 119905 is time and Δ and nabla are

Δ =

1205972

1199092

1

+

1205972

1199092

2

+ sdot sdot sdot +

1205972

1199092

119899

nabla =

120597

1199091

+

120597

1199092

+ sdot sdot sdot +

120597

119909119899

(2)

Equation (1) can be written as

120597119906119894

120597119905

+

119899

sum

119895=1

119906119895

120597119906119894

120597119909119895

= 120583Δ119906119894 119894 = 1 2 119899 (3)

2 Variational Iteration Method

According to the variational iterationmethod [2 3 10ndash14] wecan write the correction functional for (3) as

119906119899+1

= 119906119899+ int

119905

0

120582119894(120585)

[

[

120597119906119894

120597120585

+

119899

sum

119895=1

119906119895

120597119906119894

120597119909119895

minus 120583Δ119906119894]

]

119889120585 (4)

where 119894 = 1 2 119899 119906 = 119906(119909119894 120585) 120582 is a general Lagrangian

multiplier which can be found via variational theory and 119906119894

are restricted variation which means 120575119906119894= 0 The solution is

given by

119906119894(119909119895 119905) = lim

119899rarrinfin119906(119894119899)

(119909119895 119905) 119895 = 1 2 119899 (5)

3 Homotopy Perturbation Method

ApplyingHPMaccording to [4ndash6 15ndash17] for (3) we constructthe following homotopy

(1 minus 119901) [

120597119906(119894119896)

120597119905

minus

120597119906(1198940)

120597119905

]

+ 119901[

[

119899

sum

119895=1

119906(119895119896)

120597119906(119894119896)

120597119909119895

minus 120583Δ119906(119894119896)

]

]

= 0

(6)

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2016 Article ID 4146323 6 pageshttpdxdoiorg10115520164146323

2 Journal of Applied Mathematics

or

120597119906(119894119896)

120597119905

minus

120597119906(1198940)

120597119905

+ 119901[

[

120597119906(1198940)

120597119905

+

119899

sum

119895=1

119906(119895119896)

120597119906(119894119896)

120597119909119895

minus 120583Δ119906(119894119896)

]

]

= 0

(7)

where 119894 = 1 2 119899 119896 = 1 2 119901 isin [0 1] is an embed-ding parameter while 119906

(10)= 1198911(119909119895 0) 119906

(20)= 1198912(119909119895 0)

119906(1198990)

= 119891119899(119909119895 0) are initial approximations of (3)

Assume the solution of (3) has the form

119906119894=

infin

sum

ℓ=0

119901ℓ119906(119894ℓ)

(119909119895 119905) 119894 119895 = 1 2 119899 (8)

Now substituting 119906119894from (8) in (7) and comparing coeffi-

cients of terms with identical powers of 119901 we get

1199010

120597119906(1198940)

120597119905

minus

120597119906(1198940)

120597119905

= 0

1199011

120597119906(1198940)

120597119905

+ 119906(1198950)

120597119906(1198940)

120597119909119895

minus 120583Δ119906(1198940)

1199012 119906(1198951)

120597119906(1198941)

120597119909119895

minus 120583Δ119906(1198941)

1199013 119906(1198952)

120597119906(1198942)

120597119909119895

minus 120583Δ119906(1198942)

(9)

The solution of (7) is

119906119894(119909119895 119905) = 119906

(1198940)+ 119906(1198941)

+ 119906(1198942)

+ sdot sdot sdot (10)

4 Variational Homotopy Perturbation Method

Consider (3) according to [19ndash21] In HPM assume that thesolution of (3) has the form

119906119894=

infin

sum

ℓ=0

119901ℓ119906(119894ℓ)

(119909119895 119905) = V

119894 119894 119895 = 1 2 119899

119906119895=

infin

sum

ℓ=0

119901ℓ119906(119895ℓ)

(119909119895 119905) = V

119895 119894 119895 = 1 2 119899

(11)

From (11) (3) can be written as

120597V119894

120597119905

+

119899

sum

119895=1

V119895

120597V119894

120597119909119895

= 120583ΔV119894 119894 = 1 2 119899 (12)

In VIM from the correction functional for (12) we can write

V119894+1

= V0+ 119901int

119905

0

120582119894(120585)

[

[

minus

119899

sum

119895=1

V119895

120597V119894

120597119909119895

+ 120583ΔV119894]

]

119889120585 (13)

where 119894 = 1 2 119899 V = V(119909119894 120585) from (11) in (13) and by

comparing the coefficients of like powers of 119901 we get

1199010 119906(1198940)

(119909119895 0) = 119891

119894(119909119895 0)

1199011 119906(1198941)

(119909119895 119905) = int

119905

0

120582119894(120585)

sdot [minus119906(1198950)

(119909119895 120585)

120597119906(1198940)

(119909119895 120585)

120597119909119895

+ 120583Δ119906(1198940)

(119909119895 120585)] 119889120585

1199012 119906(1198942)

(119909119895 119905) = int

119905

0

120582119894(120585)

sdot [minus119906(1198951)

(119909119895 120585)

120597119906(1198941)

(119909119895 120585)

120597119909119895

+ 120583Δ119906(1198941)

(119909119895 120585)] 119889120585

1199013 119906(1198943)

(119909119895 119905) = int

119905

0

120582119894(120585)

sdot [minus119906(1198952)

(119909119895 120585)

120597119906(1198942)

(119909119895 120585)

120597119909119895

+ 120583Δ119906(1198942)

(119909119895 120585)] 119889120585

(14)

The approximations solution is given by

119906119894(119909119895 119905) = 119906

(1198940)+ 119906(1198941)

+ 119906(1198942)

+ sdot sdot sdot (15)

To demonstrate the efficiency of the methods we have solvedsome examples by VHPM as (1 + 1)-dimensional (1 + 2)-dimensional (1 + 3)-dimensional and 2-dimensional Thenwe can generalize it for (119899+1)-dimensional or 119899-dimensional

5 Application

Example 1 Consider (1 + 1)-dimensional Burgersrsquo equation[17]

120597119906

120597119905

+ 119906

120597119906

120597119909

=

1205972119906

1205971199092

(16)

with the initial condition

119906 (119909 0) = 1199060(119909) = 2119909 (17)

Journal of Applied Mathematics 3

The correction functional for (16) is

119880119899+1

= 119880119899+ 119901int

119905

0

120582 (120585) [

120597119880

120597119905

+ 119899(119909 120585)

120597119899(119909 120585)

120597119909

minus

1205972119899(119909 120585)

1205971199092

]119889120585

(18)

The general Lagrangian multiplier 120582 can be found as follows

1205821015840= 0

1 + 120582 (120585)1003816100381610038161003816120585=119905

= 0

(19)

Then 120582 = minus1Equation (11) can be written as

119880 =

infin

sum

ℓ=0

119901ℓ119906ℓ(119909 119905) (20)

Applying VHPM we have

1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot = 2119909 minus 119901int

119905

0

[(1199060+ 1199011199061+ 11990121199062

+ sdot sdot sdot)

120597

120597119909

(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[minus

1205972

1205971199092(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot)] 119889120585

(21)

Comparing the coefficient of like powers of 119901 we get

1199010 1199060= 2119909

1199011 1199061= minusint

119905

0

[1199060

1205971199060

120597119909

minus

1205972

12059711990921199060]119889120585 = minus4119909119905

1199012 1199062= minusint

119905

0

[1199061

1205971199061

120597119909

minus

1205972

12059711990921199061]119889120585 = 8119909119905

2

1199013 1199063= minusint

119905

0

[1199062

1205971199062

120597119909

minus

1205972

12059711990921199062]119889120585 = minus16119909119905

2

(22)

The approximations solution is given by

119906 (119909 119905) = 1199060+ 1199061+ 1199062+ 1199063+ sdot sdot sdot (23)

Exact solution is (119906lowast = 2119909(1 + 2119905))The results are in Table 1

Example 2 Consider (1 + 2)-dimensional Burgersrsquo equation[17]

120597119906

120597119905

= 119906

120597119906

120597119909

+

1205972119906

1205971199092+

1205972119906

1205971199102

(24)

with the initial condition

119906 (119909 119910 0) = 1199060(119909 119910) = 119909 + 119910 (25)

Table 1 Comparison of VHPM solutions with exact solution at 119905 =001 (Example 1)

119909 119906lowast(119909 119910 119905) 119906 (119909 119910 119905)

1003816100381610038161003816119906lowastminus 119906

1003816100381610038161003816

01 01960784314 01960784314 002 03921568628 03921568627 1 times 10

minus10

03 05882352942 05882352941 1 times 10minus10

04 07843137254 07843137254 005 09803921568 09803921568 006 1176470588 1176470588 007 1372549020 1372549020 008 1568627451 1568627451 009 1764705882 1764705882 0

As above we have

119880119899+1

= 1199060minus 119901int

119905

0

[minus119880119899(119909 119910 120585)

120597119880119899(119909 119910 120585)

120597119909

minus

1205972119880119899(119909 119910 120585)

1205971199092

minus

1205972119880119899(119909 119910 120585)

1205971199102

]119889120585

1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot = 119909 + 119910 minus 119901int

119905

0

[minus (1199060+ 1199011199061

+ 11990121199062+ sdot sdot sdot)

120597

120597119909

(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[minus

1205972

1205971199092(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot) minus

1205972

1205971199102(1199060

+ 1199011199061+ 11990121199062+ sdot sdot sdot)] 119889120585

(26)

Comparing the coefficient of like powers of 119901 we get

1199010 1199060= 119909 + 119910

1199011 1199061= (119909 + 119910) 119905

1199012 1199062= (119909 + 119910) 119905

2

1199013 1199063= (119909 + 119910) 119905

3

(27)

The approximations solution is given by

119906 (119909 119910 119905) = 1199060+ 1199061+ 1199062+ 1199063+ sdot sdot sdot (28)

Exact solution is (119906lowast = (119909 + 119910)(1 minus 119905))The results are in Table 2

Example 3 Consider (1 + 3)-dimensional Burgersrsquo equation[17]

120597119906

120597119905

= 119906

120597119906

120597119909

+

1205972119906

1205971199092+

1205972119906

1205971199102+

1205972119906

1205971199112

(29)

4 Journal of Applied Mathematics

Table 2 Comparison of VHPM solutions with exact solution at 119909 =

01 and 119910 = 01 (Example 2)

119905 119906lowast(119909 119910 119905) 119906(119909 119910 119905) |119906

lowastminus 119906|

001 02020202020 02020202020 0002 02040816327 02040816326 1 times 10

minus10

003 02061855670 02061855669 1 times 10minus10

004 02083333333 02083333325 8 times 10minus10

005 02105263158 02105263125 33 times 10minus9

006 02127659574 02127659475 99 times 10minus9

007 02150537634 02150537381 253 times 10minus8

008 02173913043 02173912474 569 times 10minus8

009 02197802198 02197801030 1168 times 10minus7

010 02222222222 02222220000 2222 times 10minus7

with the initial condition

119906 (119909 119910 119911 0) = 1199060(119909 119910 119911) = 119909 + 119910 + 119911 (30)

We have

119880119899+1

= 1199060minus 119901int

119905

0

[minus119880119899(119909 119910 120585)

120597119880119899(119909 119910 120585)

120597119909

minus

1205972119880119899(119909 119910 120585)

1205971199092

minus

1205972119880119899(119909 119910 120585)

1205971199102

minus

1205972119880119899(119909 119910 120585)

1205971199112

]119889120585

1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot = 119909 + 119910 + 119911 minus 119901int

119905

0

[minus (1199060

+ 1199011199061+ 11990121199062

+ sdot sdot sdot)

120597

120597119909

(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[minus

1205972

1205971199092(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot) minus

1205972

1205971199102(1199060

+ 1199011199061+ 11990121199062+ sdot sdot sdot) minus

1205972

1205971199112(1199060+ 1199011199061+ 11990121199062

+ sdot sdot sdot)] 119889120585

(31)

Comparing the coefficient of like powers of 119901 we get

1199010 1199060= 119909 + 119910 + 119911

1199011 1199061= (119909 + 119910 + 119911) 119905

1199012 1199062= (119909 + 119910 + 119911) 119905

2

1199013 1199063= (119909 + 119910 + 119911) 119905

3

(32)

Table 3 Comparison of VHPM solutions with exact solution at 119909 =

01 119910 = 01 and 119911 = 01 (Example 3)

119905 119906lowast(119909 119910 119905) 119906(119909 119910 119905) |119906

lowastminus 119906|

001 03030303030 03030303030 0002 03061224490 03061224490 0003 03092783505 03092783503 2 times 10

minus10

004 03125000000 03124999987 13 times 10minus9

005 03157894737 03157894688 49 times 10minus9

006 03191489362 03191489213 149 times 10minus8

007 03225806452 03225806072 380 times 10minus8

008 03260869565 03260868710 855 times 10minus8

009 03296703297 03296701545 1752 times 10minus7

010 03333333333 03333330000 3333 times 10minus7

The approximations solution is given by

119906 (119909 119910 119911 119905) = 1199060+ 1199061+ 1199062+ 1199063+ sdot sdot sdot (33)

Exact solution is (119906lowast = (119909 + 119910 + 119911)(1 minus 119905))The results are in Table 3

Example 4 Consider two-dimensional Burgersrsquo equations[23]

120597119906

120597119905

+ 119906

120597119906

120597119909

+ V120597119906

120597119910

=

1

119877

(

1205972119906

1205971199092+

1205972119906

1205971199102)

120597V120597119905

+ 119906

120597V120597119909

+ V120597V120597119910

=

1

119877

(

1205972V

1205971199092+

1205972V1205971199102)

(34)

with the initial conditions

1199060= 119906 (119909 119910 0) =

3

4

minus

1

4 [1 + 119890(119910minus119909)1198778

]

V0= V (119909 119910 0) =

3

4

+

1

4 [1 + 119890(119910minus119909)1198778

]

(35)

The correction functional for (34) is

119880119899+1

= 119880119899+ 119901int

119905

0

1205821(120585) [

120597119880

120597119905

+ 119899(119909119910 120585)

120597119899(119909119910 120585)120597119909

+ 119899(119909119910 120585)

120597119899(119909119910 120585)120597119909

] 119889120585 + 119901int

119905

0

1205821(120585)

sdot [minus

1

119877

(

1205972119899(119909 119910 120585)

1205971199092

+

1205972119899(119909 119910 120585)

1205971199102

)]119889120585

Journal of Applied Mathematics 5

Table 4 Comparison of VHPM solutions with exact solution at 119909 = 1 119910 = 0 and 119877 = 100 (Example 4)

119905 119906lowast(119909 119910 119905) 119906(119909 119910 119905) |119906

lowastminus 119906| Vlowast(119909 119910 119905) V(119909 119910 119905) |Vlowast minus V|

001 05000009030 05000009571 541 times 10minus8 09999990970 09999990321 649 times 10

minus8

002 05000008752 05000009862 1110 times 10minus7 09999991248 09999990031 1217 times 10

minus7

003 05000008483 05000010153 1670 times 10minus7 09999991517 09999989741 1776 times 10

minus7

004 05000008222 05000010445 2223 times 10minus7 09999991778 09999989451 2327 times 10

minus7

005 05000007969 05000010736 2767 times 10minus7 09999992031 09999989161 2870 times 10

minus7

006 05000007724 05000011027 3303 times 10minus7 09999992276 09999988871 3405 times 10

minus7

007 05000007486 05000011318 3832 times 10minus7 09999992514 09999988581 3933 times 10

minus7

008 05000007256 05000011609 4353 times 10minus7 09999992744 09999988281 4463 times 10

minus7

009 05000007032 05000011900 4868 times 10minus7 09999992968 09999987991 4977 times 10

minus7

010 05000006816 05000012191 5375 times 10minus7 09999993184 09999987701 5483 times 10

minus7

119881119899+1

= 119881119899+ 119901int

119905

0

1205822(120585) [

120597119881

120597119905

+ 119899(119909119910 120585)

120597119899(119909119910 120585)120597119909

+ 119899(119909119910 120585)

120597119899(119909119910 120585)120597119909

] 119889120585 + 119901int

119905

0

1205822(120585)

sdot [minus

1

119877

(

1205972119899(119909119910 120585)1205971199092

+

1205972119899(119909119910 120585)1205971199102

)]119889120585

(36)

The general Lagrangian multipliers are 1205821= minus1 = 120582

2

Equation (11) can be written as

119880 =

infin

sum

ℓ=0

119901ℓ119906ℓ(119909 119910 119905)

119881 =

infin

sum

ℓ=0

119901ℓVℓ(119909 119910 119905)

(37)

By VHPM we have

1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot = 119906

0minus 119901int

119905

0

[minus (1199060+ 1199011199061

+ 11990121199062+ sdot sdot sdot)

120597

120597119909

(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[(V0+ 119901V1+ 1199012V2

+ sdot sdot sdot)

120597

120597119909

(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[

1

119877

(

1205972

1205971199092(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot)

+

1205972

1205971199102(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot))] 119889120585

V0+ 119901V1+ 1199012V2+ sdot sdot sdot = V

0minus 119901int

119905

0

[minus (1199060+ 1199011199061

+ 11990121199062+ sdot sdot sdot)

120597

120597119909

(V0+ 119901V1+ 1199012V2+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[(V0+ 119901V1+ 1199012V2

+ sdot sdot sdot)

120597

120597119909

(V0+ 119901V1+ 1199012V2+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[

1

119877

(

1205972

1205971199092(V0+ 119901V1+ 1199012V2+ sdot sdot sdot)

+

1205972

1205971199102(V0+ 119901V1+ 1199012V2+ sdot sdot sdot))] 119889120585

(38)

Comparing the coefficient of like powers of 119901 we get

1199010

1199060=

3

4

minus

1

4 [1 + 119890(119910minus119909)1198778

]

V0=

3

4

+

1

4 [1 + 119890(119910minus119909)1198778

]

1199011

1199061=

1

128

119877119890minus(18)(minus119910+119909)119877

119905

[1 + 119890minus(18)(minus119910+119909)119877

]2

V1=

minus1

128

119877119890minus(18)(minus119910+119909)119877

119905

[1 + 119890minus(18)(minus119910+119909)119877

]2

1199012

1199062=

minus1

2048

1198772119890minus(14)(minus119910+119909)119877

1199052

[1 + 119890minus(18)(minus119910+119909)119877

]4

V2=

1

2048

1198772119890minus(14)(minus119910+119909)119877

1199052

[1 + 119890minus(18)(minus119910+119909)119877

]4

1199013

1199063=

1

32768

1198773119890minus(38)(minus119910+119909)119877

1199053

[1 + 119890minus(18)(minus119910+119909)119877

]6

V3=

minus1

32768

1198773119890minus(38)(minus119910+119909)119877

1199053

[1 + 119890minus(18)(minus119910+119909)119877

]6

(39)

6 Journal of Applied Mathematics

The approximations solution is given by

119906 (119909 119910 119905) = 1199060+ 1199061+ 1199062+ 1199063+ sdot sdot sdot

V (119909 119910 119905) = V0+ V1+ V2+ V3+ sdot sdot sdot

(40)

Exact solution 119906lowast= 34 minus 14(1 + 119890

(4119910minus4119909minus119905)11987732) Vlowast = 34 +

14(1 + 119890(4119910minus4119909minus119905)11987732

)The results are in Table 4

6 Conclusion

In this work the approximate solutions of 119899-dimensionalBurgersrsquo equations are obtained by combination of two pow-erful methods VIM and HPM in VHPMThe examples haveshown the efficiency and accuracy of the VHPM it reducesthe size of computation without the restrictive assumption tohandle nonlinear terms and it gives the solutions rapidly

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This paper was funded by King Abdulaziz City for Scienceand Technology (KACST) in Saudi Arabia The authorstherefore thank them for their full collaboration

References

[1] J H He ldquoA variational iteration approach to nonlinear prob-lems and its applicationsrdquoMechanical Application vol 20 no 1pp 30ndash31 1998

[2] J-H He ldquoVariational iteration method-a kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

[3] J-H He ldquoVariational iteration methodmdashsome recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[4] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[5] J-H He ldquoCoupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000

[6] J-H He ldquoHomotopy perturbation method a new nonlinearanalytical techniquerdquo Applied Mathematics and Computationvol 135 no 1 pp 73ndash79 2003

[7] A Naghipour and JManafian ldquoApplication of the Laplace Ado-mian decomposition and implicit methods for solving BurgersrsquoequationrdquoTWMS Journal of Pure and AppliedMathematics vol6 no 1 pp 68ndash77 2015

[8] T Ozis and Y Aslan ldquoThe semi-approximate approach forsolving Burgersrsquo equation with high Reynolds numberrdquo AppliedMathematics and Computation vol 163 no 1 pp 131ndash145 2005

[9] D L Young C M Fan S P Hu and S N Atluri ldquoTheEulerian-Lagrangianmethod of fundamental solutions for two-dimensional unsteady Burgersrsquo equationsrdquo Engineering Analysiswith Boundary Elements vol 32 no 5 pp 395ndash412 2008

[10] M A Abdou and A A Soliman ldquoVariational iteration methodfor solving BURgerrsquos and coupled BURgerrsquos equationsrdquo Journalof Computational and Applied Mathematics vol 181 no 2 pp245ndash251 2005

[11] A-M Wazwaz ldquoThe variational iteration method for solvinglinear and nonlinear systems of PDEsrdquoComputersampMathemat-ics with Applications vol 54 no 7-8 pp 895ndash902 2007

[12] A-MWazwaz ldquoA comparison between the variational iterationmethod and Adomian decomposition methodrdquo Journal ofComputational andAppliedMathematics vol 207 no 1 pp 129ndash136 2007

[13] J Biazar and H Aminikhah ldquoExact and numerical solutionsfor non-linear Burgerrsquos equation by VIMrdquo Mathematical andComputer Modelling vol 49 no 7-8 pp 1394ndash1400 2009

[14] B BatihaVarational ItrationalMethod and Its Applications LAPLAMBERT Academic Publishing GmbH Co KG GermangSaarbrucken Germany 2012

[15] S Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer Berlin Germany 2012

[16] K R Desai and V H Pradhan ldquoSolution of Burgerrsquos equationand coupled Burgerrsquos equations by Homotopy perturbationmethodrdquo International Journal of Engineering Research andApplications vol 2 no 3 pp 2033ndash2040 2012

[17] V K Srivastava and M K Awasthi ldquo(1 + n)-dimensional Burg-ersrsquo equation and its analytical solution a comparative study ofHPM ADM and DTMrdquo Ain Shams Engineering Journal vol 5no 2 pp 533ndash541 2014

[18] J Ahmad Z Bibi andKNoor ldquoLaplace decompositionmethodusing hersquos polynomials to burgers equationrdquo Journal of Scienceand Arts vol 2 no 27 pp 131ndash138 2014

[19] M Matinfar M Mahdavi and Z Raeisy ldquoThe implementationof variational homotopy perturbationmethod for Fisherrsquos equa-tionrdquo International Journal of Nonlinear Science vol 9 no 2 pp188ndash194 2010

[20] T Allahviranloo A Armand and S Pirmohammadi ldquoVaria-tional homotopy perturbation method an efficient scheme forsolving partial differential equations in fluid mechanicsrdquo TheJournal of Mathematics and Computer Science vol 9 no 4 pp362ndash369 2014

[21] A Daga and V Pradhan ldquoA novel approach for solving burgerrsquosequationrdquo Applications amp Applied Mathematics vol 9 no 2 pp541ndash552 2014

[22] Y Chen E Fan and M Yuen ldquoThe Hopf-Cole transformationtopological solitons and multiple fusion solutions for the n-dimensional Burgers systemrdquo Physics Letters A vol 380 no 1-2pp 9ndash14 2016

[23] S Hernandez G M Carlomagno and C A Brebbia Com-putational Methods and Experimental Measurements XVI WITPress 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

or

120597119906(119894119896)

120597119905

minus

120597119906(1198940)

120597119905

+ 119901[

[

120597119906(1198940)

120597119905

+

119899

sum

119895=1

119906(119895119896)

120597119906(119894119896)

120597119909119895

minus 120583Δ119906(119894119896)

]

]

= 0

(7)

where 119894 = 1 2 119899 119896 = 1 2 119901 isin [0 1] is an embed-ding parameter while 119906

(10)= 1198911(119909119895 0) 119906

(20)= 1198912(119909119895 0)

119906(1198990)

= 119891119899(119909119895 0) are initial approximations of (3)

Assume the solution of (3) has the form

119906119894=

infin

sum

ℓ=0

119901ℓ119906(119894ℓ)

(119909119895 119905) 119894 119895 = 1 2 119899 (8)

Now substituting 119906119894from (8) in (7) and comparing coeffi-

cients of terms with identical powers of 119901 we get

1199010

120597119906(1198940)

120597119905

minus

120597119906(1198940)

120597119905

= 0

1199011

120597119906(1198940)

120597119905

+ 119906(1198950)

120597119906(1198940)

120597119909119895

minus 120583Δ119906(1198940)

1199012 119906(1198951)

120597119906(1198941)

120597119909119895

minus 120583Δ119906(1198941)

1199013 119906(1198952)

120597119906(1198942)

120597119909119895

minus 120583Δ119906(1198942)

(9)

The solution of (7) is

119906119894(119909119895 119905) = 119906

(1198940)+ 119906(1198941)

+ 119906(1198942)

+ sdot sdot sdot (10)

4 Variational Homotopy Perturbation Method

Consider (3) according to [19ndash21] In HPM assume that thesolution of (3) has the form

119906119894=

infin

sum

ℓ=0

119901ℓ119906(119894ℓ)

(119909119895 119905) = V

119894 119894 119895 = 1 2 119899

119906119895=

infin

sum

ℓ=0

119901ℓ119906(119895ℓ)

(119909119895 119905) = V

119895 119894 119895 = 1 2 119899

(11)

From (11) (3) can be written as

120597V119894

120597119905

+

119899

sum

119895=1

V119895

120597V119894

120597119909119895

= 120583ΔV119894 119894 = 1 2 119899 (12)

In VIM from the correction functional for (12) we can write

V119894+1

= V0+ 119901int

119905

0

120582119894(120585)

[

[

minus

119899

sum

119895=1

V119895

120597V119894

120597119909119895

+ 120583ΔV119894]

]

119889120585 (13)

where 119894 = 1 2 119899 V = V(119909119894 120585) from (11) in (13) and by

comparing the coefficients of like powers of 119901 we get

1199010 119906(1198940)

(119909119895 0) = 119891

119894(119909119895 0)

1199011 119906(1198941)

(119909119895 119905) = int

119905

0

120582119894(120585)

sdot [minus119906(1198950)

(119909119895 120585)

120597119906(1198940)

(119909119895 120585)

120597119909119895

+ 120583Δ119906(1198940)

(119909119895 120585)] 119889120585

1199012 119906(1198942)

(119909119895 119905) = int

119905

0

120582119894(120585)

sdot [minus119906(1198951)

(119909119895 120585)

120597119906(1198941)

(119909119895 120585)

120597119909119895

+ 120583Δ119906(1198941)

(119909119895 120585)] 119889120585

1199013 119906(1198943)

(119909119895 119905) = int

119905

0

120582119894(120585)

sdot [minus119906(1198952)

(119909119895 120585)

120597119906(1198942)

(119909119895 120585)

120597119909119895

+ 120583Δ119906(1198942)

(119909119895 120585)] 119889120585

(14)

The approximations solution is given by

119906119894(119909119895 119905) = 119906

(1198940)+ 119906(1198941)

+ 119906(1198942)

+ sdot sdot sdot (15)

To demonstrate the efficiency of the methods we have solvedsome examples by VHPM as (1 + 1)-dimensional (1 + 2)-dimensional (1 + 3)-dimensional and 2-dimensional Thenwe can generalize it for (119899+1)-dimensional or 119899-dimensional

5 Application

Example 1 Consider (1 + 1)-dimensional Burgersrsquo equation[17]

120597119906

120597119905

+ 119906

120597119906

120597119909

=

1205972119906

1205971199092

(16)

with the initial condition

119906 (119909 0) = 1199060(119909) = 2119909 (17)

Journal of Applied Mathematics 3

The correction functional for (16) is

119880119899+1

= 119880119899+ 119901int

119905

0

120582 (120585) [

120597119880

120597119905

+ 119899(119909 120585)

120597119899(119909 120585)

120597119909

minus

1205972119899(119909 120585)

1205971199092

]119889120585

(18)

The general Lagrangian multiplier 120582 can be found as follows

1205821015840= 0

1 + 120582 (120585)1003816100381610038161003816120585=119905

= 0

(19)

Then 120582 = minus1Equation (11) can be written as

119880 =

infin

sum

ℓ=0

119901ℓ119906ℓ(119909 119905) (20)

Applying VHPM we have

1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot = 2119909 minus 119901int

119905

0

[(1199060+ 1199011199061+ 11990121199062

+ sdot sdot sdot)

120597

120597119909

(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[minus

1205972

1205971199092(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot)] 119889120585

(21)

Comparing the coefficient of like powers of 119901 we get

1199010 1199060= 2119909

1199011 1199061= minusint

119905

0

[1199060

1205971199060

120597119909

minus

1205972

12059711990921199060]119889120585 = minus4119909119905

1199012 1199062= minusint

119905

0

[1199061

1205971199061

120597119909

minus

1205972

12059711990921199061]119889120585 = 8119909119905

2

1199013 1199063= minusint

119905

0

[1199062

1205971199062

120597119909

minus

1205972

12059711990921199062]119889120585 = minus16119909119905

2

(22)

The approximations solution is given by

119906 (119909 119905) = 1199060+ 1199061+ 1199062+ 1199063+ sdot sdot sdot (23)

Exact solution is (119906lowast = 2119909(1 + 2119905))The results are in Table 1

Example 2 Consider (1 + 2)-dimensional Burgersrsquo equation[17]

120597119906

120597119905

= 119906

120597119906

120597119909

+

1205972119906

1205971199092+

1205972119906

1205971199102

(24)

with the initial condition

119906 (119909 119910 0) = 1199060(119909 119910) = 119909 + 119910 (25)

Table 1 Comparison of VHPM solutions with exact solution at 119905 =001 (Example 1)

119909 119906lowast(119909 119910 119905) 119906 (119909 119910 119905)

1003816100381610038161003816119906lowastminus 119906

1003816100381610038161003816

01 01960784314 01960784314 002 03921568628 03921568627 1 times 10

minus10

03 05882352942 05882352941 1 times 10minus10

04 07843137254 07843137254 005 09803921568 09803921568 006 1176470588 1176470588 007 1372549020 1372549020 008 1568627451 1568627451 009 1764705882 1764705882 0

As above we have

119880119899+1

= 1199060minus 119901int

119905

0

[minus119880119899(119909 119910 120585)

120597119880119899(119909 119910 120585)

120597119909

minus

1205972119880119899(119909 119910 120585)

1205971199092

minus

1205972119880119899(119909 119910 120585)

1205971199102

]119889120585

1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot = 119909 + 119910 minus 119901int

119905

0

[minus (1199060+ 1199011199061

+ 11990121199062+ sdot sdot sdot)

120597

120597119909

(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[minus

1205972

1205971199092(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot) minus

1205972

1205971199102(1199060

+ 1199011199061+ 11990121199062+ sdot sdot sdot)] 119889120585

(26)

Comparing the coefficient of like powers of 119901 we get

1199010 1199060= 119909 + 119910

1199011 1199061= (119909 + 119910) 119905

1199012 1199062= (119909 + 119910) 119905

2

1199013 1199063= (119909 + 119910) 119905

3

(27)

The approximations solution is given by

119906 (119909 119910 119905) = 1199060+ 1199061+ 1199062+ 1199063+ sdot sdot sdot (28)

Exact solution is (119906lowast = (119909 + 119910)(1 minus 119905))The results are in Table 2

Example 3 Consider (1 + 3)-dimensional Burgersrsquo equation[17]

120597119906

120597119905

= 119906

120597119906

120597119909

+

1205972119906

1205971199092+

1205972119906

1205971199102+

1205972119906

1205971199112

(29)

4 Journal of Applied Mathematics

Table 2 Comparison of VHPM solutions with exact solution at 119909 =

01 and 119910 = 01 (Example 2)

119905 119906lowast(119909 119910 119905) 119906(119909 119910 119905) |119906

lowastminus 119906|

001 02020202020 02020202020 0002 02040816327 02040816326 1 times 10

minus10

003 02061855670 02061855669 1 times 10minus10

004 02083333333 02083333325 8 times 10minus10

005 02105263158 02105263125 33 times 10minus9

006 02127659574 02127659475 99 times 10minus9

007 02150537634 02150537381 253 times 10minus8

008 02173913043 02173912474 569 times 10minus8

009 02197802198 02197801030 1168 times 10minus7

010 02222222222 02222220000 2222 times 10minus7

with the initial condition

119906 (119909 119910 119911 0) = 1199060(119909 119910 119911) = 119909 + 119910 + 119911 (30)

We have

119880119899+1

= 1199060minus 119901int

119905

0

[minus119880119899(119909 119910 120585)

120597119880119899(119909 119910 120585)

120597119909

minus

1205972119880119899(119909 119910 120585)

1205971199092

minus

1205972119880119899(119909 119910 120585)

1205971199102

minus

1205972119880119899(119909 119910 120585)

1205971199112

]119889120585

1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot = 119909 + 119910 + 119911 minus 119901int

119905

0

[minus (1199060

+ 1199011199061+ 11990121199062

+ sdot sdot sdot)

120597

120597119909

(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[minus

1205972

1205971199092(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot) minus

1205972

1205971199102(1199060

+ 1199011199061+ 11990121199062+ sdot sdot sdot) minus

1205972

1205971199112(1199060+ 1199011199061+ 11990121199062

+ sdot sdot sdot)] 119889120585

(31)

Comparing the coefficient of like powers of 119901 we get

1199010 1199060= 119909 + 119910 + 119911

1199011 1199061= (119909 + 119910 + 119911) 119905

1199012 1199062= (119909 + 119910 + 119911) 119905

2

1199013 1199063= (119909 + 119910 + 119911) 119905

3

(32)

Table 3 Comparison of VHPM solutions with exact solution at 119909 =

01 119910 = 01 and 119911 = 01 (Example 3)

119905 119906lowast(119909 119910 119905) 119906(119909 119910 119905) |119906

lowastminus 119906|

001 03030303030 03030303030 0002 03061224490 03061224490 0003 03092783505 03092783503 2 times 10

minus10

004 03125000000 03124999987 13 times 10minus9

005 03157894737 03157894688 49 times 10minus9

006 03191489362 03191489213 149 times 10minus8

007 03225806452 03225806072 380 times 10minus8

008 03260869565 03260868710 855 times 10minus8

009 03296703297 03296701545 1752 times 10minus7

010 03333333333 03333330000 3333 times 10minus7

The approximations solution is given by

119906 (119909 119910 119911 119905) = 1199060+ 1199061+ 1199062+ 1199063+ sdot sdot sdot (33)

Exact solution is (119906lowast = (119909 + 119910 + 119911)(1 minus 119905))The results are in Table 3

Example 4 Consider two-dimensional Burgersrsquo equations[23]

120597119906

120597119905

+ 119906

120597119906

120597119909

+ V120597119906

120597119910

=

1

119877

(

1205972119906

1205971199092+

1205972119906

1205971199102)

120597V120597119905

+ 119906

120597V120597119909

+ V120597V120597119910

=

1

119877

(

1205972V

1205971199092+

1205972V1205971199102)

(34)

with the initial conditions

1199060= 119906 (119909 119910 0) =

3

4

minus

1

4 [1 + 119890(119910minus119909)1198778

]

V0= V (119909 119910 0) =

3

4

+

1

4 [1 + 119890(119910minus119909)1198778

]

(35)

The correction functional for (34) is

119880119899+1

= 119880119899+ 119901int

119905

0

1205821(120585) [

120597119880

120597119905

+ 119899(119909119910 120585)

120597119899(119909119910 120585)120597119909

+ 119899(119909119910 120585)

120597119899(119909119910 120585)120597119909

] 119889120585 + 119901int

119905

0

1205821(120585)

sdot [minus

1

119877

(

1205972119899(119909 119910 120585)

1205971199092

+

1205972119899(119909 119910 120585)

1205971199102

)]119889120585

Journal of Applied Mathematics 5

Table 4 Comparison of VHPM solutions with exact solution at 119909 = 1 119910 = 0 and 119877 = 100 (Example 4)

119905 119906lowast(119909 119910 119905) 119906(119909 119910 119905) |119906

lowastminus 119906| Vlowast(119909 119910 119905) V(119909 119910 119905) |Vlowast minus V|

001 05000009030 05000009571 541 times 10minus8 09999990970 09999990321 649 times 10

minus8

002 05000008752 05000009862 1110 times 10minus7 09999991248 09999990031 1217 times 10

minus7

003 05000008483 05000010153 1670 times 10minus7 09999991517 09999989741 1776 times 10

minus7

004 05000008222 05000010445 2223 times 10minus7 09999991778 09999989451 2327 times 10

minus7

005 05000007969 05000010736 2767 times 10minus7 09999992031 09999989161 2870 times 10

minus7

006 05000007724 05000011027 3303 times 10minus7 09999992276 09999988871 3405 times 10

minus7

007 05000007486 05000011318 3832 times 10minus7 09999992514 09999988581 3933 times 10

minus7

008 05000007256 05000011609 4353 times 10minus7 09999992744 09999988281 4463 times 10

minus7

009 05000007032 05000011900 4868 times 10minus7 09999992968 09999987991 4977 times 10

minus7

010 05000006816 05000012191 5375 times 10minus7 09999993184 09999987701 5483 times 10

minus7

119881119899+1

= 119881119899+ 119901int

119905

0

1205822(120585) [

120597119881

120597119905

+ 119899(119909119910 120585)

120597119899(119909119910 120585)120597119909

+ 119899(119909119910 120585)

120597119899(119909119910 120585)120597119909

] 119889120585 + 119901int

119905

0

1205822(120585)

sdot [minus

1

119877

(

1205972119899(119909119910 120585)1205971199092

+

1205972119899(119909119910 120585)1205971199102

)]119889120585

(36)

The general Lagrangian multipliers are 1205821= minus1 = 120582

2

Equation (11) can be written as

119880 =

infin

sum

ℓ=0

119901ℓ119906ℓ(119909 119910 119905)

119881 =

infin

sum

ℓ=0

119901ℓVℓ(119909 119910 119905)

(37)

By VHPM we have

1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot = 119906

0minus 119901int

119905

0

[minus (1199060+ 1199011199061

+ 11990121199062+ sdot sdot sdot)

120597

120597119909

(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[(V0+ 119901V1+ 1199012V2

+ sdot sdot sdot)

120597

120597119909

(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[

1

119877

(

1205972

1205971199092(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot)

+

1205972

1205971199102(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot))] 119889120585

V0+ 119901V1+ 1199012V2+ sdot sdot sdot = V

0minus 119901int

119905

0

[minus (1199060+ 1199011199061

+ 11990121199062+ sdot sdot sdot)

120597

120597119909

(V0+ 119901V1+ 1199012V2+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[(V0+ 119901V1+ 1199012V2

+ sdot sdot sdot)

120597

120597119909

(V0+ 119901V1+ 1199012V2+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[

1

119877

(

1205972

1205971199092(V0+ 119901V1+ 1199012V2+ sdot sdot sdot)

+

1205972

1205971199102(V0+ 119901V1+ 1199012V2+ sdot sdot sdot))] 119889120585

(38)

Comparing the coefficient of like powers of 119901 we get

1199010

1199060=

3

4

minus

1

4 [1 + 119890(119910minus119909)1198778

]

V0=

3

4

+

1

4 [1 + 119890(119910minus119909)1198778

]

1199011

1199061=

1

128

119877119890minus(18)(minus119910+119909)119877

119905

[1 + 119890minus(18)(minus119910+119909)119877

]2

V1=

minus1

128

119877119890minus(18)(minus119910+119909)119877

119905

[1 + 119890minus(18)(minus119910+119909)119877

]2

1199012

1199062=

minus1

2048

1198772119890minus(14)(minus119910+119909)119877

1199052

[1 + 119890minus(18)(minus119910+119909)119877

]4

V2=

1

2048

1198772119890minus(14)(minus119910+119909)119877

1199052

[1 + 119890minus(18)(minus119910+119909)119877

]4

1199013

1199063=

1

32768

1198773119890minus(38)(minus119910+119909)119877

1199053

[1 + 119890minus(18)(minus119910+119909)119877

]6

V3=

minus1

32768

1198773119890minus(38)(minus119910+119909)119877

1199053

[1 + 119890minus(18)(minus119910+119909)119877

]6

(39)

6 Journal of Applied Mathematics

The approximations solution is given by

119906 (119909 119910 119905) = 1199060+ 1199061+ 1199062+ 1199063+ sdot sdot sdot

V (119909 119910 119905) = V0+ V1+ V2+ V3+ sdot sdot sdot

(40)

Exact solution 119906lowast= 34 minus 14(1 + 119890

(4119910minus4119909minus119905)11987732) Vlowast = 34 +

14(1 + 119890(4119910minus4119909minus119905)11987732

)The results are in Table 4

6 Conclusion

In this work the approximate solutions of 119899-dimensionalBurgersrsquo equations are obtained by combination of two pow-erful methods VIM and HPM in VHPMThe examples haveshown the efficiency and accuracy of the VHPM it reducesthe size of computation without the restrictive assumption tohandle nonlinear terms and it gives the solutions rapidly

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This paper was funded by King Abdulaziz City for Scienceand Technology (KACST) in Saudi Arabia The authorstherefore thank them for their full collaboration

References

[1] J H He ldquoA variational iteration approach to nonlinear prob-lems and its applicationsrdquoMechanical Application vol 20 no 1pp 30ndash31 1998

[2] J-H He ldquoVariational iteration method-a kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

[3] J-H He ldquoVariational iteration methodmdashsome recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[4] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[5] J-H He ldquoCoupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000

[6] J-H He ldquoHomotopy perturbation method a new nonlinearanalytical techniquerdquo Applied Mathematics and Computationvol 135 no 1 pp 73ndash79 2003

[7] A Naghipour and JManafian ldquoApplication of the Laplace Ado-mian decomposition and implicit methods for solving BurgersrsquoequationrdquoTWMS Journal of Pure and AppliedMathematics vol6 no 1 pp 68ndash77 2015

[8] T Ozis and Y Aslan ldquoThe semi-approximate approach forsolving Burgersrsquo equation with high Reynolds numberrdquo AppliedMathematics and Computation vol 163 no 1 pp 131ndash145 2005

[9] D L Young C M Fan S P Hu and S N Atluri ldquoTheEulerian-Lagrangianmethod of fundamental solutions for two-dimensional unsteady Burgersrsquo equationsrdquo Engineering Analysiswith Boundary Elements vol 32 no 5 pp 395ndash412 2008

[10] M A Abdou and A A Soliman ldquoVariational iteration methodfor solving BURgerrsquos and coupled BURgerrsquos equationsrdquo Journalof Computational and Applied Mathematics vol 181 no 2 pp245ndash251 2005

[11] A-M Wazwaz ldquoThe variational iteration method for solvinglinear and nonlinear systems of PDEsrdquoComputersampMathemat-ics with Applications vol 54 no 7-8 pp 895ndash902 2007

[12] A-MWazwaz ldquoA comparison between the variational iterationmethod and Adomian decomposition methodrdquo Journal ofComputational andAppliedMathematics vol 207 no 1 pp 129ndash136 2007

[13] J Biazar and H Aminikhah ldquoExact and numerical solutionsfor non-linear Burgerrsquos equation by VIMrdquo Mathematical andComputer Modelling vol 49 no 7-8 pp 1394ndash1400 2009

[14] B BatihaVarational ItrationalMethod and Its Applications LAPLAMBERT Academic Publishing GmbH Co KG GermangSaarbrucken Germany 2012

[15] S Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer Berlin Germany 2012

[16] K R Desai and V H Pradhan ldquoSolution of Burgerrsquos equationand coupled Burgerrsquos equations by Homotopy perturbationmethodrdquo International Journal of Engineering Research andApplications vol 2 no 3 pp 2033ndash2040 2012

[17] V K Srivastava and M K Awasthi ldquo(1 + n)-dimensional Burg-ersrsquo equation and its analytical solution a comparative study ofHPM ADM and DTMrdquo Ain Shams Engineering Journal vol 5no 2 pp 533ndash541 2014

[18] J Ahmad Z Bibi andKNoor ldquoLaplace decompositionmethodusing hersquos polynomials to burgers equationrdquo Journal of Scienceand Arts vol 2 no 27 pp 131ndash138 2014

[19] M Matinfar M Mahdavi and Z Raeisy ldquoThe implementationof variational homotopy perturbationmethod for Fisherrsquos equa-tionrdquo International Journal of Nonlinear Science vol 9 no 2 pp188ndash194 2010

[20] T Allahviranloo A Armand and S Pirmohammadi ldquoVaria-tional homotopy perturbation method an efficient scheme forsolving partial differential equations in fluid mechanicsrdquo TheJournal of Mathematics and Computer Science vol 9 no 4 pp362ndash369 2014

[21] A Daga and V Pradhan ldquoA novel approach for solving burgerrsquosequationrdquo Applications amp Applied Mathematics vol 9 no 2 pp541ndash552 2014

[22] Y Chen E Fan and M Yuen ldquoThe Hopf-Cole transformationtopological solitons and multiple fusion solutions for the n-dimensional Burgers systemrdquo Physics Letters A vol 380 no 1-2pp 9ndash14 2016

[23] S Hernandez G M Carlomagno and C A Brebbia Com-putational Methods and Experimental Measurements XVI WITPress 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

The correction functional for (16) is

119880119899+1

= 119880119899+ 119901int

119905

0

120582 (120585) [

120597119880

120597119905

+ 119899(119909 120585)

120597119899(119909 120585)

120597119909

minus

1205972119899(119909 120585)

1205971199092

]119889120585

(18)

The general Lagrangian multiplier 120582 can be found as follows

1205821015840= 0

1 + 120582 (120585)1003816100381610038161003816120585=119905

= 0

(19)

Then 120582 = minus1Equation (11) can be written as

119880 =

infin

sum

ℓ=0

119901ℓ119906ℓ(119909 119905) (20)

Applying VHPM we have

1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot = 2119909 minus 119901int

119905

0

[(1199060+ 1199011199061+ 11990121199062

+ sdot sdot sdot)

120597

120597119909

(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[minus

1205972

1205971199092(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot)] 119889120585

(21)

Comparing the coefficient of like powers of 119901 we get

1199010 1199060= 2119909

1199011 1199061= minusint

119905

0

[1199060

1205971199060

120597119909

minus

1205972

12059711990921199060]119889120585 = minus4119909119905

1199012 1199062= minusint

119905

0

[1199061

1205971199061

120597119909

minus

1205972

12059711990921199061]119889120585 = 8119909119905

2

1199013 1199063= minusint

119905

0

[1199062

1205971199062

120597119909

minus

1205972

12059711990921199062]119889120585 = minus16119909119905

2

(22)

The approximations solution is given by

119906 (119909 119905) = 1199060+ 1199061+ 1199062+ 1199063+ sdot sdot sdot (23)

Exact solution is (119906lowast = 2119909(1 + 2119905))The results are in Table 1

Example 2 Consider (1 + 2)-dimensional Burgersrsquo equation[17]

120597119906

120597119905

= 119906

120597119906

120597119909

+

1205972119906

1205971199092+

1205972119906

1205971199102

(24)

with the initial condition

119906 (119909 119910 0) = 1199060(119909 119910) = 119909 + 119910 (25)

Table 1 Comparison of VHPM solutions with exact solution at 119905 =001 (Example 1)

119909 119906lowast(119909 119910 119905) 119906 (119909 119910 119905)

1003816100381610038161003816119906lowastminus 119906

1003816100381610038161003816

01 01960784314 01960784314 002 03921568628 03921568627 1 times 10

minus10

03 05882352942 05882352941 1 times 10minus10

04 07843137254 07843137254 005 09803921568 09803921568 006 1176470588 1176470588 007 1372549020 1372549020 008 1568627451 1568627451 009 1764705882 1764705882 0

As above we have

119880119899+1

= 1199060minus 119901int

119905

0

[minus119880119899(119909 119910 120585)

120597119880119899(119909 119910 120585)

120597119909

minus

1205972119880119899(119909 119910 120585)

1205971199092

minus

1205972119880119899(119909 119910 120585)

1205971199102

]119889120585

1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot = 119909 + 119910 minus 119901int

119905

0

[minus (1199060+ 1199011199061

+ 11990121199062+ sdot sdot sdot)

120597

120597119909

(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[minus

1205972

1205971199092(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot) minus

1205972

1205971199102(1199060

+ 1199011199061+ 11990121199062+ sdot sdot sdot)] 119889120585

(26)

Comparing the coefficient of like powers of 119901 we get

1199010 1199060= 119909 + 119910

1199011 1199061= (119909 + 119910) 119905

1199012 1199062= (119909 + 119910) 119905

2

1199013 1199063= (119909 + 119910) 119905

3

(27)

The approximations solution is given by

119906 (119909 119910 119905) = 1199060+ 1199061+ 1199062+ 1199063+ sdot sdot sdot (28)

Exact solution is (119906lowast = (119909 + 119910)(1 minus 119905))The results are in Table 2

Example 3 Consider (1 + 3)-dimensional Burgersrsquo equation[17]

120597119906

120597119905

= 119906

120597119906

120597119909

+

1205972119906

1205971199092+

1205972119906

1205971199102+

1205972119906

1205971199112

(29)

4 Journal of Applied Mathematics

Table 2 Comparison of VHPM solutions with exact solution at 119909 =

01 and 119910 = 01 (Example 2)

119905 119906lowast(119909 119910 119905) 119906(119909 119910 119905) |119906

lowastminus 119906|

001 02020202020 02020202020 0002 02040816327 02040816326 1 times 10

minus10

003 02061855670 02061855669 1 times 10minus10

004 02083333333 02083333325 8 times 10minus10

005 02105263158 02105263125 33 times 10minus9

006 02127659574 02127659475 99 times 10minus9

007 02150537634 02150537381 253 times 10minus8

008 02173913043 02173912474 569 times 10minus8

009 02197802198 02197801030 1168 times 10minus7

010 02222222222 02222220000 2222 times 10minus7

with the initial condition

119906 (119909 119910 119911 0) = 1199060(119909 119910 119911) = 119909 + 119910 + 119911 (30)

We have

119880119899+1

= 1199060minus 119901int

119905

0

[minus119880119899(119909 119910 120585)

120597119880119899(119909 119910 120585)

120597119909

minus

1205972119880119899(119909 119910 120585)

1205971199092

minus

1205972119880119899(119909 119910 120585)

1205971199102

minus

1205972119880119899(119909 119910 120585)

1205971199112

]119889120585

1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot = 119909 + 119910 + 119911 minus 119901int

119905

0

[minus (1199060

+ 1199011199061+ 11990121199062

+ sdot sdot sdot)

120597

120597119909

(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[minus

1205972

1205971199092(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot) minus

1205972

1205971199102(1199060

+ 1199011199061+ 11990121199062+ sdot sdot sdot) minus

1205972

1205971199112(1199060+ 1199011199061+ 11990121199062

+ sdot sdot sdot)] 119889120585

(31)

Comparing the coefficient of like powers of 119901 we get

1199010 1199060= 119909 + 119910 + 119911

1199011 1199061= (119909 + 119910 + 119911) 119905

1199012 1199062= (119909 + 119910 + 119911) 119905

2

1199013 1199063= (119909 + 119910 + 119911) 119905

3

(32)

Table 3 Comparison of VHPM solutions with exact solution at 119909 =

01 119910 = 01 and 119911 = 01 (Example 3)

119905 119906lowast(119909 119910 119905) 119906(119909 119910 119905) |119906

lowastminus 119906|

001 03030303030 03030303030 0002 03061224490 03061224490 0003 03092783505 03092783503 2 times 10

minus10

004 03125000000 03124999987 13 times 10minus9

005 03157894737 03157894688 49 times 10minus9

006 03191489362 03191489213 149 times 10minus8

007 03225806452 03225806072 380 times 10minus8

008 03260869565 03260868710 855 times 10minus8

009 03296703297 03296701545 1752 times 10minus7

010 03333333333 03333330000 3333 times 10minus7

The approximations solution is given by

119906 (119909 119910 119911 119905) = 1199060+ 1199061+ 1199062+ 1199063+ sdot sdot sdot (33)

Exact solution is (119906lowast = (119909 + 119910 + 119911)(1 minus 119905))The results are in Table 3

Example 4 Consider two-dimensional Burgersrsquo equations[23]

120597119906

120597119905

+ 119906

120597119906

120597119909

+ V120597119906

120597119910

=

1

119877

(

1205972119906

1205971199092+

1205972119906

1205971199102)

120597V120597119905

+ 119906

120597V120597119909

+ V120597V120597119910

=

1

119877

(

1205972V

1205971199092+

1205972V1205971199102)

(34)

with the initial conditions

1199060= 119906 (119909 119910 0) =

3

4

minus

1

4 [1 + 119890(119910minus119909)1198778

]

V0= V (119909 119910 0) =

3

4

+

1

4 [1 + 119890(119910minus119909)1198778

]

(35)

The correction functional for (34) is

119880119899+1

= 119880119899+ 119901int

119905

0

1205821(120585) [

120597119880

120597119905

+ 119899(119909119910 120585)

120597119899(119909119910 120585)120597119909

+ 119899(119909119910 120585)

120597119899(119909119910 120585)120597119909

] 119889120585 + 119901int

119905

0

1205821(120585)

sdot [minus

1

119877

(

1205972119899(119909 119910 120585)

1205971199092

+

1205972119899(119909 119910 120585)

1205971199102

)]119889120585

Journal of Applied Mathematics 5

Table 4 Comparison of VHPM solutions with exact solution at 119909 = 1 119910 = 0 and 119877 = 100 (Example 4)

119905 119906lowast(119909 119910 119905) 119906(119909 119910 119905) |119906

lowastminus 119906| Vlowast(119909 119910 119905) V(119909 119910 119905) |Vlowast minus V|

001 05000009030 05000009571 541 times 10minus8 09999990970 09999990321 649 times 10

minus8

002 05000008752 05000009862 1110 times 10minus7 09999991248 09999990031 1217 times 10

minus7

003 05000008483 05000010153 1670 times 10minus7 09999991517 09999989741 1776 times 10

minus7

004 05000008222 05000010445 2223 times 10minus7 09999991778 09999989451 2327 times 10

minus7

005 05000007969 05000010736 2767 times 10minus7 09999992031 09999989161 2870 times 10

minus7

006 05000007724 05000011027 3303 times 10minus7 09999992276 09999988871 3405 times 10

minus7

007 05000007486 05000011318 3832 times 10minus7 09999992514 09999988581 3933 times 10

minus7

008 05000007256 05000011609 4353 times 10minus7 09999992744 09999988281 4463 times 10

minus7

009 05000007032 05000011900 4868 times 10minus7 09999992968 09999987991 4977 times 10

minus7

010 05000006816 05000012191 5375 times 10minus7 09999993184 09999987701 5483 times 10

minus7

119881119899+1

= 119881119899+ 119901int

119905

0

1205822(120585) [

120597119881

120597119905

+ 119899(119909119910 120585)

120597119899(119909119910 120585)120597119909

+ 119899(119909119910 120585)

120597119899(119909119910 120585)120597119909

] 119889120585 + 119901int

119905

0

1205822(120585)

sdot [minus

1

119877

(

1205972119899(119909119910 120585)1205971199092

+

1205972119899(119909119910 120585)1205971199102

)]119889120585

(36)

The general Lagrangian multipliers are 1205821= minus1 = 120582

2

Equation (11) can be written as

119880 =

infin

sum

ℓ=0

119901ℓ119906ℓ(119909 119910 119905)

119881 =

infin

sum

ℓ=0

119901ℓVℓ(119909 119910 119905)

(37)

By VHPM we have

1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot = 119906

0minus 119901int

119905

0

[minus (1199060+ 1199011199061

+ 11990121199062+ sdot sdot sdot)

120597

120597119909

(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[(V0+ 119901V1+ 1199012V2

+ sdot sdot sdot)

120597

120597119909

(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[

1

119877

(

1205972

1205971199092(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot)

+

1205972

1205971199102(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot))] 119889120585

V0+ 119901V1+ 1199012V2+ sdot sdot sdot = V

0minus 119901int

119905

0

[minus (1199060+ 1199011199061

+ 11990121199062+ sdot sdot sdot)

120597

120597119909

(V0+ 119901V1+ 1199012V2+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[(V0+ 119901V1+ 1199012V2

+ sdot sdot sdot)

120597

120597119909

(V0+ 119901V1+ 1199012V2+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[

1

119877

(

1205972

1205971199092(V0+ 119901V1+ 1199012V2+ sdot sdot sdot)

+

1205972

1205971199102(V0+ 119901V1+ 1199012V2+ sdot sdot sdot))] 119889120585

(38)

Comparing the coefficient of like powers of 119901 we get

1199010

1199060=

3

4

minus

1

4 [1 + 119890(119910minus119909)1198778

]

V0=

3

4

+

1

4 [1 + 119890(119910minus119909)1198778

]

1199011

1199061=

1

128

119877119890minus(18)(minus119910+119909)119877

119905

[1 + 119890minus(18)(minus119910+119909)119877

]2

V1=

minus1

128

119877119890minus(18)(minus119910+119909)119877

119905

[1 + 119890minus(18)(minus119910+119909)119877

]2

1199012

1199062=

minus1

2048

1198772119890minus(14)(minus119910+119909)119877

1199052

[1 + 119890minus(18)(minus119910+119909)119877

]4

V2=

1

2048

1198772119890minus(14)(minus119910+119909)119877

1199052

[1 + 119890minus(18)(minus119910+119909)119877

]4

1199013

1199063=

1

32768

1198773119890minus(38)(minus119910+119909)119877

1199053

[1 + 119890minus(18)(minus119910+119909)119877

]6

V3=

minus1

32768

1198773119890minus(38)(minus119910+119909)119877

1199053

[1 + 119890minus(18)(minus119910+119909)119877

]6

(39)

6 Journal of Applied Mathematics

The approximations solution is given by

119906 (119909 119910 119905) = 1199060+ 1199061+ 1199062+ 1199063+ sdot sdot sdot

V (119909 119910 119905) = V0+ V1+ V2+ V3+ sdot sdot sdot

(40)

Exact solution 119906lowast= 34 minus 14(1 + 119890

(4119910minus4119909minus119905)11987732) Vlowast = 34 +

14(1 + 119890(4119910minus4119909minus119905)11987732

)The results are in Table 4

6 Conclusion

In this work the approximate solutions of 119899-dimensionalBurgersrsquo equations are obtained by combination of two pow-erful methods VIM and HPM in VHPMThe examples haveshown the efficiency and accuracy of the VHPM it reducesthe size of computation without the restrictive assumption tohandle nonlinear terms and it gives the solutions rapidly

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This paper was funded by King Abdulaziz City for Scienceand Technology (KACST) in Saudi Arabia The authorstherefore thank them for their full collaboration

References

[1] J H He ldquoA variational iteration approach to nonlinear prob-lems and its applicationsrdquoMechanical Application vol 20 no 1pp 30ndash31 1998

[2] J-H He ldquoVariational iteration method-a kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

[3] J-H He ldquoVariational iteration methodmdashsome recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[4] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[5] J-H He ldquoCoupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000

[6] J-H He ldquoHomotopy perturbation method a new nonlinearanalytical techniquerdquo Applied Mathematics and Computationvol 135 no 1 pp 73ndash79 2003

[7] A Naghipour and JManafian ldquoApplication of the Laplace Ado-mian decomposition and implicit methods for solving BurgersrsquoequationrdquoTWMS Journal of Pure and AppliedMathematics vol6 no 1 pp 68ndash77 2015

[8] T Ozis and Y Aslan ldquoThe semi-approximate approach forsolving Burgersrsquo equation with high Reynolds numberrdquo AppliedMathematics and Computation vol 163 no 1 pp 131ndash145 2005

[9] D L Young C M Fan S P Hu and S N Atluri ldquoTheEulerian-Lagrangianmethod of fundamental solutions for two-dimensional unsteady Burgersrsquo equationsrdquo Engineering Analysiswith Boundary Elements vol 32 no 5 pp 395ndash412 2008

[10] M A Abdou and A A Soliman ldquoVariational iteration methodfor solving BURgerrsquos and coupled BURgerrsquos equationsrdquo Journalof Computational and Applied Mathematics vol 181 no 2 pp245ndash251 2005

[11] A-M Wazwaz ldquoThe variational iteration method for solvinglinear and nonlinear systems of PDEsrdquoComputersampMathemat-ics with Applications vol 54 no 7-8 pp 895ndash902 2007

[12] A-MWazwaz ldquoA comparison between the variational iterationmethod and Adomian decomposition methodrdquo Journal ofComputational andAppliedMathematics vol 207 no 1 pp 129ndash136 2007

[13] J Biazar and H Aminikhah ldquoExact and numerical solutionsfor non-linear Burgerrsquos equation by VIMrdquo Mathematical andComputer Modelling vol 49 no 7-8 pp 1394ndash1400 2009

[14] B BatihaVarational ItrationalMethod and Its Applications LAPLAMBERT Academic Publishing GmbH Co KG GermangSaarbrucken Germany 2012

[15] S Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer Berlin Germany 2012

[16] K R Desai and V H Pradhan ldquoSolution of Burgerrsquos equationand coupled Burgerrsquos equations by Homotopy perturbationmethodrdquo International Journal of Engineering Research andApplications vol 2 no 3 pp 2033ndash2040 2012

[17] V K Srivastava and M K Awasthi ldquo(1 + n)-dimensional Burg-ersrsquo equation and its analytical solution a comparative study ofHPM ADM and DTMrdquo Ain Shams Engineering Journal vol 5no 2 pp 533ndash541 2014

[18] J Ahmad Z Bibi andKNoor ldquoLaplace decompositionmethodusing hersquos polynomials to burgers equationrdquo Journal of Scienceand Arts vol 2 no 27 pp 131ndash138 2014

[19] M Matinfar M Mahdavi and Z Raeisy ldquoThe implementationof variational homotopy perturbationmethod for Fisherrsquos equa-tionrdquo International Journal of Nonlinear Science vol 9 no 2 pp188ndash194 2010

[20] T Allahviranloo A Armand and S Pirmohammadi ldquoVaria-tional homotopy perturbation method an efficient scheme forsolving partial differential equations in fluid mechanicsrdquo TheJournal of Mathematics and Computer Science vol 9 no 4 pp362ndash369 2014

[21] A Daga and V Pradhan ldquoA novel approach for solving burgerrsquosequationrdquo Applications amp Applied Mathematics vol 9 no 2 pp541ndash552 2014

[22] Y Chen E Fan and M Yuen ldquoThe Hopf-Cole transformationtopological solitons and multiple fusion solutions for the n-dimensional Burgers systemrdquo Physics Letters A vol 380 no 1-2pp 9ndash14 2016

[23] S Hernandez G M Carlomagno and C A Brebbia Com-putational Methods and Experimental Measurements XVI WITPress 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

Table 2 Comparison of VHPM solutions with exact solution at 119909 =

01 and 119910 = 01 (Example 2)

119905 119906lowast(119909 119910 119905) 119906(119909 119910 119905) |119906

lowastminus 119906|

001 02020202020 02020202020 0002 02040816327 02040816326 1 times 10

minus10

003 02061855670 02061855669 1 times 10minus10

004 02083333333 02083333325 8 times 10minus10

005 02105263158 02105263125 33 times 10minus9

006 02127659574 02127659475 99 times 10minus9

007 02150537634 02150537381 253 times 10minus8

008 02173913043 02173912474 569 times 10minus8

009 02197802198 02197801030 1168 times 10minus7

010 02222222222 02222220000 2222 times 10minus7

with the initial condition

119906 (119909 119910 119911 0) = 1199060(119909 119910 119911) = 119909 + 119910 + 119911 (30)

We have

119880119899+1

= 1199060minus 119901int

119905

0

[minus119880119899(119909 119910 120585)

120597119880119899(119909 119910 120585)

120597119909

minus

1205972119880119899(119909 119910 120585)

1205971199092

minus

1205972119880119899(119909 119910 120585)

1205971199102

minus

1205972119880119899(119909 119910 120585)

1205971199112

]119889120585

1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot = 119909 + 119910 + 119911 minus 119901int

119905

0

[minus (1199060

+ 1199011199061+ 11990121199062

+ sdot sdot sdot)

120597

120597119909

(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[minus

1205972

1205971199092(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot) minus

1205972

1205971199102(1199060

+ 1199011199061+ 11990121199062+ sdot sdot sdot) minus

1205972

1205971199112(1199060+ 1199011199061+ 11990121199062

+ sdot sdot sdot)] 119889120585

(31)

Comparing the coefficient of like powers of 119901 we get

1199010 1199060= 119909 + 119910 + 119911

1199011 1199061= (119909 + 119910 + 119911) 119905

1199012 1199062= (119909 + 119910 + 119911) 119905

2

1199013 1199063= (119909 + 119910 + 119911) 119905

3

(32)

Table 3 Comparison of VHPM solutions with exact solution at 119909 =

01 119910 = 01 and 119911 = 01 (Example 3)

119905 119906lowast(119909 119910 119905) 119906(119909 119910 119905) |119906

lowastminus 119906|

001 03030303030 03030303030 0002 03061224490 03061224490 0003 03092783505 03092783503 2 times 10

minus10

004 03125000000 03124999987 13 times 10minus9

005 03157894737 03157894688 49 times 10minus9

006 03191489362 03191489213 149 times 10minus8

007 03225806452 03225806072 380 times 10minus8

008 03260869565 03260868710 855 times 10minus8

009 03296703297 03296701545 1752 times 10minus7

010 03333333333 03333330000 3333 times 10minus7

The approximations solution is given by

119906 (119909 119910 119911 119905) = 1199060+ 1199061+ 1199062+ 1199063+ sdot sdot sdot (33)

Exact solution is (119906lowast = (119909 + 119910 + 119911)(1 minus 119905))The results are in Table 3

Example 4 Consider two-dimensional Burgersrsquo equations[23]

120597119906

120597119905

+ 119906

120597119906

120597119909

+ V120597119906

120597119910

=

1

119877

(

1205972119906

1205971199092+

1205972119906

1205971199102)

120597V120597119905

+ 119906

120597V120597119909

+ V120597V120597119910

=

1

119877

(

1205972V

1205971199092+

1205972V1205971199102)

(34)

with the initial conditions

1199060= 119906 (119909 119910 0) =

3

4

minus

1

4 [1 + 119890(119910minus119909)1198778

]

V0= V (119909 119910 0) =

3

4

+

1

4 [1 + 119890(119910minus119909)1198778

]

(35)

The correction functional for (34) is

119880119899+1

= 119880119899+ 119901int

119905

0

1205821(120585) [

120597119880

120597119905

+ 119899(119909119910 120585)

120597119899(119909119910 120585)120597119909

+ 119899(119909119910 120585)

120597119899(119909119910 120585)120597119909

] 119889120585 + 119901int

119905

0

1205821(120585)

sdot [minus

1

119877

(

1205972119899(119909 119910 120585)

1205971199092

+

1205972119899(119909 119910 120585)

1205971199102

)]119889120585

Journal of Applied Mathematics 5

Table 4 Comparison of VHPM solutions with exact solution at 119909 = 1 119910 = 0 and 119877 = 100 (Example 4)

119905 119906lowast(119909 119910 119905) 119906(119909 119910 119905) |119906

lowastminus 119906| Vlowast(119909 119910 119905) V(119909 119910 119905) |Vlowast minus V|

001 05000009030 05000009571 541 times 10minus8 09999990970 09999990321 649 times 10

minus8

002 05000008752 05000009862 1110 times 10minus7 09999991248 09999990031 1217 times 10

minus7

003 05000008483 05000010153 1670 times 10minus7 09999991517 09999989741 1776 times 10

minus7

004 05000008222 05000010445 2223 times 10minus7 09999991778 09999989451 2327 times 10

minus7

005 05000007969 05000010736 2767 times 10minus7 09999992031 09999989161 2870 times 10

minus7

006 05000007724 05000011027 3303 times 10minus7 09999992276 09999988871 3405 times 10

minus7

007 05000007486 05000011318 3832 times 10minus7 09999992514 09999988581 3933 times 10

minus7

008 05000007256 05000011609 4353 times 10minus7 09999992744 09999988281 4463 times 10

minus7

009 05000007032 05000011900 4868 times 10minus7 09999992968 09999987991 4977 times 10

minus7

010 05000006816 05000012191 5375 times 10minus7 09999993184 09999987701 5483 times 10

minus7

119881119899+1

= 119881119899+ 119901int

119905

0

1205822(120585) [

120597119881

120597119905

+ 119899(119909119910 120585)

120597119899(119909119910 120585)120597119909

+ 119899(119909119910 120585)

120597119899(119909119910 120585)120597119909

] 119889120585 + 119901int

119905

0

1205822(120585)

sdot [minus

1

119877

(

1205972119899(119909119910 120585)1205971199092

+

1205972119899(119909119910 120585)1205971199102

)]119889120585

(36)

The general Lagrangian multipliers are 1205821= minus1 = 120582

2

Equation (11) can be written as

119880 =

infin

sum

ℓ=0

119901ℓ119906ℓ(119909 119910 119905)

119881 =

infin

sum

ℓ=0

119901ℓVℓ(119909 119910 119905)

(37)

By VHPM we have

1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot = 119906

0minus 119901int

119905

0

[minus (1199060+ 1199011199061

+ 11990121199062+ sdot sdot sdot)

120597

120597119909

(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[(V0+ 119901V1+ 1199012V2

+ sdot sdot sdot)

120597

120597119909

(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[

1

119877

(

1205972

1205971199092(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot)

+

1205972

1205971199102(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot))] 119889120585

V0+ 119901V1+ 1199012V2+ sdot sdot sdot = V

0minus 119901int

119905

0

[minus (1199060+ 1199011199061

+ 11990121199062+ sdot sdot sdot)

120597

120597119909

(V0+ 119901V1+ 1199012V2+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[(V0+ 119901V1+ 1199012V2

+ sdot sdot sdot)

120597

120597119909

(V0+ 119901V1+ 1199012V2+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[

1

119877

(

1205972

1205971199092(V0+ 119901V1+ 1199012V2+ sdot sdot sdot)

+

1205972

1205971199102(V0+ 119901V1+ 1199012V2+ sdot sdot sdot))] 119889120585

(38)

Comparing the coefficient of like powers of 119901 we get

1199010

1199060=

3

4

minus

1

4 [1 + 119890(119910minus119909)1198778

]

V0=

3

4

+

1

4 [1 + 119890(119910minus119909)1198778

]

1199011

1199061=

1

128

119877119890minus(18)(minus119910+119909)119877

119905

[1 + 119890minus(18)(minus119910+119909)119877

]2

V1=

minus1

128

119877119890minus(18)(minus119910+119909)119877

119905

[1 + 119890minus(18)(minus119910+119909)119877

]2

1199012

1199062=

minus1

2048

1198772119890minus(14)(minus119910+119909)119877

1199052

[1 + 119890minus(18)(minus119910+119909)119877

]4

V2=

1

2048

1198772119890minus(14)(minus119910+119909)119877

1199052

[1 + 119890minus(18)(minus119910+119909)119877

]4

1199013

1199063=

1

32768

1198773119890minus(38)(minus119910+119909)119877

1199053

[1 + 119890minus(18)(minus119910+119909)119877

]6

V3=

minus1

32768

1198773119890minus(38)(minus119910+119909)119877

1199053

[1 + 119890minus(18)(minus119910+119909)119877

]6

(39)

6 Journal of Applied Mathematics

The approximations solution is given by

119906 (119909 119910 119905) = 1199060+ 1199061+ 1199062+ 1199063+ sdot sdot sdot

V (119909 119910 119905) = V0+ V1+ V2+ V3+ sdot sdot sdot

(40)

Exact solution 119906lowast= 34 minus 14(1 + 119890

(4119910minus4119909minus119905)11987732) Vlowast = 34 +

14(1 + 119890(4119910minus4119909minus119905)11987732

)The results are in Table 4

6 Conclusion

In this work the approximate solutions of 119899-dimensionalBurgersrsquo equations are obtained by combination of two pow-erful methods VIM and HPM in VHPMThe examples haveshown the efficiency and accuracy of the VHPM it reducesthe size of computation without the restrictive assumption tohandle nonlinear terms and it gives the solutions rapidly

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This paper was funded by King Abdulaziz City for Scienceand Technology (KACST) in Saudi Arabia The authorstherefore thank them for their full collaboration

References

[1] J H He ldquoA variational iteration approach to nonlinear prob-lems and its applicationsrdquoMechanical Application vol 20 no 1pp 30ndash31 1998

[2] J-H He ldquoVariational iteration method-a kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

[3] J-H He ldquoVariational iteration methodmdashsome recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[4] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[5] J-H He ldquoCoupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000

[6] J-H He ldquoHomotopy perturbation method a new nonlinearanalytical techniquerdquo Applied Mathematics and Computationvol 135 no 1 pp 73ndash79 2003

[7] A Naghipour and JManafian ldquoApplication of the Laplace Ado-mian decomposition and implicit methods for solving BurgersrsquoequationrdquoTWMS Journal of Pure and AppliedMathematics vol6 no 1 pp 68ndash77 2015

[8] T Ozis and Y Aslan ldquoThe semi-approximate approach forsolving Burgersrsquo equation with high Reynolds numberrdquo AppliedMathematics and Computation vol 163 no 1 pp 131ndash145 2005

[9] D L Young C M Fan S P Hu and S N Atluri ldquoTheEulerian-Lagrangianmethod of fundamental solutions for two-dimensional unsteady Burgersrsquo equationsrdquo Engineering Analysiswith Boundary Elements vol 32 no 5 pp 395ndash412 2008

[10] M A Abdou and A A Soliman ldquoVariational iteration methodfor solving BURgerrsquos and coupled BURgerrsquos equationsrdquo Journalof Computational and Applied Mathematics vol 181 no 2 pp245ndash251 2005

[11] A-M Wazwaz ldquoThe variational iteration method for solvinglinear and nonlinear systems of PDEsrdquoComputersampMathemat-ics with Applications vol 54 no 7-8 pp 895ndash902 2007

[12] A-MWazwaz ldquoA comparison between the variational iterationmethod and Adomian decomposition methodrdquo Journal ofComputational andAppliedMathematics vol 207 no 1 pp 129ndash136 2007

[13] J Biazar and H Aminikhah ldquoExact and numerical solutionsfor non-linear Burgerrsquos equation by VIMrdquo Mathematical andComputer Modelling vol 49 no 7-8 pp 1394ndash1400 2009

[14] B BatihaVarational ItrationalMethod and Its Applications LAPLAMBERT Academic Publishing GmbH Co KG GermangSaarbrucken Germany 2012

[15] S Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer Berlin Germany 2012

[16] K R Desai and V H Pradhan ldquoSolution of Burgerrsquos equationand coupled Burgerrsquos equations by Homotopy perturbationmethodrdquo International Journal of Engineering Research andApplications vol 2 no 3 pp 2033ndash2040 2012

[17] V K Srivastava and M K Awasthi ldquo(1 + n)-dimensional Burg-ersrsquo equation and its analytical solution a comparative study ofHPM ADM and DTMrdquo Ain Shams Engineering Journal vol 5no 2 pp 533ndash541 2014

[18] J Ahmad Z Bibi andKNoor ldquoLaplace decompositionmethodusing hersquos polynomials to burgers equationrdquo Journal of Scienceand Arts vol 2 no 27 pp 131ndash138 2014

[19] M Matinfar M Mahdavi and Z Raeisy ldquoThe implementationof variational homotopy perturbationmethod for Fisherrsquos equa-tionrdquo International Journal of Nonlinear Science vol 9 no 2 pp188ndash194 2010

[20] T Allahviranloo A Armand and S Pirmohammadi ldquoVaria-tional homotopy perturbation method an efficient scheme forsolving partial differential equations in fluid mechanicsrdquo TheJournal of Mathematics and Computer Science vol 9 no 4 pp362ndash369 2014

[21] A Daga and V Pradhan ldquoA novel approach for solving burgerrsquosequationrdquo Applications amp Applied Mathematics vol 9 no 2 pp541ndash552 2014

[22] Y Chen E Fan and M Yuen ldquoThe Hopf-Cole transformationtopological solitons and multiple fusion solutions for the n-dimensional Burgers systemrdquo Physics Letters A vol 380 no 1-2pp 9ndash14 2016

[23] S Hernandez G M Carlomagno and C A Brebbia Com-putational Methods and Experimental Measurements XVI WITPress 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

Table 4 Comparison of VHPM solutions with exact solution at 119909 = 1 119910 = 0 and 119877 = 100 (Example 4)

119905 119906lowast(119909 119910 119905) 119906(119909 119910 119905) |119906

lowastminus 119906| Vlowast(119909 119910 119905) V(119909 119910 119905) |Vlowast minus V|

001 05000009030 05000009571 541 times 10minus8 09999990970 09999990321 649 times 10

minus8

002 05000008752 05000009862 1110 times 10minus7 09999991248 09999990031 1217 times 10

minus7

003 05000008483 05000010153 1670 times 10minus7 09999991517 09999989741 1776 times 10

minus7

004 05000008222 05000010445 2223 times 10minus7 09999991778 09999989451 2327 times 10

minus7

005 05000007969 05000010736 2767 times 10minus7 09999992031 09999989161 2870 times 10

minus7

006 05000007724 05000011027 3303 times 10minus7 09999992276 09999988871 3405 times 10

minus7

007 05000007486 05000011318 3832 times 10minus7 09999992514 09999988581 3933 times 10

minus7

008 05000007256 05000011609 4353 times 10minus7 09999992744 09999988281 4463 times 10

minus7

009 05000007032 05000011900 4868 times 10minus7 09999992968 09999987991 4977 times 10

minus7

010 05000006816 05000012191 5375 times 10minus7 09999993184 09999987701 5483 times 10

minus7

119881119899+1

= 119881119899+ 119901int

119905

0

1205822(120585) [

120597119881

120597119905

+ 119899(119909119910 120585)

120597119899(119909119910 120585)120597119909

+ 119899(119909119910 120585)

120597119899(119909119910 120585)120597119909

] 119889120585 + 119901int

119905

0

1205822(120585)

sdot [minus

1

119877

(

1205972119899(119909119910 120585)1205971199092

+

1205972119899(119909119910 120585)1205971199102

)]119889120585

(36)

The general Lagrangian multipliers are 1205821= minus1 = 120582

2

Equation (11) can be written as

119880 =

infin

sum

ℓ=0

119901ℓ119906ℓ(119909 119910 119905)

119881 =

infin

sum

ℓ=0

119901ℓVℓ(119909 119910 119905)

(37)

By VHPM we have

1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot = 119906

0minus 119901int

119905

0

[minus (1199060+ 1199011199061

+ 11990121199062+ sdot sdot sdot)

120597

120597119909

(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[(V0+ 119901V1+ 1199012V2

+ sdot sdot sdot)

120597

120597119909

(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[

1

119877

(

1205972

1205971199092(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot)

+

1205972

1205971199102(1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot))] 119889120585

V0+ 119901V1+ 1199012V2+ sdot sdot sdot = V

0minus 119901int

119905

0

[minus (1199060+ 1199011199061

+ 11990121199062+ sdot sdot sdot)

120597

120597119909

(V0+ 119901V1+ 1199012V2+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[(V0+ 119901V1+ 1199012V2

+ sdot sdot sdot)

120597

120597119909

(V0+ 119901V1+ 1199012V2+ sdot sdot sdot )] 119889120585

minus 119901int

119905

0

[

1

119877

(

1205972

1205971199092(V0+ 119901V1+ 1199012V2+ sdot sdot sdot)

+

1205972

1205971199102(V0+ 119901V1+ 1199012V2+ sdot sdot sdot))] 119889120585

(38)

Comparing the coefficient of like powers of 119901 we get

1199010

1199060=

3

4

minus

1

4 [1 + 119890(119910minus119909)1198778

]

V0=

3

4

+

1

4 [1 + 119890(119910minus119909)1198778

]

1199011

1199061=

1

128

119877119890minus(18)(minus119910+119909)119877

119905

[1 + 119890minus(18)(minus119910+119909)119877

]2

V1=

minus1

128

119877119890minus(18)(minus119910+119909)119877

119905

[1 + 119890minus(18)(minus119910+119909)119877

]2

1199012

1199062=

minus1

2048

1198772119890minus(14)(minus119910+119909)119877

1199052

[1 + 119890minus(18)(minus119910+119909)119877

]4

V2=

1

2048

1198772119890minus(14)(minus119910+119909)119877

1199052

[1 + 119890minus(18)(minus119910+119909)119877

]4

1199013

1199063=

1

32768

1198773119890minus(38)(minus119910+119909)119877

1199053

[1 + 119890minus(18)(minus119910+119909)119877

]6

V3=

minus1

32768

1198773119890minus(38)(minus119910+119909)119877

1199053

[1 + 119890minus(18)(minus119910+119909)119877

]6

(39)

6 Journal of Applied Mathematics

The approximations solution is given by

119906 (119909 119910 119905) = 1199060+ 1199061+ 1199062+ 1199063+ sdot sdot sdot

V (119909 119910 119905) = V0+ V1+ V2+ V3+ sdot sdot sdot

(40)

Exact solution 119906lowast= 34 minus 14(1 + 119890

(4119910minus4119909minus119905)11987732) Vlowast = 34 +

14(1 + 119890(4119910minus4119909minus119905)11987732

)The results are in Table 4

6 Conclusion

In this work the approximate solutions of 119899-dimensionalBurgersrsquo equations are obtained by combination of two pow-erful methods VIM and HPM in VHPMThe examples haveshown the efficiency and accuracy of the VHPM it reducesthe size of computation without the restrictive assumption tohandle nonlinear terms and it gives the solutions rapidly

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This paper was funded by King Abdulaziz City for Scienceand Technology (KACST) in Saudi Arabia The authorstherefore thank them for their full collaboration

References

[1] J H He ldquoA variational iteration approach to nonlinear prob-lems and its applicationsrdquoMechanical Application vol 20 no 1pp 30ndash31 1998

[2] J-H He ldquoVariational iteration method-a kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

[3] J-H He ldquoVariational iteration methodmdashsome recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[4] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[5] J-H He ldquoCoupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000

[6] J-H He ldquoHomotopy perturbation method a new nonlinearanalytical techniquerdquo Applied Mathematics and Computationvol 135 no 1 pp 73ndash79 2003

[7] A Naghipour and JManafian ldquoApplication of the Laplace Ado-mian decomposition and implicit methods for solving BurgersrsquoequationrdquoTWMS Journal of Pure and AppliedMathematics vol6 no 1 pp 68ndash77 2015

[8] T Ozis and Y Aslan ldquoThe semi-approximate approach forsolving Burgersrsquo equation with high Reynolds numberrdquo AppliedMathematics and Computation vol 163 no 1 pp 131ndash145 2005

[9] D L Young C M Fan S P Hu and S N Atluri ldquoTheEulerian-Lagrangianmethod of fundamental solutions for two-dimensional unsteady Burgersrsquo equationsrdquo Engineering Analysiswith Boundary Elements vol 32 no 5 pp 395ndash412 2008

[10] M A Abdou and A A Soliman ldquoVariational iteration methodfor solving BURgerrsquos and coupled BURgerrsquos equationsrdquo Journalof Computational and Applied Mathematics vol 181 no 2 pp245ndash251 2005

[11] A-M Wazwaz ldquoThe variational iteration method for solvinglinear and nonlinear systems of PDEsrdquoComputersampMathemat-ics with Applications vol 54 no 7-8 pp 895ndash902 2007

[12] A-MWazwaz ldquoA comparison between the variational iterationmethod and Adomian decomposition methodrdquo Journal ofComputational andAppliedMathematics vol 207 no 1 pp 129ndash136 2007

[13] J Biazar and H Aminikhah ldquoExact and numerical solutionsfor non-linear Burgerrsquos equation by VIMrdquo Mathematical andComputer Modelling vol 49 no 7-8 pp 1394ndash1400 2009

[14] B BatihaVarational ItrationalMethod and Its Applications LAPLAMBERT Academic Publishing GmbH Co KG GermangSaarbrucken Germany 2012

[15] S Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer Berlin Germany 2012

[16] K R Desai and V H Pradhan ldquoSolution of Burgerrsquos equationand coupled Burgerrsquos equations by Homotopy perturbationmethodrdquo International Journal of Engineering Research andApplications vol 2 no 3 pp 2033ndash2040 2012

[17] V K Srivastava and M K Awasthi ldquo(1 + n)-dimensional Burg-ersrsquo equation and its analytical solution a comparative study ofHPM ADM and DTMrdquo Ain Shams Engineering Journal vol 5no 2 pp 533ndash541 2014

[18] J Ahmad Z Bibi andKNoor ldquoLaplace decompositionmethodusing hersquos polynomials to burgers equationrdquo Journal of Scienceand Arts vol 2 no 27 pp 131ndash138 2014

[19] M Matinfar M Mahdavi and Z Raeisy ldquoThe implementationof variational homotopy perturbationmethod for Fisherrsquos equa-tionrdquo International Journal of Nonlinear Science vol 9 no 2 pp188ndash194 2010

[20] T Allahviranloo A Armand and S Pirmohammadi ldquoVaria-tional homotopy perturbation method an efficient scheme forsolving partial differential equations in fluid mechanicsrdquo TheJournal of Mathematics and Computer Science vol 9 no 4 pp362ndash369 2014

[21] A Daga and V Pradhan ldquoA novel approach for solving burgerrsquosequationrdquo Applications amp Applied Mathematics vol 9 no 2 pp541ndash552 2014

[22] Y Chen E Fan and M Yuen ldquoThe Hopf-Cole transformationtopological solitons and multiple fusion solutions for the n-dimensional Burgers systemrdquo Physics Letters A vol 380 no 1-2pp 9ndash14 2016

[23] S Hernandez G M Carlomagno and C A Brebbia Com-putational Methods and Experimental Measurements XVI WITPress 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

The approximations solution is given by

119906 (119909 119910 119905) = 1199060+ 1199061+ 1199062+ 1199063+ sdot sdot sdot

V (119909 119910 119905) = V0+ V1+ V2+ V3+ sdot sdot sdot

(40)

Exact solution 119906lowast= 34 minus 14(1 + 119890

(4119910minus4119909minus119905)11987732) Vlowast = 34 +

14(1 + 119890(4119910minus4119909minus119905)11987732

)The results are in Table 4

6 Conclusion

In this work the approximate solutions of 119899-dimensionalBurgersrsquo equations are obtained by combination of two pow-erful methods VIM and HPM in VHPMThe examples haveshown the efficiency and accuracy of the VHPM it reducesthe size of computation without the restrictive assumption tohandle nonlinear terms and it gives the solutions rapidly

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This paper was funded by King Abdulaziz City for Scienceand Technology (KACST) in Saudi Arabia The authorstherefore thank them for their full collaboration

References

[1] J H He ldquoA variational iteration approach to nonlinear prob-lems and its applicationsrdquoMechanical Application vol 20 no 1pp 30ndash31 1998

[2] J-H He ldquoVariational iteration method-a kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

[3] J-H He ldquoVariational iteration methodmdashsome recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[4] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[5] J-H He ldquoCoupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000

[6] J-H He ldquoHomotopy perturbation method a new nonlinearanalytical techniquerdquo Applied Mathematics and Computationvol 135 no 1 pp 73ndash79 2003

[7] A Naghipour and JManafian ldquoApplication of the Laplace Ado-mian decomposition and implicit methods for solving BurgersrsquoequationrdquoTWMS Journal of Pure and AppliedMathematics vol6 no 1 pp 68ndash77 2015

[8] T Ozis and Y Aslan ldquoThe semi-approximate approach forsolving Burgersrsquo equation with high Reynolds numberrdquo AppliedMathematics and Computation vol 163 no 1 pp 131ndash145 2005

[9] D L Young C M Fan S P Hu and S N Atluri ldquoTheEulerian-Lagrangianmethod of fundamental solutions for two-dimensional unsteady Burgersrsquo equationsrdquo Engineering Analysiswith Boundary Elements vol 32 no 5 pp 395ndash412 2008

[10] M A Abdou and A A Soliman ldquoVariational iteration methodfor solving BURgerrsquos and coupled BURgerrsquos equationsrdquo Journalof Computational and Applied Mathematics vol 181 no 2 pp245ndash251 2005

[11] A-M Wazwaz ldquoThe variational iteration method for solvinglinear and nonlinear systems of PDEsrdquoComputersampMathemat-ics with Applications vol 54 no 7-8 pp 895ndash902 2007

[12] A-MWazwaz ldquoA comparison between the variational iterationmethod and Adomian decomposition methodrdquo Journal ofComputational andAppliedMathematics vol 207 no 1 pp 129ndash136 2007

[13] J Biazar and H Aminikhah ldquoExact and numerical solutionsfor non-linear Burgerrsquos equation by VIMrdquo Mathematical andComputer Modelling vol 49 no 7-8 pp 1394ndash1400 2009

[14] B BatihaVarational ItrationalMethod and Its Applications LAPLAMBERT Academic Publishing GmbH Co KG GermangSaarbrucken Germany 2012

[15] S Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer Berlin Germany 2012

[16] K R Desai and V H Pradhan ldquoSolution of Burgerrsquos equationand coupled Burgerrsquos equations by Homotopy perturbationmethodrdquo International Journal of Engineering Research andApplications vol 2 no 3 pp 2033ndash2040 2012

[17] V K Srivastava and M K Awasthi ldquo(1 + n)-dimensional Burg-ersrsquo equation and its analytical solution a comparative study ofHPM ADM and DTMrdquo Ain Shams Engineering Journal vol 5no 2 pp 533ndash541 2014

[18] J Ahmad Z Bibi andKNoor ldquoLaplace decompositionmethodusing hersquos polynomials to burgers equationrdquo Journal of Scienceand Arts vol 2 no 27 pp 131ndash138 2014

[19] M Matinfar M Mahdavi and Z Raeisy ldquoThe implementationof variational homotopy perturbationmethod for Fisherrsquos equa-tionrdquo International Journal of Nonlinear Science vol 9 no 2 pp188ndash194 2010

[20] T Allahviranloo A Armand and S Pirmohammadi ldquoVaria-tional homotopy perturbation method an efficient scheme forsolving partial differential equations in fluid mechanicsrdquo TheJournal of Mathematics and Computer Science vol 9 no 4 pp362ndash369 2014

[21] A Daga and V Pradhan ldquoA novel approach for solving burgerrsquosequationrdquo Applications amp Applied Mathematics vol 9 no 2 pp541ndash552 2014

[22] Y Chen E Fan and M Yuen ldquoThe Hopf-Cole transformationtopological solitons and multiple fusion solutions for the n-dimensional Burgers systemrdquo Physics Letters A vol 380 no 1-2pp 9ndash14 2016

[23] S Hernandez G M Carlomagno and C A Brebbia Com-putational Methods and Experimental Measurements XVI WITPress 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of