Research ArticleA New Efficient Method for Solving Two-Dimensional NonlinearSystem of Burgerrsquos Differential Equations

Shams A Ahmed 1 Mohamed Elbadri 1 and Mohamed Z Mohamed 2

1Department of Mathematic Faculty of Sciences and Arts Jouf University Tubarjal Saudi Arabia2Department of Mathematic Deanship of Preparatory Programs University of Hail Hail Saudi Arabia

Correspondence should be addressed to Shams A Ahmed shamsalden20hotmailcom

Received 29 October 2019 Revised 29 December 2019 Accepted 7 January 2020 Published 11 February 2020

Academic Editor Kunquan Lan

Copyright copy 2020 Shams A Ahmed et al )is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

In this work the Sumudu decomposition method (SDM) is utilized to obtain the approximate solution of two-dimensionalnonlinear system of Burgerrsquos differential equations )is method is considered to be an effective tool in solving many problemsOur results have shown that the SDM offers a much better approximation for solving several numbers of systems of two-dimensional nonlinear Burgerrsquos differential equations To clarify the facility and accuracy of the strategy two examplesare provided

1 Introduction

Burgerrsquos equation is one of the foremost necessary partialdifferential equations in fluid mechanics )is equationdemonstrates the coupling between diffusion and convec-tion processes Burgerrsquos equation describes the structure ofshock waves traffic flow and acoustic transmission Ad-ditionally like this it also appears in varied areas of appliedmathematics and physics such as modelling of gas dynamics[1ndash5] Recently many numerical and analytical methodshave been used to study the two-dimensional Burgerrsquosequation such as the differential transformation method [6]homotopy perturbation method [7] homotopy analysismethod [8] variational iteration method [9] Adomiandecomposition method [10ndash12] cubic B-spline differentialquadrature method [13] finite difference method [14] finiteelement [15] and local discontinuous Galerkin finite ele-ment method [16] and also mathematicians have usedtransformmethods coupled with analytical methods [17ndash30]to solve PDEs )e Sumudu decomposition method (SDM)is one of these methods and it has been successfully used tosolve intricate problems in engineering mathematics and

applied science [31ndash35] )e SDM was first introduced byKumar [36] to solve nonlinear partial differential equationsthat show in all aspects of applied science and engineering)is method is an elegant combination of the Sumudutransform method and the Adomian decompositionmethod )e SDM method generates the solution in a seriesform whose components are determined by a recursiverelationship

In the current study we consider the system of two-dimensional nonlinear Burgerrsquos equations [9]

θt + θθρ + αθσ 1R

θρρ + θσσ1113872 1113873

αt + θαρ + αασ 1R

αρρ + ασσ1113872 1113873

(1)

with the initial conditions

θ(ρ σ 0) w(ρ σ) ρ σ isin E

α(ρ σ 0) h(ρ σ) ρ σ isin E(2)

HindawiAbstract and Applied AnalysisVolume 2020 Article ID 7413859 7 pageshttpsdoiorg10115520207413859

and the boundary conditions

θ(ρ σ t) w1(ρ σ t) ρ σ isin zE

α(ρ σ t) h1(ρ σ t) ρ σ isin zE (3)

where E (ρ σ) | ale ρle b ale σ le b1113864 1113865 and zE is its bound-ary θ(ρ σ t) and α(ρ σ t) are the velocity components to bedetermined w h w1 and h1 are the known functions and R

is the Reynolds number)e major objective of this work is to get analytical and

numerical solutions of the system of two-dimensionalnonlinear Burgerrsquos equations (1) by using SDM)is work isorganized as follows the analysis of the method is given inSection 2 )e application of SDM to two examples is givenin Section 3 Concluding remarks are given in the lastsection

2 Analysis of the Method

Now to obtain the approximate solution of equation (1)apply the Sumudu transformation to equation (1) and usingthe given condition (2) gives

S[ θ(ρ σ u)] w(ρ σ) minus uS θθρ + αθσ1113960 1113961 + uS1Rnabla2θ1113872 11138731113876 1113877

S[α(ρ σ u)] h(ρ σ) minus uS θαρ + αασ1113960 1113961 + uS1Rnabla2α1113872 11138731113876 1113877

(4)

where nabla2 (z2zρ2) + (z2zσ2) Apply the inverse operatorSminus 1 to both sides of the equation (4) and it gives

θ(ρ σ t) w(ρ σ) minus Sminus 1

uS θθρ + αθσ1113960 11139611113960 1113961 + Sminus 1

uS1Rnabla2θ1113872 11138731113876 11138771113876 1113877

α(ρ σ t) h(ρ σ) minus Sminus 1

uS θαρ + αασ1113960 11139611113960 1113961 + Sminus 1

uS1Rnabla2α1113872 11138731113876 11138771113876 1113877

(5)

)e Adomian decomposition method suggests that thelinear terms θ(ρ σ t) and α(ρ σ t) and the nonlinear termsθθρ αθσ θαρ and αασ are decomposed by an infinite seriesof components

θ(ρ σ t) 1113944

infin

n0θn(ρ σ t)

θθρ 1113944infin

n0An

αθσ 1113944infin

n0Bn

α(ρ σ t) 1113944infin

n0αn(ρ σ t)

θαρ 1113944infin

n0Cn

αασ 1113944infin

n0Cn

(6)

For some Adomian polynomials An(θ) are given by

An θ0 θ1 θ2 θn( 1113857 1n

dn

dλn N 1113944

infin

n0λnθn

⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦

λ0

n 0 1 2

(7)

Substituting equation (6) into both sides of equation (5)leads to

1113944

infin

n0θn(ρ σ t) w(ρ σ) minus S

minus 1uS 1113944infin

n0An

⎡⎣ ⎤⎦ + 1113944infin

n0Bn

⎡⎣ ⎤⎦⎡⎣ ⎤⎦⎡⎣ ⎤⎦

+ Sminus 1

uS1Rnabla2 1113944

infin

n0θn(ρ σ t)⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦⎡⎢⎢⎣ ⎤⎥⎥⎦

1113944

infin

n0αn(ρ σ t) h(ρ σ) minus S

minus 1uS 1113944infin

n0Cn

⎡⎣ ⎤⎦ + 1113944infin

n0Dn

⎡⎣ ⎤⎦⎡⎣ ⎤⎦⎡⎣ ⎤⎦

+ Sminus 1

uS1Rnabla2 1113944

infin

n0αn(ρ σ t)⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦⎡⎢⎢⎣ ⎤⎥⎥⎦

(8)

To construct the recursive relation needed for the de-termination of the components (θ0 θ1 θ2 θn) and(α0 α1 α2 αn) it is important to note that the Adomianmethod suggests that the zeroth components θ0 and α0 areusually defined by the functions w(ρ σ) and h(ρ σ)

Accordingly the formal recursive relation is defined in(Figures 1 and 2)

θ0 (ρ σ t) w(ρ σ)

θk+1(ρ σ t) minus Sminus 1

uS Ak + Bk1113858 11138591113858 1113859

+ Sminus 1

uS1Rnabla2 θk( 11138571113872 11138731113876 11138771113876 1113877 kge 0

α0(ρ σ t) h(ρ σ)

αk+1(x y t) minus Sminus 1

uS Ck + Dk1113858 11138591113858 1113859

+ Sminus 1

uS1Rnabla2 αk( 11138571113872 11138731113876 11138771113876 1113877 kge 0

(9)

Having determined these components substitute it intoθ(ρ σ t) 1113936

infinn0θn(ρ σ t) and α(ρ σ t) 1113936

infinn0αn(ρ σ t) to

obtain the solution in a series form

3 Application

In this part two examples are provided to illustrate themethod

Example 1 Consider the system of two-dimensional Bur-gerrsquos equation (1) with the following initial conditions [9]

2 Abstract and Applied Analysis

θ(ρ σ 0) ρ + σ ρ σ isin E

α(ρ σ 0) ρ minus σ ρ σ isin E(10)

Solution Subsequent to the discussion presented above thesystem of equation (8) becomes

1113944

infin

n0θn(ρ σ t) ρ + σ minus S

minus 1uS 1113944infin

n0An

⎡⎣ ⎤⎦ + 1113944infin

n0Bn

⎡⎣ ⎤⎦⎡⎣ ⎤⎦⎡⎣ ⎤⎦

+ Sminus 1

uS1Rnabla2 1113944

infin

n0θn(ρ σ t)⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦⎡⎢⎢⎣ ⎤⎥⎥⎦

1113944

infin

n0αn(ρ σ t) ρ minus σ minus S

minus 1uS 1113944infin

n0Cn

⎡⎣ ⎤⎦ + 1113944infin

n0Dn

⎡⎣ ⎤⎦⎡⎣ ⎤⎦⎡⎣ ⎤⎦

+ Sminus 1

uS1Rnabla2 1113944

infin

n0αn(ρ σ t)⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦⎡⎢⎢⎣ ⎤⎥⎥⎦

(11)

)e recursive relation can be constructed from equation(11) given by

θ0 (ρ σ t) ρ + σ

θk+1(ρ σ t) minus Sminus 1

uS Ak + Bk1113858 11138591113858 1113859

+ Sminus 1

uS1Rnabla2 θk( 11138571113872 11138731113876 11138771113876 1113877 kge 0

α0(ρ σ t) ρ minus σ

αk+1(x y t) minus Sminus 1

uS Ck + Dk1113858 11138591113858 1113859

+ Sminus 1

uS1Rnabla2 αk( 11138571113872 11138731113876 11138771113876 1113877 kge 0

(12)

We get the next couple of components and upon settingR 1 we have

θ0 α0( 1113857 (ρ + σ ρ minus σ )

θ1 α1( 1113857 (minus 2ρt minus 2σt)

θ2 α2( 1113857 2ρt2

+ 2σt2 2ρt

2minus 2σt

21113872 1113873

θ3 α3( 1113857 minus 4ρt3 minus 4σt

31113872 1113873

θ4 α4( 1113857 4ρt4

+ 4σt4 4ρt

4minus 4σt

41113872 1113873

(13)

00 0005 05

10 10

07

06

05

(a)

07

06

0500 00

05 0510 10

(b)

Figure 1 Distributions of approximation solutions for θ(ρ σ t) at (a) t 001 and (b) t 05 with R 100 for Example 2

00

05

10

08

09

10

00

05

10

(a)

10

09

08

00 00

05 05

10 10

(b)

Figure 2 Distributions of approximation solutions for α(ρ σ t) at (a) t 001 and (b) t 05 with R 100 for Example 2

Abstract and Applied Analysis 3

and so on Consequently the solution in a series form isgiven by

(θ α) ρ 1 + 2t2 + 4t4 + middot middot middot( 1113857 minus 2ρt 1 + 2t2 + middot middot middot( 1113857 + σ 1 + 2t2 + 4t4 + middot middot middot( 1113857

ρ 1 + 2t2 + 4t4 + middot middot middot( 1113857 minus 2σt 1 + 2t2 + middot middot middot( 1113857 minus σ 1 + 2t2 + 4t4 + middot middot middot( 11138571113888 1113889 (14)

and in a closed form it is

( θ(ρ σ t) α(ρ σ t)) ρ + σ minus 2ρt

1 minus 2t2ρ minus σ minus 2σt

1 minus 2t21113874 1113875 (15)

which is the exact solution of two-dimensional Burgerrsquosequations [9]

Example 2 Consider another system of Burgerrsquos equations(1) with the following initial conditions [9]

Table 1 )e (SDM) results for θ(ρ σ t) for first four approximations for R 1 with mesh points ρ 01 and σ 01 for Example 2

t Exact θ(ρ σ t) SDM θ(ρ σ t) |θExact minus θSDM|

005 06249023437698682 06249023437698682 001 06248046876589456 06248046876589457 11102230E minus 16015 06247070317864406 06247070317864418 12212453E minus 1502 06246093762715608 06246093762715658 49960036E minus 15025 06245117212335117 06245117212335268 15099033E minus 1403 06244140667914967 06244140667915343 37636561E minus 14035 06243164130647162 06243164130647978 81601392E minus 1404 06242187601723671 06242187601725259 15876189E minus 13045 06241211082336422 06241211082339287 28654856E minus 1305 06240234573677299 0624023457368215 48505644E minus 13

Table 2 )e (SDM) results for α(ρ σ t) for first four approximations for R 1 with mesh points ρ 01 and σ 01 for Example 2

t Exact α(ρ σ t) SDM α(ρ σ t) |αExact minus αSDM|

005 0875097656230132 0875097656230132 001 0875195312341054 0875195312341054 11102230E minus 16015 0875292968213559 0875292968213558 12212453E minus 1502 0875390623728439 0875390623728434 49960036E minus 15025 0875488278766488 0875488278766473 15099033E minus 1403 0875585933208503 0875585933208466 37636561E minus 14035 0875683586935284 0875683586935202 81601392E minus 1404 0875781239827633 0875781239827474 15887291E minus 13045 0875878891766358 0875878891766071 28654856E minus 1305 087597654263227 0875976542631785 48505644E minus 13

Table 3 )e (SDM) results for θ(ρ σ t) for first four approximations and for R 1 with mesh points ρ 03 and σ 01 for Example 2

t Exact θ(ρ σ t) SDM θ(ρ σ t) |θExact minus θSDM|

005 06233399413556532 06233399413556536 44408921E minus 1601 06232423033622646 06232423033622709 63282712E minus 15015 06231446675140766 06231446675141092 32529535E minus 1402 06230470339301989 06230470339303031 10424994E minus 13025 06229494027297301 06229494027299877 25757174E minus 1303 06228517740317571 06228517740322975 54045657E minus 13035 06227541479553547 06227541479563675 10127454E minus 1204 06226565246195848 06226565144551626 17474910E minus 12045 0622558904143496 06225588896714487 28308467E minus 1205 06224612866461229 0622461286650486 43630655E minus 12

4 Abstract and Applied Analysis

θ(ρ σ 0) 34

minus1

4 1 + e(R(σminus ρ)8)( 1113857

α(ρ σ 0) 34

+1

4 1 + e(R(σminus ρ)8)( 1113857

(16)

with the exact solutions

θ(ρ σ t) 34

minus1

4 1 + e(R(4σminus 4ρminus t)32)( 1113857

α(ρ σ t) 34

+1

4 1 + e(R(4σminus 4ρminus t)32)( 1113857

(17)

Solution Using the previous aforesaid discussion we get

θ0 (ρ σ t) 34

minus1

4 1 + e(R(minus ρ+σ)8)( 1113857

α0(ρ σ t) 34

+1

4 1 + e(R(minus ρ+σ)8)( 1113857

θ1 (ρ σ t) minuse(18)R(minus ρ+σ)Rt

64 1 + e(18)R(minus ρ+σ)( 11138573 minus

e(14)R(minus ρ+σ)Rt

64 1 + e(18)R(minus ρ+σ)( 11138573 +

e(18)R(minus ρ+σ)Rt

128 1 + e(18)R(minus ρ+σ)( 11138572

α1(ρ σ t) e(18)R(minus ρ+σ)Rt

64 1 + e(18)R(minus ρ+σ)( 11138573 +

e(14)R(minus ρ+σ)Rt

64 1 + e(18)R(minus ρ+σ)( 11138573 minus

e(18)R(minus ρ+σ)Rt

128 1 + e(18)R(minus ρ+σ)( 11138572

θ2 (ρ σ t) minuse(Rρ8)+(Rσ8) minus e(Rρ8) + e(Rσ8)( 1113857R2t2

8192 e(Rρ8) + e(Rσ8)( 11138573

α2(ρ σ t) e(Rρ8)+(Rσ8) minus e(Rρ8) + e(Rσ8)( 1113857R2t2

8192 e(Rρ8) + e(Rσ8)( 11138573

θ3 (ρ σ t) minuse(Rρ8)+(Rσ8) e(Rρ4) + e(Rσ4) minus 4e(Rρ8)+(Rσ8)( 1113857R3t3

786432 e(Rρ8)+(Rσ8)( 11138574

α3(ρ σ t) e(Rρ8)+(Rσ8) e(Rρ4) + e(Rρ4) minus 4e(Rρ8)+(Rσ8)( 1113857R3t3

786432 e(Rρ8)+(Rσ8)( 11138574

(18)

)erefore the solution θ(ρ σ t) and α(ρ σ t) in theseries form is given by

θ(ρ σ t) θ0(ρ σ t) + θ1(ρ σ t) + θ2(ρ σ t) + θ3(ρ σ t)

α(ρ σ t) α0(ρ σ t) + α1(ρ σ t) + α2(ρ σ t) + α3(ρ σ t)

(19)

Numerical outcomes shown in Tables 1ndash4 illustrate thatthe accuracy of SDM agrees good with the exact solutions ofthe system of two-dimensional Burgerrsquos equation and ab-solute errors are very small for the present choice of ρ σ Rand t

Table 4 )e (SDM) results for α(ρ σ t) for first four approximations and for R 1 with mesh points ρ 03 and σ 01 for Example 2

t Exact α(ρ σ t) SDM α(ρ σ t) |αExact minus αSDM|

005 0876660058644347 0876660058644346 44408921E minus 1601 0876757696637735 0876757696637729 63282712E minus 15015 0876855332485923 0876855332485891 32529535E minus 1402 0876952966069801 0876952966069697 10424994E minus 13025 087705059727027 0877050597270012 25757174E minus 1303 0877148225968243 0877148225967703 54045657E minus 13035 0877245852044645 0877245852043633 10127454E minus 1204 0877343475380415 0877343485544838 17474910E minus 12045 0877441095856504 0877441095853673 28309577E minus 1205 0877538713353877 0877538713349514 43630655E minus 12

Abstract and Applied Analysis 5

4 Conclusion

In this paper SDM had been successfully applied to find thesolutions of the system of two-dimensional nonlinearBurgerrsquos equations )e numerical studies showed that SDMoffers accurate results for two-dimensional nonlinear Bur-gerrsquos equations in comparison with another analyticalmethods )is fact is shown in the second example)erefore this method may be a favourable method to solveother nonlinear partial differential equations

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

References

[1] J M Burger ldquoA mathematical model illustrating the theory ofturbulencerdquo Advances in Applied Mechanics vol 1 pp 171ndash199 1948

[2] J D Cole ldquoOn a quasi-linear parabolic equation occurring inaerodynamicsrdquo Quarterly of Applied Mathematics vol 9no 3 pp 225ndash236 1951

[3] E Hopf ldquo)e partial differential equation ut + uux μuxxrdquoCommunications on Pure and Applied Mathematics vol 3no 3 pp 201ndash230 1950

[4] P D Lax Hyperbolic Systems of Conservation Laws and theMathematical 1eory of Shock Waves SIAM PhiladelphiaPA USA 1973

[5] J Smoller Shock Waves and ReactionndashDiffusion EquationsSpringer-Verlag Berlin Germany 1983

[6] R Abazari and A Borhanifar ldquoNumerical study of the so-lution of the Burgers and coupled Burgers equations by adifferential transformation methodrdquo Computers amp Mathe-matics with Applications vol 59 no 8 pp 2711ndash2722 2010

[7] A Molabahrami F Khani and S Hamedi-Nezhad ldquoSolitonsolutions of the two-dimensional KdV-Burgers equation byhomotopy perturbation methodrdquo Physics Letters A vol 370no 5-6 pp 433ndash436 2007

[8] M Inc ldquoOn numerical solution of Burgerrsquos equation byhomotopy analysis methodrdquo Physics Letters A vol 372 no 4pp 356ndash360 2008

[9] J Biazar and H Aminikhah ldquoExact and numerical solutionsfor non-linear Burgerrsquos equation by VIMrdquoMathematical andComputer Modelling vol 49 pp 1394ndash1400 2009

[10] M Dehghan A Hamidi andM Shakourifar ldquo)e solution ofcoupled Burgerrsquos equations using Adomianndashpade techniquerdquoApplied Mathematics and Computation vol 189 pp 1034ndash1047 2007

[11] A Gorguis ldquoA comparison between Cole-Hopf transfor-mation and the decomposition method for solving Burgerrsquosequationsrdquo Applied Mathematics and Computation vol 173no 1 pp 126ndash136 2006

[12] A Alharbi and E S Fahmy ldquoAdmndashpade solutions for gen-eralized Burgerrsquos and BurgerrsquosndashHuxley systems with twocoupled equationsrdquo Journal of Computational and AppliedMathematics vol 233 no 8 pp 2071ndash2080 2010

[13] B K Singh and P Kumar ldquoA novel approach for numericalcomputation of Burgerrsquos equation in (1+1) and (2+1)

dimensionsrdquo Alexandria Engineering Journal vol 55 no 4pp 3331ndash3344 2016

[14] A H A Ali and G A Gardner ldquoA collocation solution forBurgerrsquos equation using cubic B-spline finite elementsrdquoComputer Methods in Applied Mechanics and Engineeringvol 100 no 3 pp 325ndash337 1992

[15] M Basto V Semiao and F Calheiros ldquoDynamics andsynchronization of numerical solutions of the Burgerrsquosequationrdquo Journal of Computational and Applied Mathe-matics vol 231 no 2 pp 793ndash806 2009

[16] G Zhao X Yu and R Zhang ldquo)e new numerical method forsolving the system of two-dimensional Burgerrsquos equationsrdquoComputers amp Mathematics with Applications vol 62 no 8pp 3279ndash3291 2011

[17] M A Gondal S I Batool and M Khan ldquoA novel fractionalLaplace decomposition method for chaotic systems and thegeneration of chaotic sequencesrdquo Journal of Vibration andControl vol 20 no 16 pp 2530ndash2535 2014

[18] M Khan F Soleymani and M A Gondal ldquoA new analyticalsolution procedure for the motion of a spherical particle in aplane Couette flowrdquo Zeitschrift fur Naturforschung A vol 68no 5 pp 319ndash326 2013

[19] A Salah M Khan and M A Gondal ldquoA novel solutionprocedure for fuzzy fractional heat equations by homotopyanalysis transformmethodrdquoNeural Comput amp Applic vol 23no 2 pp 269ndash271 2013

[20] M Khan M A Gondal and K Omrani ldquoA new analyticalapproach to two-dimensional magneto-hydrodynamics flowover a nonlinear porous stretching sheet by Laplace Padedecomposition methodrdquo International Journal of Results inMathematics vol 63 pp 289ndash301 2013

[21] M A Gondal A Salah M Khan and S I Batool ldquoA novelanalytical solution of a fractional diffusion problem byhomotopy analysis transform methodrdquo Neural Comput ampApplic vol 23 no 6 pp 1643ndash1647 2013

[22] M Khan M A Gondal and S I Batool ldquoA new modifiedLaplace decomposition method for higher order boundaryvalue problemsrdquo Computational and Mathematical Organi-zation 1eory vol 19 no 4 pp 446ndash459 2013

[23] M Khan and M A Gondal ldquoA reliable treatment of Abelrsquossecond kind singular integral equationsrdquo Applied Mathe-matics Letters vol 25 no 11 pp 1666ndash1670 2012

[24] M Khan M A Gondal and S Kumar ldquoA new analyticalsolution procedure for nonlinear integral equationsrdquo Math-ematical and Computer Modelling vol 55 no 7-8pp 1892ndash1897 2012

[25] M Khan and M A Gondal ldquoNew computational dynamicsfor magnetohydrodynamics flow over a nonlinear stretchingsheetrdquo Zeitschrift fur Naturforschung A vol 67 no 5pp 262ndash266 2012

[26] M Khan M A Gondal and S I Batool ldquoA novel analyticalimplementation of nonlinear volterra integral equationsrdquoZeitschrift Fur Naturforschung vol 67 no 12 pp 674 ndash 6782012

[27] M Khan M A Gondal I Hussain and S Karimi Vanani ldquoAnew comparative study between homotopy analysis transformmethod and homotopy perturbation transform method on asemi infinite domainrdquo Mathematical and Computer Model-ling vol 55 no 3-4 pp 1143ndash1150 2012

[28] M Khan and M Hussain ldquoApplication of Laplace decom-position method on semi-infinite domainrdquo Numerical Algo-rithms vol 56 no 2 pp 211ndash218 2011

[29] M Khan and M A Gondal ldquoHomotopy perturbation padetransform method for blasius flow equation using Hersquos

6 Abstract and Applied Analysis

polynomialsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 12 no 1ndash7 2011

[30] M A Gondal andM Khan ldquoHomotopy perturbationmethodfor nonlinear exponential boundary layer equation usingLaplace transformation Hersquos polynomials and Pade tech-nologyrdquo International Journal of Nonlinear Science and Nu-merical Simulation vol 11 no 12 pp 1145ndash1153 2010

[31] S A Ahmed ldquoA comparison between modified Sumududecomposition method and homotopy perturbation methodrdquoApplied Mathematics vol 9 no 3 pp 199ndash206 2018

[32] S Ahmed and T Elzaki ldquo)e solution of nonlinear Volterraintegro-differential equations of second kind by combineSumudu transforms and Adomian decomposition methodrdquoInternational Journal of Advanced and Innovative Researchvol 2 no 12 pp 90ndash93 2013

[33] S Ahmed and T Elzaki ldquoA comparative study of Sumududecomposition method and Sumudu projected differentialtransform methodrdquo World Applied Sciences Journal vol 31no 10 pp 1704ndash1709 2014

[34] S Ahmed and T Elzaki ldquoSolution of heat and wavemdashlikeequations by adomian decomposition Sumudu transformmethodrdquo British Journal of Mathematics amp Computer Sciencevol 8 no 2 pp 101ndash111 2015

[35] S Ahmed and T Elzaki ldquoOn the comparative study inte-gromdashdifferential equations using difference numericalmethodsrdquo Journal of King Saud UniversitymdashScience vol 32no 1 pp 84ndash89 2018

[36] D Kumar J Singh and S Rathore ldquoSumudu decompositionmethod for nonlinear equationsrdquo International MathematicalForum vol 7 no 11 pp 515ndash521 2012

Abstract and Applied Analysis 7

θ(ρ σ 0) ρ + σ ρ σ isin E

α(ρ σ 0) ρ minus σ ρ σ isin E(10)

Solution Subsequent to the discussion presented above thesystem of equation (8) becomes

1113944

infin

n0θn(ρ σ t) ρ + σ minus S

minus 1uS 1113944infin

n0An

⎡⎣ ⎤⎦ + 1113944infin

n0Bn

⎡⎣ ⎤⎦⎡⎣ ⎤⎦⎡⎣ ⎤⎦

+ Sminus 1

uS1Rnabla2 1113944

infin

n0θn(ρ σ t)⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦⎡⎢⎢⎣ ⎤⎥⎥⎦

1113944

infin

n0αn(ρ σ t) ρ minus σ minus S

minus 1uS 1113944infin

n0Cn

⎡⎣ ⎤⎦ + 1113944infin

n0Dn

⎡⎣ ⎤⎦⎡⎣ ⎤⎦⎡⎣ ⎤⎦

+ Sminus 1

uS1Rnabla2 1113944

infin

n0αn(ρ σ t)⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦⎡⎢⎢⎣ ⎤⎥⎥⎦

(11)

)e recursive relation can be constructed from equation(11) given by

θ0 (ρ σ t) ρ + σ

θk+1(ρ σ t) minus Sminus 1

uS Ak + Bk1113858 11138591113858 1113859

+ Sminus 1

uS1Rnabla2 θk( 11138571113872 11138731113876 11138771113876 1113877 kge 0

α0(ρ σ t) ρ minus σ

αk+1(x y t) minus Sminus 1

uS Ck + Dk1113858 11138591113858 1113859

+ Sminus 1

uS1Rnabla2 αk( 11138571113872 11138731113876 11138771113876 1113877 kge 0

(12)

We get the next couple of components and upon settingR 1 we have

θ0 α0( 1113857 (ρ + σ ρ minus σ )

θ1 α1( 1113857 (minus 2ρt minus 2σt)

θ2 α2( 1113857 2ρt2

+ 2σt2 2ρt

2minus 2σt

21113872 1113873

θ3 α3( 1113857 minus 4ρt3 minus 4σt

31113872 1113873

θ4 α4( 1113857 4ρt4

+ 4σt4 4ρt

4minus 4σt

41113872 1113873

(13)

00 0005 05

10 10

07

06

05

(a)

07

06

0500 00

05 0510 10

(b)

Figure 1 Distributions of approximation solutions for θ(ρ σ t) at (a) t 001 and (b) t 05 with R 100 for Example 2

00

05

10

08

09

10

00

05

10

(a)

10

09

08

00 00

05 05

10 10

(b)

Figure 2 Distributions of approximation solutions for α(ρ σ t) at (a) t 001 and (b) t 05 with R 100 for Example 2

Abstract and Applied Analysis 3

and so on Consequently the solution in a series form isgiven by

(θ α) ρ 1 + 2t2 + 4t4 + middot middot middot( 1113857 minus 2ρt 1 + 2t2 + middot middot middot( 1113857 + σ 1 + 2t2 + 4t4 + middot middot middot( 1113857

ρ 1 + 2t2 + 4t4 + middot middot middot( 1113857 minus 2σt 1 + 2t2 + middot middot middot( 1113857 minus σ 1 + 2t2 + 4t4 + middot middot middot( 11138571113888 1113889 (14)

and in a closed form it is

( θ(ρ σ t) α(ρ σ t)) ρ + σ minus 2ρt

1 minus 2t2ρ minus σ minus 2σt

1 minus 2t21113874 1113875 (15)

which is the exact solution of two-dimensional Burgerrsquosequations [9]

Example 2 Consider another system of Burgerrsquos equations(1) with the following initial conditions [9]

Table 1 )e (SDM) results for θ(ρ σ t) for first four approximations for R 1 with mesh points ρ 01 and σ 01 for Example 2

t Exact θ(ρ σ t) SDM θ(ρ σ t) |θExact minus θSDM|

005 06249023437698682 06249023437698682 001 06248046876589456 06248046876589457 11102230E minus 16015 06247070317864406 06247070317864418 12212453E minus 1502 06246093762715608 06246093762715658 49960036E minus 15025 06245117212335117 06245117212335268 15099033E minus 1403 06244140667914967 06244140667915343 37636561E minus 14035 06243164130647162 06243164130647978 81601392E minus 1404 06242187601723671 06242187601725259 15876189E minus 13045 06241211082336422 06241211082339287 28654856E minus 1305 06240234573677299 0624023457368215 48505644E minus 13

Table 2 )e (SDM) results for α(ρ σ t) for first four approximations for R 1 with mesh points ρ 01 and σ 01 for Example 2

t Exact α(ρ σ t) SDM α(ρ σ t) |αExact minus αSDM|

005 0875097656230132 0875097656230132 001 0875195312341054 0875195312341054 11102230E minus 16015 0875292968213559 0875292968213558 12212453E minus 1502 0875390623728439 0875390623728434 49960036E minus 15025 0875488278766488 0875488278766473 15099033E minus 1403 0875585933208503 0875585933208466 37636561E minus 14035 0875683586935284 0875683586935202 81601392E minus 1404 0875781239827633 0875781239827474 15887291E minus 13045 0875878891766358 0875878891766071 28654856E minus 1305 087597654263227 0875976542631785 48505644E minus 13

Table 3 )e (SDM) results for θ(ρ σ t) for first four approximations and for R 1 with mesh points ρ 03 and σ 01 for Example 2

t Exact θ(ρ σ t) SDM θ(ρ σ t) |θExact minus θSDM|

005 06233399413556532 06233399413556536 44408921E minus 1601 06232423033622646 06232423033622709 63282712E minus 15015 06231446675140766 06231446675141092 32529535E minus 1402 06230470339301989 06230470339303031 10424994E minus 13025 06229494027297301 06229494027299877 25757174E minus 1303 06228517740317571 06228517740322975 54045657E minus 13035 06227541479553547 06227541479563675 10127454E minus 1204 06226565246195848 06226565144551626 17474910E minus 12045 0622558904143496 06225588896714487 28308467E minus 1205 06224612866461229 0622461286650486 43630655E minus 12

4 Abstract and Applied Analysis

θ(ρ σ 0) 34

minus1

4 1 + e(R(σminus ρ)8)( 1113857

α(ρ σ 0) 34

+1

4 1 + e(R(σminus ρ)8)( 1113857

(16)

with the exact solutions

θ(ρ σ t) 34

minus1

4 1 + e(R(4σminus 4ρminus t)32)( 1113857

α(ρ σ t) 34

+1

4 1 + e(R(4σminus 4ρminus t)32)( 1113857

(17)

Solution Using the previous aforesaid discussion we get

θ0 (ρ σ t) 34

minus1

4 1 + e(R(minus ρ+σ)8)( 1113857

α0(ρ σ t) 34

+1

4 1 + e(R(minus ρ+σ)8)( 1113857

θ1 (ρ σ t) minuse(18)R(minus ρ+σ)Rt

64 1 + e(18)R(minus ρ+σ)( 11138573 minus

e(14)R(minus ρ+σ)Rt

64 1 + e(18)R(minus ρ+σ)( 11138573 +

e(18)R(minus ρ+σ)Rt

128 1 + e(18)R(minus ρ+σ)( 11138572

α1(ρ σ t) e(18)R(minus ρ+σ)Rt

64 1 + e(18)R(minus ρ+σ)( 11138573 +

e(14)R(minus ρ+σ)Rt

64 1 + e(18)R(minus ρ+σ)( 11138573 minus

e(18)R(minus ρ+σ)Rt

128 1 + e(18)R(minus ρ+σ)( 11138572

θ2 (ρ σ t) minuse(Rρ8)+(Rσ8) minus e(Rρ8) + e(Rσ8)( 1113857R2t2

8192 e(Rρ8) + e(Rσ8)( 11138573

α2(ρ σ t) e(Rρ8)+(Rσ8) minus e(Rρ8) + e(Rσ8)( 1113857R2t2

8192 e(Rρ8) + e(Rσ8)( 11138573

θ3 (ρ σ t) minuse(Rρ8)+(Rσ8) e(Rρ4) + e(Rσ4) minus 4e(Rρ8)+(Rσ8)( 1113857R3t3

786432 e(Rρ8)+(Rσ8)( 11138574

α3(ρ σ t) e(Rρ8)+(Rσ8) e(Rρ4) + e(Rρ4) minus 4e(Rρ8)+(Rσ8)( 1113857R3t3

786432 e(Rρ8)+(Rσ8)( 11138574

(18)

)erefore the solution θ(ρ σ t) and α(ρ σ t) in theseries form is given by

θ(ρ σ t) θ0(ρ σ t) + θ1(ρ σ t) + θ2(ρ σ t) + θ3(ρ σ t)

α(ρ σ t) α0(ρ σ t) + α1(ρ σ t) + α2(ρ σ t) + α3(ρ σ t)

(19)

Numerical outcomes shown in Tables 1ndash4 illustrate thatthe accuracy of SDM agrees good with the exact solutions ofthe system of two-dimensional Burgerrsquos equation and ab-solute errors are very small for the present choice of ρ σ Rand t

Table 4 )e (SDM) results for α(ρ σ t) for first four approximations and for R 1 with mesh points ρ 03 and σ 01 for Example 2

t Exact α(ρ σ t) SDM α(ρ σ t) |αExact minus αSDM|

005 0876660058644347 0876660058644346 44408921E minus 1601 0876757696637735 0876757696637729 63282712E minus 15015 0876855332485923 0876855332485891 32529535E minus 1402 0876952966069801 0876952966069697 10424994E minus 13025 087705059727027 0877050597270012 25757174E minus 1303 0877148225968243 0877148225967703 54045657E minus 13035 0877245852044645 0877245852043633 10127454E minus 1204 0877343475380415 0877343485544838 17474910E minus 12045 0877441095856504 0877441095853673 28309577E minus 1205 0877538713353877 0877538713349514 43630655E minus 12

Abstract and Applied Analysis 5

4 Conclusion

In this paper SDM had been successfully applied to find thesolutions of the system of two-dimensional nonlinearBurgerrsquos equations )e numerical studies showed that SDMoffers accurate results for two-dimensional nonlinear Bur-gerrsquos equations in comparison with another analyticalmethods )is fact is shown in the second example)erefore this method may be a favourable method to solveother nonlinear partial differential equations

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

References

[1] J M Burger ldquoA mathematical model illustrating the theory ofturbulencerdquo Advances in Applied Mechanics vol 1 pp 171ndash199 1948

[2] J D Cole ldquoOn a quasi-linear parabolic equation occurring inaerodynamicsrdquo Quarterly of Applied Mathematics vol 9no 3 pp 225ndash236 1951

[3] E Hopf ldquo)e partial differential equation ut + uux μuxxrdquoCommunications on Pure and Applied Mathematics vol 3no 3 pp 201ndash230 1950

[4] P D Lax Hyperbolic Systems of Conservation Laws and theMathematical 1eory of Shock Waves SIAM PhiladelphiaPA USA 1973

[5] J Smoller Shock Waves and ReactionndashDiffusion EquationsSpringer-Verlag Berlin Germany 1983

[6] R Abazari and A Borhanifar ldquoNumerical study of the so-lution of the Burgers and coupled Burgers equations by adifferential transformation methodrdquo Computers amp Mathe-matics with Applications vol 59 no 8 pp 2711ndash2722 2010

[7] A Molabahrami F Khani and S Hamedi-Nezhad ldquoSolitonsolutions of the two-dimensional KdV-Burgers equation byhomotopy perturbation methodrdquo Physics Letters A vol 370no 5-6 pp 433ndash436 2007

[8] M Inc ldquoOn numerical solution of Burgerrsquos equation byhomotopy analysis methodrdquo Physics Letters A vol 372 no 4pp 356ndash360 2008

[9] J Biazar and H Aminikhah ldquoExact and numerical solutionsfor non-linear Burgerrsquos equation by VIMrdquoMathematical andComputer Modelling vol 49 pp 1394ndash1400 2009

[10] M Dehghan A Hamidi andM Shakourifar ldquo)e solution ofcoupled Burgerrsquos equations using Adomianndashpade techniquerdquoApplied Mathematics and Computation vol 189 pp 1034ndash1047 2007

[11] A Gorguis ldquoA comparison between Cole-Hopf transfor-mation and the decomposition method for solving Burgerrsquosequationsrdquo Applied Mathematics and Computation vol 173no 1 pp 126ndash136 2006

[12] A Alharbi and E S Fahmy ldquoAdmndashpade solutions for gen-eralized Burgerrsquos and BurgerrsquosndashHuxley systems with twocoupled equationsrdquo Journal of Computational and AppliedMathematics vol 233 no 8 pp 2071ndash2080 2010

[13] B K Singh and P Kumar ldquoA novel approach for numericalcomputation of Burgerrsquos equation in (1+1) and (2+1)

dimensionsrdquo Alexandria Engineering Journal vol 55 no 4pp 3331ndash3344 2016

[14] A H A Ali and G A Gardner ldquoA collocation solution forBurgerrsquos equation using cubic B-spline finite elementsrdquoComputer Methods in Applied Mechanics and Engineeringvol 100 no 3 pp 325ndash337 1992

[15] M Basto V Semiao and F Calheiros ldquoDynamics andsynchronization of numerical solutions of the Burgerrsquosequationrdquo Journal of Computational and Applied Mathe-matics vol 231 no 2 pp 793ndash806 2009

[16] G Zhao X Yu and R Zhang ldquo)e new numerical method forsolving the system of two-dimensional Burgerrsquos equationsrdquoComputers amp Mathematics with Applications vol 62 no 8pp 3279ndash3291 2011

[17] M A Gondal S I Batool and M Khan ldquoA novel fractionalLaplace decomposition method for chaotic systems and thegeneration of chaotic sequencesrdquo Journal of Vibration andControl vol 20 no 16 pp 2530ndash2535 2014

[18] M Khan F Soleymani and M A Gondal ldquoA new analyticalsolution procedure for the motion of a spherical particle in aplane Couette flowrdquo Zeitschrift fur Naturforschung A vol 68no 5 pp 319ndash326 2013

[19] A Salah M Khan and M A Gondal ldquoA novel solutionprocedure for fuzzy fractional heat equations by homotopyanalysis transformmethodrdquoNeural Comput amp Applic vol 23no 2 pp 269ndash271 2013

[20] M Khan M A Gondal and K Omrani ldquoA new analyticalapproach to two-dimensional magneto-hydrodynamics flowover a nonlinear porous stretching sheet by Laplace Padedecomposition methodrdquo International Journal of Results inMathematics vol 63 pp 289ndash301 2013

[21] M A Gondal A Salah M Khan and S I Batool ldquoA novelanalytical solution of a fractional diffusion problem byhomotopy analysis transform methodrdquo Neural Comput ampApplic vol 23 no 6 pp 1643ndash1647 2013

[22] M Khan M A Gondal and S I Batool ldquoA new modifiedLaplace decomposition method for higher order boundaryvalue problemsrdquo Computational and Mathematical Organi-zation 1eory vol 19 no 4 pp 446ndash459 2013

[23] M Khan and M A Gondal ldquoA reliable treatment of Abelrsquossecond kind singular integral equationsrdquo Applied Mathe-matics Letters vol 25 no 11 pp 1666ndash1670 2012

[24] M Khan M A Gondal and S Kumar ldquoA new analyticalsolution procedure for nonlinear integral equationsrdquo Math-ematical and Computer Modelling vol 55 no 7-8pp 1892ndash1897 2012

[25] M Khan and M A Gondal ldquoNew computational dynamicsfor magnetohydrodynamics flow over a nonlinear stretchingsheetrdquo Zeitschrift fur Naturforschung A vol 67 no 5pp 262ndash266 2012

[26] M Khan M A Gondal and S I Batool ldquoA novel analyticalimplementation of nonlinear volterra integral equationsrdquoZeitschrift Fur Naturforschung vol 67 no 12 pp 674 ndash 6782012

[27] M Khan M A Gondal I Hussain and S Karimi Vanani ldquoAnew comparative study between homotopy analysis transformmethod and homotopy perturbation transform method on asemi infinite domainrdquo Mathematical and Computer Model-ling vol 55 no 3-4 pp 1143ndash1150 2012

[28] M Khan and M Hussain ldquoApplication of Laplace decom-position method on semi-infinite domainrdquo Numerical Algo-rithms vol 56 no 2 pp 211ndash218 2011

[29] M Khan and M A Gondal ldquoHomotopy perturbation padetransform method for blasius flow equation using Hersquos

6 Abstract and Applied Analysis

polynomialsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 12 no 1ndash7 2011

[30] M A Gondal andM Khan ldquoHomotopy perturbationmethodfor nonlinear exponential boundary layer equation usingLaplace transformation Hersquos polynomials and Pade tech-nologyrdquo International Journal of Nonlinear Science and Nu-merical Simulation vol 11 no 12 pp 1145ndash1153 2010

[31] S A Ahmed ldquoA comparison between modified Sumududecomposition method and homotopy perturbation methodrdquoApplied Mathematics vol 9 no 3 pp 199ndash206 2018

[32] S Ahmed and T Elzaki ldquo)e solution of nonlinear Volterraintegro-differential equations of second kind by combineSumudu transforms and Adomian decomposition methodrdquoInternational Journal of Advanced and Innovative Researchvol 2 no 12 pp 90ndash93 2013

[33] S Ahmed and T Elzaki ldquoA comparative study of Sumududecomposition method and Sumudu projected differentialtransform methodrdquo World Applied Sciences Journal vol 31no 10 pp 1704ndash1709 2014

[34] S Ahmed and T Elzaki ldquoSolution of heat and wavemdashlikeequations by adomian decomposition Sumudu transformmethodrdquo British Journal of Mathematics amp Computer Sciencevol 8 no 2 pp 101ndash111 2015

[35] S Ahmed and T Elzaki ldquoOn the comparative study inte-gromdashdifferential equations using difference numericalmethodsrdquo Journal of King Saud UniversitymdashScience vol 32no 1 pp 84ndash89 2018

[36] D Kumar J Singh and S Rathore ldquoSumudu decompositionmethod for nonlinear equationsrdquo International MathematicalForum vol 7 no 11 pp 515ndash521 2012

Abstract and Applied Analysis 7

and so on Consequently the solution in a series form isgiven by

(θ α) ρ 1 + 2t2 + 4t4 + middot middot middot( 1113857 minus 2ρt 1 + 2t2 + middot middot middot( 1113857 + σ 1 + 2t2 + 4t4 + middot middot middot( 1113857

ρ 1 + 2t2 + 4t4 + middot middot middot( 1113857 minus 2σt 1 + 2t2 + middot middot middot( 1113857 minus σ 1 + 2t2 + 4t4 + middot middot middot( 11138571113888 1113889 (14)

and in a closed form it is

( θ(ρ σ t) α(ρ σ t)) ρ + σ minus 2ρt

1 minus 2t2ρ minus σ minus 2σt

1 minus 2t21113874 1113875 (15)

which is the exact solution of two-dimensional Burgerrsquosequations [9]

Example 2 Consider another system of Burgerrsquos equations(1) with the following initial conditions [9]

Table 1 )e (SDM) results for θ(ρ σ t) for first four approximations for R 1 with mesh points ρ 01 and σ 01 for Example 2

t Exact θ(ρ σ t) SDM θ(ρ σ t) |θExact minus θSDM|

005 06249023437698682 06249023437698682 001 06248046876589456 06248046876589457 11102230E minus 16015 06247070317864406 06247070317864418 12212453E minus 1502 06246093762715608 06246093762715658 49960036E minus 15025 06245117212335117 06245117212335268 15099033E minus 1403 06244140667914967 06244140667915343 37636561E minus 14035 06243164130647162 06243164130647978 81601392E minus 1404 06242187601723671 06242187601725259 15876189E minus 13045 06241211082336422 06241211082339287 28654856E minus 1305 06240234573677299 0624023457368215 48505644E minus 13

Table 2 )e (SDM) results for α(ρ σ t) for first four approximations for R 1 with mesh points ρ 01 and σ 01 for Example 2

t Exact α(ρ σ t) SDM α(ρ σ t) |αExact minus αSDM|

005 0875097656230132 0875097656230132 001 0875195312341054 0875195312341054 11102230E minus 16015 0875292968213559 0875292968213558 12212453E minus 1502 0875390623728439 0875390623728434 49960036E minus 15025 0875488278766488 0875488278766473 15099033E minus 1403 0875585933208503 0875585933208466 37636561E minus 14035 0875683586935284 0875683586935202 81601392E minus 1404 0875781239827633 0875781239827474 15887291E minus 13045 0875878891766358 0875878891766071 28654856E minus 1305 087597654263227 0875976542631785 48505644E minus 13

Table 3 )e (SDM) results for θ(ρ σ t) for first four approximations and for R 1 with mesh points ρ 03 and σ 01 for Example 2

t Exact θ(ρ σ t) SDM θ(ρ σ t) |θExact minus θSDM|

005 06233399413556532 06233399413556536 44408921E minus 1601 06232423033622646 06232423033622709 63282712E minus 15015 06231446675140766 06231446675141092 32529535E minus 1402 06230470339301989 06230470339303031 10424994E minus 13025 06229494027297301 06229494027299877 25757174E minus 1303 06228517740317571 06228517740322975 54045657E minus 13035 06227541479553547 06227541479563675 10127454E minus 1204 06226565246195848 06226565144551626 17474910E minus 12045 0622558904143496 06225588896714487 28308467E minus 1205 06224612866461229 0622461286650486 43630655E minus 12

4 Abstract and Applied Analysis

θ(ρ σ 0) 34

minus1

4 1 + e(R(σminus ρ)8)( 1113857

α(ρ σ 0) 34

+1

4 1 + e(R(σminus ρ)8)( 1113857

(16)

with the exact solutions

θ(ρ σ t) 34

minus1

4 1 + e(R(4σminus 4ρminus t)32)( 1113857

α(ρ σ t) 34

+1

4 1 + e(R(4σminus 4ρminus t)32)( 1113857

(17)

Solution Using the previous aforesaid discussion we get

θ0 (ρ σ t) 34

minus1

4 1 + e(R(minus ρ+σ)8)( 1113857

α0(ρ σ t) 34

+1

4 1 + e(R(minus ρ+σ)8)( 1113857

θ1 (ρ σ t) minuse(18)R(minus ρ+σ)Rt

64 1 + e(18)R(minus ρ+σ)( 11138573 minus

e(14)R(minus ρ+σ)Rt

64 1 + e(18)R(minus ρ+σ)( 11138573 +

e(18)R(minus ρ+σ)Rt

128 1 + e(18)R(minus ρ+σ)( 11138572

α1(ρ σ t) e(18)R(minus ρ+σ)Rt

64 1 + e(18)R(minus ρ+σ)( 11138573 +

e(14)R(minus ρ+σ)Rt

64 1 + e(18)R(minus ρ+σ)( 11138573 minus

e(18)R(minus ρ+σ)Rt

128 1 + e(18)R(minus ρ+σ)( 11138572

θ2 (ρ σ t) minuse(Rρ8)+(Rσ8) minus e(Rρ8) + e(Rσ8)( 1113857R2t2

8192 e(Rρ8) + e(Rσ8)( 11138573

α2(ρ σ t) e(Rρ8)+(Rσ8) minus e(Rρ8) + e(Rσ8)( 1113857R2t2

8192 e(Rρ8) + e(Rσ8)( 11138573

θ3 (ρ σ t) minuse(Rρ8)+(Rσ8) e(Rρ4) + e(Rσ4) minus 4e(Rρ8)+(Rσ8)( 1113857R3t3

786432 e(Rρ8)+(Rσ8)( 11138574

α3(ρ σ t) e(Rρ8)+(Rσ8) e(Rρ4) + e(Rρ4) minus 4e(Rρ8)+(Rσ8)( 1113857R3t3

786432 e(Rρ8)+(Rσ8)( 11138574

(18)

)erefore the solution θ(ρ σ t) and α(ρ σ t) in theseries form is given by

θ(ρ σ t) θ0(ρ σ t) + θ1(ρ σ t) + θ2(ρ σ t) + θ3(ρ σ t)

α(ρ σ t) α0(ρ σ t) + α1(ρ σ t) + α2(ρ σ t) + α3(ρ σ t)

(19)

Numerical outcomes shown in Tables 1ndash4 illustrate thatthe accuracy of SDM agrees good with the exact solutions ofthe system of two-dimensional Burgerrsquos equation and ab-solute errors are very small for the present choice of ρ σ Rand t

Table 4 )e (SDM) results for α(ρ σ t) for first four approximations and for R 1 with mesh points ρ 03 and σ 01 for Example 2

t Exact α(ρ σ t) SDM α(ρ σ t) |αExact minus αSDM|

005 0876660058644347 0876660058644346 44408921E minus 1601 0876757696637735 0876757696637729 63282712E minus 15015 0876855332485923 0876855332485891 32529535E minus 1402 0876952966069801 0876952966069697 10424994E minus 13025 087705059727027 0877050597270012 25757174E minus 1303 0877148225968243 0877148225967703 54045657E minus 13035 0877245852044645 0877245852043633 10127454E minus 1204 0877343475380415 0877343485544838 17474910E minus 12045 0877441095856504 0877441095853673 28309577E minus 1205 0877538713353877 0877538713349514 43630655E minus 12

Abstract and Applied Analysis 5

4 Conclusion

In this paper SDM had been successfully applied to find thesolutions of the system of two-dimensional nonlinearBurgerrsquos equations )e numerical studies showed that SDMoffers accurate results for two-dimensional nonlinear Bur-gerrsquos equations in comparison with another analyticalmethods )is fact is shown in the second example)erefore this method may be a favourable method to solveother nonlinear partial differential equations

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

References

[1] J M Burger ldquoA mathematical model illustrating the theory ofturbulencerdquo Advances in Applied Mechanics vol 1 pp 171ndash199 1948

[2] J D Cole ldquoOn a quasi-linear parabolic equation occurring inaerodynamicsrdquo Quarterly of Applied Mathematics vol 9no 3 pp 225ndash236 1951

[3] E Hopf ldquo)e partial differential equation ut + uux μuxxrdquoCommunications on Pure and Applied Mathematics vol 3no 3 pp 201ndash230 1950

[4] P D Lax Hyperbolic Systems of Conservation Laws and theMathematical 1eory of Shock Waves SIAM PhiladelphiaPA USA 1973

[5] J Smoller Shock Waves and ReactionndashDiffusion EquationsSpringer-Verlag Berlin Germany 1983

[6] R Abazari and A Borhanifar ldquoNumerical study of the so-lution of the Burgers and coupled Burgers equations by adifferential transformation methodrdquo Computers amp Mathe-matics with Applications vol 59 no 8 pp 2711ndash2722 2010

[7] A Molabahrami F Khani and S Hamedi-Nezhad ldquoSolitonsolutions of the two-dimensional KdV-Burgers equation byhomotopy perturbation methodrdquo Physics Letters A vol 370no 5-6 pp 433ndash436 2007

[8] M Inc ldquoOn numerical solution of Burgerrsquos equation byhomotopy analysis methodrdquo Physics Letters A vol 372 no 4pp 356ndash360 2008

[9] J Biazar and H Aminikhah ldquoExact and numerical solutionsfor non-linear Burgerrsquos equation by VIMrdquoMathematical andComputer Modelling vol 49 pp 1394ndash1400 2009

[10] M Dehghan A Hamidi andM Shakourifar ldquo)e solution ofcoupled Burgerrsquos equations using Adomianndashpade techniquerdquoApplied Mathematics and Computation vol 189 pp 1034ndash1047 2007

[11] A Gorguis ldquoA comparison between Cole-Hopf transfor-mation and the decomposition method for solving Burgerrsquosequationsrdquo Applied Mathematics and Computation vol 173no 1 pp 126ndash136 2006

[12] A Alharbi and E S Fahmy ldquoAdmndashpade solutions for gen-eralized Burgerrsquos and BurgerrsquosndashHuxley systems with twocoupled equationsrdquo Journal of Computational and AppliedMathematics vol 233 no 8 pp 2071ndash2080 2010

[13] B K Singh and P Kumar ldquoA novel approach for numericalcomputation of Burgerrsquos equation in (1+1) and (2+1)

dimensionsrdquo Alexandria Engineering Journal vol 55 no 4pp 3331ndash3344 2016

[14] A H A Ali and G A Gardner ldquoA collocation solution forBurgerrsquos equation using cubic B-spline finite elementsrdquoComputer Methods in Applied Mechanics and Engineeringvol 100 no 3 pp 325ndash337 1992

[15] M Basto V Semiao and F Calheiros ldquoDynamics andsynchronization of numerical solutions of the Burgerrsquosequationrdquo Journal of Computational and Applied Mathe-matics vol 231 no 2 pp 793ndash806 2009

[16] G Zhao X Yu and R Zhang ldquo)e new numerical method forsolving the system of two-dimensional Burgerrsquos equationsrdquoComputers amp Mathematics with Applications vol 62 no 8pp 3279ndash3291 2011

[17] M A Gondal S I Batool and M Khan ldquoA novel fractionalLaplace decomposition method for chaotic systems and thegeneration of chaotic sequencesrdquo Journal of Vibration andControl vol 20 no 16 pp 2530ndash2535 2014

[18] M Khan F Soleymani and M A Gondal ldquoA new analyticalsolution procedure for the motion of a spherical particle in aplane Couette flowrdquo Zeitschrift fur Naturforschung A vol 68no 5 pp 319ndash326 2013

[19] A Salah M Khan and M A Gondal ldquoA novel solutionprocedure for fuzzy fractional heat equations by homotopyanalysis transformmethodrdquoNeural Comput amp Applic vol 23no 2 pp 269ndash271 2013

[20] M Khan M A Gondal and K Omrani ldquoA new analyticalapproach to two-dimensional magneto-hydrodynamics flowover a nonlinear porous stretching sheet by Laplace Padedecomposition methodrdquo International Journal of Results inMathematics vol 63 pp 289ndash301 2013

[21] M A Gondal A Salah M Khan and S I Batool ldquoA novelanalytical solution of a fractional diffusion problem byhomotopy analysis transform methodrdquo Neural Comput ampApplic vol 23 no 6 pp 1643ndash1647 2013

[22] M Khan M A Gondal and S I Batool ldquoA new modifiedLaplace decomposition method for higher order boundaryvalue problemsrdquo Computational and Mathematical Organi-zation 1eory vol 19 no 4 pp 446ndash459 2013

[23] M Khan and M A Gondal ldquoA reliable treatment of Abelrsquossecond kind singular integral equationsrdquo Applied Mathe-matics Letters vol 25 no 11 pp 1666ndash1670 2012

[24] M Khan M A Gondal and S Kumar ldquoA new analyticalsolution procedure for nonlinear integral equationsrdquo Math-ematical and Computer Modelling vol 55 no 7-8pp 1892ndash1897 2012

[25] M Khan and M A Gondal ldquoNew computational dynamicsfor magnetohydrodynamics flow over a nonlinear stretchingsheetrdquo Zeitschrift fur Naturforschung A vol 67 no 5pp 262ndash266 2012

[26] M Khan M A Gondal and S I Batool ldquoA novel analyticalimplementation of nonlinear volterra integral equationsrdquoZeitschrift Fur Naturforschung vol 67 no 12 pp 674 ndash 6782012

[27] M Khan M A Gondal I Hussain and S Karimi Vanani ldquoAnew comparative study between homotopy analysis transformmethod and homotopy perturbation transform method on asemi infinite domainrdquo Mathematical and Computer Model-ling vol 55 no 3-4 pp 1143ndash1150 2012

[28] M Khan and M Hussain ldquoApplication of Laplace decom-position method on semi-infinite domainrdquo Numerical Algo-rithms vol 56 no 2 pp 211ndash218 2011

[29] M Khan and M A Gondal ldquoHomotopy perturbation padetransform method for blasius flow equation using Hersquos

6 Abstract and Applied Analysis

polynomialsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 12 no 1ndash7 2011

[30] M A Gondal andM Khan ldquoHomotopy perturbationmethodfor nonlinear exponential boundary layer equation usingLaplace transformation Hersquos polynomials and Pade tech-nologyrdquo International Journal of Nonlinear Science and Nu-merical Simulation vol 11 no 12 pp 1145ndash1153 2010

[31] S A Ahmed ldquoA comparison between modified Sumududecomposition method and homotopy perturbation methodrdquoApplied Mathematics vol 9 no 3 pp 199ndash206 2018

[32] S Ahmed and T Elzaki ldquo)e solution of nonlinear Volterraintegro-differential equations of second kind by combineSumudu transforms and Adomian decomposition methodrdquoInternational Journal of Advanced and Innovative Researchvol 2 no 12 pp 90ndash93 2013

[33] S Ahmed and T Elzaki ldquoA comparative study of Sumududecomposition method and Sumudu projected differentialtransform methodrdquo World Applied Sciences Journal vol 31no 10 pp 1704ndash1709 2014

[34] S Ahmed and T Elzaki ldquoSolution of heat and wavemdashlikeequations by adomian decomposition Sumudu transformmethodrdquo British Journal of Mathematics amp Computer Sciencevol 8 no 2 pp 101ndash111 2015

[35] S Ahmed and T Elzaki ldquoOn the comparative study inte-gromdashdifferential equations using difference numericalmethodsrdquo Journal of King Saud UniversitymdashScience vol 32no 1 pp 84ndash89 2018

[36] D Kumar J Singh and S Rathore ldquoSumudu decompositionmethod for nonlinear equationsrdquo International MathematicalForum vol 7 no 11 pp 515ndash521 2012

Abstract and Applied Analysis 7

θ(ρ σ 0) 34

minus1

4 1 + e(R(σminus ρ)8)( 1113857

α(ρ σ 0) 34

+1

4 1 + e(R(σminus ρ)8)( 1113857

(16)

with the exact solutions

θ(ρ σ t) 34

minus1

4 1 + e(R(4σminus 4ρminus t)32)( 1113857

α(ρ σ t) 34

+1

4 1 + e(R(4σminus 4ρminus t)32)( 1113857

(17)

Solution Using the previous aforesaid discussion we get

θ0 (ρ σ t) 34

minus1

4 1 + e(R(minus ρ+σ)8)( 1113857

α0(ρ σ t) 34

+1

4 1 + e(R(minus ρ+σ)8)( 1113857

θ1 (ρ σ t) minuse(18)R(minus ρ+σ)Rt

64 1 + e(18)R(minus ρ+σ)( 11138573 minus

e(14)R(minus ρ+σ)Rt

64 1 + e(18)R(minus ρ+σ)( 11138573 +

e(18)R(minus ρ+σ)Rt

128 1 + e(18)R(minus ρ+σ)( 11138572

α1(ρ σ t) e(18)R(minus ρ+σ)Rt

64 1 + e(18)R(minus ρ+σ)( 11138573 +

e(14)R(minus ρ+σ)Rt

64 1 + e(18)R(minus ρ+σ)( 11138573 minus

e(18)R(minus ρ+σ)Rt

128 1 + e(18)R(minus ρ+σ)( 11138572

θ2 (ρ σ t) minuse(Rρ8)+(Rσ8) minus e(Rρ8) + e(Rσ8)( 1113857R2t2

8192 e(Rρ8) + e(Rσ8)( 11138573

α2(ρ σ t) e(Rρ8)+(Rσ8) minus e(Rρ8) + e(Rσ8)( 1113857R2t2

8192 e(Rρ8) + e(Rσ8)( 11138573

θ3 (ρ σ t) minuse(Rρ8)+(Rσ8) e(Rρ4) + e(Rσ4) minus 4e(Rρ8)+(Rσ8)( 1113857R3t3

786432 e(Rρ8)+(Rσ8)( 11138574

α3(ρ σ t) e(Rρ8)+(Rσ8) e(Rρ4) + e(Rρ4) minus 4e(Rρ8)+(Rσ8)( 1113857R3t3

786432 e(Rρ8)+(Rσ8)( 11138574

(18)

)erefore the solution θ(ρ σ t) and α(ρ σ t) in theseries form is given by

θ(ρ σ t) θ0(ρ σ t) + θ1(ρ σ t) + θ2(ρ σ t) + θ3(ρ σ t)

α(ρ σ t) α0(ρ σ t) + α1(ρ σ t) + α2(ρ σ t) + α3(ρ σ t)

(19)

Numerical outcomes shown in Tables 1ndash4 illustrate thatthe accuracy of SDM agrees good with the exact solutions ofthe system of two-dimensional Burgerrsquos equation and ab-solute errors are very small for the present choice of ρ σ Rand t

Table 4 )e (SDM) results for α(ρ σ t) for first four approximations and for R 1 with mesh points ρ 03 and σ 01 for Example 2

t Exact α(ρ σ t) SDM α(ρ σ t) |αExact minus αSDM|

005 0876660058644347 0876660058644346 44408921E minus 1601 0876757696637735 0876757696637729 63282712E minus 15015 0876855332485923 0876855332485891 32529535E minus 1402 0876952966069801 0876952966069697 10424994E minus 13025 087705059727027 0877050597270012 25757174E minus 1303 0877148225968243 0877148225967703 54045657E minus 13035 0877245852044645 0877245852043633 10127454E minus 1204 0877343475380415 0877343485544838 17474910E minus 12045 0877441095856504 0877441095853673 28309577E minus 1205 0877538713353877 0877538713349514 43630655E minus 12

Abstract and Applied Analysis 5

4 Conclusion

In this paper SDM had been successfully applied to find thesolutions of the system of two-dimensional nonlinearBurgerrsquos equations )e numerical studies showed that SDMoffers accurate results for two-dimensional nonlinear Bur-gerrsquos equations in comparison with another analyticalmethods )is fact is shown in the second example)erefore this method may be a favourable method to solveother nonlinear partial differential equations

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

References

[1] J M Burger ldquoA mathematical model illustrating the theory ofturbulencerdquo Advances in Applied Mechanics vol 1 pp 171ndash199 1948

[2] J D Cole ldquoOn a quasi-linear parabolic equation occurring inaerodynamicsrdquo Quarterly of Applied Mathematics vol 9no 3 pp 225ndash236 1951

[3] E Hopf ldquo)e partial differential equation ut + uux μuxxrdquoCommunications on Pure and Applied Mathematics vol 3no 3 pp 201ndash230 1950

[4] P D Lax Hyperbolic Systems of Conservation Laws and theMathematical 1eory of Shock Waves SIAM PhiladelphiaPA USA 1973

[5] J Smoller Shock Waves and ReactionndashDiffusion EquationsSpringer-Verlag Berlin Germany 1983

[6] R Abazari and A Borhanifar ldquoNumerical study of the so-lution of the Burgers and coupled Burgers equations by adifferential transformation methodrdquo Computers amp Mathe-matics with Applications vol 59 no 8 pp 2711ndash2722 2010

[7] A Molabahrami F Khani and S Hamedi-Nezhad ldquoSolitonsolutions of the two-dimensional KdV-Burgers equation byhomotopy perturbation methodrdquo Physics Letters A vol 370no 5-6 pp 433ndash436 2007

[8] M Inc ldquoOn numerical solution of Burgerrsquos equation byhomotopy analysis methodrdquo Physics Letters A vol 372 no 4pp 356ndash360 2008

[9] J Biazar and H Aminikhah ldquoExact and numerical solutionsfor non-linear Burgerrsquos equation by VIMrdquoMathematical andComputer Modelling vol 49 pp 1394ndash1400 2009

[10] M Dehghan A Hamidi andM Shakourifar ldquo)e solution ofcoupled Burgerrsquos equations using Adomianndashpade techniquerdquoApplied Mathematics and Computation vol 189 pp 1034ndash1047 2007

[11] A Gorguis ldquoA comparison between Cole-Hopf transfor-mation and the decomposition method for solving Burgerrsquosequationsrdquo Applied Mathematics and Computation vol 173no 1 pp 126ndash136 2006

[12] A Alharbi and E S Fahmy ldquoAdmndashpade solutions for gen-eralized Burgerrsquos and BurgerrsquosndashHuxley systems with twocoupled equationsrdquo Journal of Computational and AppliedMathematics vol 233 no 8 pp 2071ndash2080 2010

[13] B K Singh and P Kumar ldquoA novel approach for numericalcomputation of Burgerrsquos equation in (1+1) and (2+1)

dimensionsrdquo Alexandria Engineering Journal vol 55 no 4pp 3331ndash3344 2016

[14] A H A Ali and G A Gardner ldquoA collocation solution forBurgerrsquos equation using cubic B-spline finite elementsrdquoComputer Methods in Applied Mechanics and Engineeringvol 100 no 3 pp 325ndash337 1992

[15] M Basto V Semiao and F Calheiros ldquoDynamics andsynchronization of numerical solutions of the Burgerrsquosequationrdquo Journal of Computational and Applied Mathe-matics vol 231 no 2 pp 793ndash806 2009

[16] G Zhao X Yu and R Zhang ldquo)e new numerical method forsolving the system of two-dimensional Burgerrsquos equationsrdquoComputers amp Mathematics with Applications vol 62 no 8pp 3279ndash3291 2011

[17] M A Gondal S I Batool and M Khan ldquoA novel fractionalLaplace decomposition method for chaotic systems and thegeneration of chaotic sequencesrdquo Journal of Vibration andControl vol 20 no 16 pp 2530ndash2535 2014

[18] M Khan F Soleymani and M A Gondal ldquoA new analyticalsolution procedure for the motion of a spherical particle in aplane Couette flowrdquo Zeitschrift fur Naturforschung A vol 68no 5 pp 319ndash326 2013

[19] A Salah M Khan and M A Gondal ldquoA novel solutionprocedure for fuzzy fractional heat equations by homotopyanalysis transformmethodrdquoNeural Comput amp Applic vol 23no 2 pp 269ndash271 2013

[20] M Khan M A Gondal and K Omrani ldquoA new analyticalapproach to two-dimensional magneto-hydrodynamics flowover a nonlinear porous stretching sheet by Laplace Padedecomposition methodrdquo International Journal of Results inMathematics vol 63 pp 289ndash301 2013

[21] M A Gondal A Salah M Khan and S I Batool ldquoA novelanalytical solution of a fractional diffusion problem byhomotopy analysis transform methodrdquo Neural Comput ampApplic vol 23 no 6 pp 1643ndash1647 2013

[22] M Khan M A Gondal and S I Batool ldquoA new modifiedLaplace decomposition method for higher order boundaryvalue problemsrdquo Computational and Mathematical Organi-zation 1eory vol 19 no 4 pp 446ndash459 2013

[23] M Khan and M A Gondal ldquoA reliable treatment of Abelrsquossecond kind singular integral equationsrdquo Applied Mathe-matics Letters vol 25 no 11 pp 1666ndash1670 2012

[24] M Khan M A Gondal and S Kumar ldquoA new analyticalsolution procedure for nonlinear integral equationsrdquo Math-ematical and Computer Modelling vol 55 no 7-8pp 1892ndash1897 2012

[25] M Khan and M A Gondal ldquoNew computational dynamicsfor magnetohydrodynamics flow over a nonlinear stretchingsheetrdquo Zeitschrift fur Naturforschung A vol 67 no 5pp 262ndash266 2012

[26] M Khan M A Gondal and S I Batool ldquoA novel analyticalimplementation of nonlinear volterra integral equationsrdquoZeitschrift Fur Naturforschung vol 67 no 12 pp 674 ndash 6782012

[27] M Khan M A Gondal I Hussain and S Karimi Vanani ldquoAnew comparative study between homotopy analysis transformmethod and homotopy perturbation transform method on asemi infinite domainrdquo Mathematical and Computer Model-ling vol 55 no 3-4 pp 1143ndash1150 2012

[28] M Khan and M Hussain ldquoApplication of Laplace decom-position method on semi-infinite domainrdquo Numerical Algo-rithms vol 56 no 2 pp 211ndash218 2011

[29] M Khan and M A Gondal ldquoHomotopy perturbation padetransform method for blasius flow equation using Hersquos

6 Abstract and Applied Analysis

polynomialsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 12 no 1ndash7 2011

[30] M A Gondal andM Khan ldquoHomotopy perturbationmethodfor nonlinear exponential boundary layer equation usingLaplace transformation Hersquos polynomials and Pade tech-nologyrdquo International Journal of Nonlinear Science and Nu-merical Simulation vol 11 no 12 pp 1145ndash1153 2010

[31] S A Ahmed ldquoA comparison between modified Sumududecomposition method and homotopy perturbation methodrdquoApplied Mathematics vol 9 no 3 pp 199ndash206 2018

[32] S Ahmed and T Elzaki ldquo)e solution of nonlinear Volterraintegro-differential equations of second kind by combineSumudu transforms and Adomian decomposition methodrdquoInternational Journal of Advanced and Innovative Researchvol 2 no 12 pp 90ndash93 2013

[33] S Ahmed and T Elzaki ldquoA comparative study of Sumududecomposition method and Sumudu projected differentialtransform methodrdquo World Applied Sciences Journal vol 31no 10 pp 1704ndash1709 2014

[34] S Ahmed and T Elzaki ldquoSolution of heat and wavemdashlikeequations by adomian decomposition Sumudu transformmethodrdquo British Journal of Mathematics amp Computer Sciencevol 8 no 2 pp 101ndash111 2015

[35] S Ahmed and T Elzaki ldquoOn the comparative study inte-gromdashdifferential equations using difference numericalmethodsrdquo Journal of King Saud UniversitymdashScience vol 32no 1 pp 84ndash89 2018

[36] D Kumar J Singh and S Rathore ldquoSumudu decompositionmethod for nonlinear equationsrdquo International MathematicalForum vol 7 no 11 pp 515ndash521 2012

Abstract and Applied Analysis 7

4 Conclusion

In this paper SDM had been successfully applied to find thesolutions of the system of two-dimensional nonlinearBurgerrsquos equations )e numerical studies showed that SDMoffers accurate results for two-dimensional nonlinear Bur-gerrsquos equations in comparison with another analyticalmethods )is fact is shown in the second example)erefore this method may be a favourable method to solveother nonlinear partial differential equations

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

References

[1] J M Burger ldquoA mathematical model illustrating the theory ofturbulencerdquo Advances in Applied Mechanics vol 1 pp 171ndash199 1948

[2] J D Cole ldquoOn a quasi-linear parabolic equation occurring inaerodynamicsrdquo Quarterly of Applied Mathematics vol 9no 3 pp 225ndash236 1951

[3] E Hopf ldquo)e partial differential equation ut + uux μuxxrdquoCommunications on Pure and Applied Mathematics vol 3no 3 pp 201ndash230 1950

[4] P D Lax Hyperbolic Systems of Conservation Laws and theMathematical 1eory of Shock Waves SIAM PhiladelphiaPA USA 1973

[5] J Smoller Shock Waves and ReactionndashDiffusion EquationsSpringer-Verlag Berlin Germany 1983

[6] R Abazari and A Borhanifar ldquoNumerical study of the so-lution of the Burgers and coupled Burgers equations by adifferential transformation methodrdquo Computers amp Mathe-matics with Applications vol 59 no 8 pp 2711ndash2722 2010

[7] A Molabahrami F Khani and S Hamedi-Nezhad ldquoSolitonsolutions of the two-dimensional KdV-Burgers equation byhomotopy perturbation methodrdquo Physics Letters A vol 370no 5-6 pp 433ndash436 2007

[8] M Inc ldquoOn numerical solution of Burgerrsquos equation byhomotopy analysis methodrdquo Physics Letters A vol 372 no 4pp 356ndash360 2008

[9] J Biazar and H Aminikhah ldquoExact and numerical solutionsfor non-linear Burgerrsquos equation by VIMrdquoMathematical andComputer Modelling vol 49 pp 1394ndash1400 2009

[10] M Dehghan A Hamidi andM Shakourifar ldquo)e solution ofcoupled Burgerrsquos equations using Adomianndashpade techniquerdquoApplied Mathematics and Computation vol 189 pp 1034ndash1047 2007

[11] A Gorguis ldquoA comparison between Cole-Hopf transfor-mation and the decomposition method for solving Burgerrsquosequationsrdquo Applied Mathematics and Computation vol 173no 1 pp 126ndash136 2006

[12] A Alharbi and E S Fahmy ldquoAdmndashpade solutions for gen-eralized Burgerrsquos and BurgerrsquosndashHuxley systems with twocoupled equationsrdquo Journal of Computational and AppliedMathematics vol 233 no 8 pp 2071ndash2080 2010

[13] B K Singh and P Kumar ldquoA novel approach for numericalcomputation of Burgerrsquos equation in (1+1) and (2+1)

dimensionsrdquo Alexandria Engineering Journal vol 55 no 4pp 3331ndash3344 2016

[14] A H A Ali and G A Gardner ldquoA collocation solution forBurgerrsquos equation using cubic B-spline finite elementsrdquoComputer Methods in Applied Mechanics and Engineeringvol 100 no 3 pp 325ndash337 1992

[15] M Basto V Semiao and F Calheiros ldquoDynamics andsynchronization of numerical solutions of the Burgerrsquosequationrdquo Journal of Computational and Applied Mathe-matics vol 231 no 2 pp 793ndash806 2009

[16] G Zhao X Yu and R Zhang ldquo)e new numerical method forsolving the system of two-dimensional Burgerrsquos equationsrdquoComputers amp Mathematics with Applications vol 62 no 8pp 3279ndash3291 2011

[17] M A Gondal S I Batool and M Khan ldquoA novel fractionalLaplace decomposition method for chaotic systems and thegeneration of chaotic sequencesrdquo Journal of Vibration andControl vol 20 no 16 pp 2530ndash2535 2014

[18] M Khan F Soleymani and M A Gondal ldquoA new analyticalsolution procedure for the motion of a spherical particle in aplane Couette flowrdquo Zeitschrift fur Naturforschung A vol 68no 5 pp 319ndash326 2013

[19] A Salah M Khan and M A Gondal ldquoA novel solutionprocedure for fuzzy fractional heat equations by homotopyanalysis transformmethodrdquoNeural Comput amp Applic vol 23no 2 pp 269ndash271 2013

[20] M Khan M A Gondal and K Omrani ldquoA new analyticalapproach to two-dimensional magneto-hydrodynamics flowover a nonlinear porous stretching sheet by Laplace Padedecomposition methodrdquo International Journal of Results inMathematics vol 63 pp 289ndash301 2013

[21] M A Gondal A Salah M Khan and S I Batool ldquoA novelanalytical solution of a fractional diffusion problem byhomotopy analysis transform methodrdquo Neural Comput ampApplic vol 23 no 6 pp 1643ndash1647 2013

[22] M Khan M A Gondal and S I Batool ldquoA new modifiedLaplace decomposition method for higher order boundaryvalue problemsrdquo Computational and Mathematical Organi-zation 1eory vol 19 no 4 pp 446ndash459 2013

[23] M Khan and M A Gondal ldquoA reliable treatment of Abelrsquossecond kind singular integral equationsrdquo Applied Mathe-matics Letters vol 25 no 11 pp 1666ndash1670 2012

[24] M Khan M A Gondal and S Kumar ldquoA new analyticalsolution procedure for nonlinear integral equationsrdquo Math-ematical and Computer Modelling vol 55 no 7-8pp 1892ndash1897 2012

[25] M Khan and M A Gondal ldquoNew computational dynamicsfor magnetohydrodynamics flow over a nonlinear stretchingsheetrdquo Zeitschrift fur Naturforschung A vol 67 no 5pp 262ndash266 2012

[26] M Khan M A Gondal and S I Batool ldquoA novel analyticalimplementation of nonlinear volterra integral equationsrdquoZeitschrift Fur Naturforschung vol 67 no 12 pp 674 ndash 6782012

[27] M Khan M A Gondal I Hussain and S Karimi Vanani ldquoAnew comparative study between homotopy analysis transformmethod and homotopy perturbation transform method on asemi infinite domainrdquo Mathematical and Computer Model-ling vol 55 no 3-4 pp 1143ndash1150 2012

[28] M Khan and M Hussain ldquoApplication of Laplace decom-position method on semi-infinite domainrdquo Numerical Algo-rithms vol 56 no 2 pp 211ndash218 2011

[29] M Khan and M A Gondal ldquoHomotopy perturbation padetransform method for blasius flow equation using Hersquos

6 Abstract and Applied Analysis

polynomialsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 12 no 1ndash7 2011

[30] M A Gondal andM Khan ldquoHomotopy perturbationmethodfor nonlinear exponential boundary layer equation usingLaplace transformation Hersquos polynomials and Pade tech-nologyrdquo International Journal of Nonlinear Science and Nu-merical Simulation vol 11 no 12 pp 1145ndash1153 2010

[31] S A Ahmed ldquoA comparison between modified Sumududecomposition method and homotopy perturbation methodrdquoApplied Mathematics vol 9 no 3 pp 199ndash206 2018

[32] S Ahmed and T Elzaki ldquo)e solution of nonlinear Volterraintegro-differential equations of second kind by combineSumudu transforms and Adomian decomposition methodrdquoInternational Journal of Advanced and Innovative Researchvol 2 no 12 pp 90ndash93 2013

[33] S Ahmed and T Elzaki ldquoA comparative study of Sumududecomposition method and Sumudu projected differentialtransform methodrdquo World Applied Sciences Journal vol 31no 10 pp 1704ndash1709 2014

[34] S Ahmed and T Elzaki ldquoSolution of heat and wavemdashlikeequations by adomian decomposition Sumudu transformmethodrdquo British Journal of Mathematics amp Computer Sciencevol 8 no 2 pp 101ndash111 2015

[35] S Ahmed and T Elzaki ldquoOn the comparative study inte-gromdashdifferential equations using difference numericalmethodsrdquo Journal of King Saud UniversitymdashScience vol 32no 1 pp 84ndash89 2018

[36] D Kumar J Singh and S Rathore ldquoSumudu decompositionmethod for nonlinear equationsrdquo International MathematicalForum vol 7 no 11 pp 515ndash521 2012

Abstract and Applied Analysis 7

polynomialsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 12 no 1ndash7 2011

[30] M A Gondal andM Khan ldquoHomotopy perturbationmethodfor nonlinear exponential boundary layer equation usingLaplace transformation Hersquos polynomials and Pade tech-nologyrdquo International Journal of Nonlinear Science and Nu-merical Simulation vol 11 no 12 pp 1145ndash1153 2010

[31] S A Ahmed ldquoA comparison between modified Sumududecomposition method and homotopy perturbation methodrdquoApplied Mathematics vol 9 no 3 pp 199ndash206 2018

[32] S Ahmed and T Elzaki ldquo)e solution of nonlinear Volterraintegro-differential equations of second kind by combineSumudu transforms and Adomian decomposition methodrdquoInternational Journal of Advanced and Innovative Researchvol 2 no 12 pp 90ndash93 2013

[33] S Ahmed and T Elzaki ldquoA comparative study of Sumududecomposition method and Sumudu projected differentialtransform methodrdquo World Applied Sciences Journal vol 31no 10 pp 1704ndash1709 2014

[34] S Ahmed and T Elzaki ldquoSolution of heat and wavemdashlikeequations by adomian decomposition Sumudu transformmethodrdquo British Journal of Mathematics amp Computer Sciencevol 8 no 2 pp 101ndash111 2015

[35] S Ahmed and T Elzaki ldquoOn the comparative study inte-gromdashdifferential equations using difference numericalmethodsrdquo Journal of King Saud UniversitymdashScience vol 32no 1 pp 84ndash89 2018

[36] D Kumar J Singh and S Rathore ldquoSumudu decompositionmethod for nonlinear equationsrdquo International MathematicalForum vol 7 no 11 pp 515ndash521 2012

Abstract and Applied Analysis 7