The Improved Modified Decomposition Method for the Treatment of Systems of Singular Nonlinear PartialDifferentialEquationswithInitialConditions

DALIR Nemat and NOURAZAR S.Salman

Department of Mechanical Engineering,Amirkabir University of Technology,Tehran 158754413,Iran.

Received 25 October 2012;Accepted 29 June 2014

The Improved Modified Decomposition Method for the Treatment of Systems of Singular Nonlinear PartialDifferentialEquationswithInitialConditions

DALIR Nemat and NOURAZAR S.Salman?

Department of Mechanical Engineering,Amirkabir University of Technology,Tehran 158754413,Iran.

Received 25 October 2012;Accepted 29 June 2014

.New developed inverse differential operators incorporated into the semianalytical treatment of the modified decomposition method(MDM)are used to solve the systems of first and second-order singular nonlinear partial differential equations (PDEs)with initial conditions arising in physics.The new proposed method is called the improved modified decomposition method(IMDM),and is used to the treatment of a few case study initial-value problems.The results obtained by the IMDM are in full agreement with the existing exact analytical solutions.

Systems of singular nonlinear PDEs;initial conditions;IMDM;inverse operators.

1 Introduction

Systemsof singular nonlinearpartial differential equations(PDEs)appearin a lot of cases in physics such as cylindrical and spherical Kortweg-de-Vries equation,Cauchy momentum equations,and the reaction-diffusion equations.Nevertheless,these systems can not be easily solved.Various numerical methods have been introduced for the solution of the systems of singular nonlinear PDEs,but they are usually developed for specific problems,and do not have the capability to generally solve these systems.

The development of semi-analytical methods have opened a new horizon on solving various mathematical and physical equations in recent years.These methods give approximate analytical series solutions.Homotopy perturbation method(HPM),Homotopy analysis method(HAM),Adomian decomposition method(ADM),modified decomposition method(MDM)and differential transform method(DTM)are a few of thesemi-analytical methods.In the semi-analytical Adomian decompositionmethod(ADM) and modified decomposition method(MDM),an inverse linear differential operator is developed for the equation in hand and then infinite series are substituted for the dependent variable and any nonlinear terms available in the equation.The ADM and the MDM have been applied for solving various mathematical and physical equations.However, theyhave previouslyneverbeenusedforsolving thesystemsofsingular nonlinearPDEs.

Wazwaz[1]applied the decomposition method to systems of PDEs and to reactiondiffusion Brusselator equation.In another paper,Wazwaz[2]applied the MDM to the problem of transient flow of gas through a porous medium.Wazwaz[3]used the MDM for the analytic treatment of variable-coefficint fourth-order parabolic PDEs.Wazwaz[4, 5]presented an ADM algorithm for solving differential equations of Lane-Emden type and a ADM treatment for mixed Volterra-Fredholm integral equations.Wazwaz[6]also used the ADM for solving the Bratu-type equations.Ray[7]applied the MDM for the solution of the coupled Klein-Gordon-Schrodinger equation.Noor et al.[8]used MDM for solving initial and boundary-value problems using Pade approximants.Muttaqi and co-workers[9]applied the MDM for solving the nonlinear Voltera-Fredholm integrodifferentialequation.KhanandHussain[10]appliedthecombinedLaplace transformationdecomposition method on the semi-infinite domain.Khan et al.[11]proposed the Auxiliary Laplace Parameter Method(ALPM)using Adomian polynomials and Laplace transformation for solving nonlinear differential equations.In a paper by Mosta and Sibanda [12],a novel numerical approach,based on a new application of the successive linearizationmethod(SLM),waspresentedforthesolutionofaclassofsingularnonlinearboundaryvalue problems arising in physiology.Khan and Gondal[13]constructed a new mechanism for the solution of Abel’s type singular integral equations,i.e.the two-step Laplace decomposition algorithm(TSLDA).Kumar and Singh[14]discussed the physical processes of a thin liquid film evolution over a wet surface down a vertical wall,derived the classical thinfilm equation,whichis anonlinear third-ordersingularordinary differential equation,and then obtained series solution by employing ADM.

In the present study,new inverse developed differential operators incorporated into the MDM are used to solve the first and second-ordersystemsof singular nonlinear PDEs with initial conditions arising in physics.The results of the solutions are compared with the existing exact solutions of the systems of singular nonlinear PDEs,which indicate excellent agreement.

2 Application of IMDM to systems of singular nonlinear PDEs

2.1 Systems of general first-order singular nonlinear PDEs

We consider the system of general first-order(in t)coupled singular nonlinear PDEs as follows:

where x and t are the independent variables,u and v are the dependent variables,F and G are nonlinear functions of u,v,ux,vx,and n1and n2are real constants,n1＞0,n2＞0. The non-homogeneous initial conditions are as follows:

Defining the linear operators

the left hand side of Eq.(2.1)is rewritten as

and Eq.(2.1)becomes as:

in the following forms:

The inverse differential operators (2.4), which are defined in the present work, introducethe IMDM for solving the systems of general first-order singular nonlinear PDEs withinitial conditions. Applying the inverse operators (2.4) to the differential Eq. (2.3) gives:

Due to the ADM,the dependent variables u and v and the nonlinear terms F and G in Eq.(2.6)should be substituted by the following infinite series[11]:

where,in the infinite series of F and G,Amand Bmwhich are called the Adomian polynomials are defined as[7]:

The substitution of the infinite series(2.7)in Eq.(2.6)results in:

Due to the MDM,from Eq.(2.9),the relations for u0(x,t),u1(x,t),v0(x,t),v1(x,t)and the recurrence relations for um+1(x,t),vm+1(x,t)take the following forms[11]:

2.2 Systems of general second-order singular nonlinear PDEs

We consider system of general second-order(in t)coupled singular nonlinear PDEs as follows:

Here x and t are the independent variables,u and v are the dependent variables,F and G are the nonlinear functions of u,v,ux,vx,uxx,vxx,and n1and n2are real constants, n1＞0,n2＞0.The non-homogeneous initial conditions for the the system of Eq.(2.11)are as follows:

By defining the following linear operator

the left hand-side of Eq. (2.11) can be rewritten as

and Eq. (2.11) is written as:

in the following forms:

The inverse differential operators(2.14),proposed in the present work,improve the MDM for solving the systems of general second-order coupled singular nonlinear PDEs with initial conditions.Applying the inverse differential operators of Eq.(2.14)to Eq. (2.13)gives:

where e(x)and g(x)are obtained as the result of initial conditions.Using the infinite series(2.7),due to the ADM,Eq.(2.15)can be rewritten as[11]:

where the Adomian polynomials Amand Bmare defined in Eq.(2.8).According to the MDM,from Eq.(2.16),the relations for u0(x,t),u1(x,t),v0(x,t),v1(x,t),and the recurrence relations for um+1(x,t),vm+1(x,t)are written as[11]:

3 Examples of systems of singular nonlinear PDEs solved by the IMDM

3.1 Examples of systems of first-order singular nonlinear PDEs

Example 3.1.We consider the following system of singular nonlinear initial-value problem(IVP)in first-order PDEs:

By using the infinite series (2.7) for the dependent variables u and v and the nonlinearterms uvx and vux, Eq. (3.2) can be rewritten as:

Use of the relations for u0(x,t),u1(x,t),v0(x,t),v1(x,t),and the recurrence relations for um+1(x,t),vm+1(x,t),due to the MDM,Eq.(3.3)gives:

The Adomian polynomials Am’s and Bm’s,by the implementation in the symbolic software Mathematica,are obtained as[11]:

and

Also the expressions for the solution terms,andare obtained as follows:

and

Thus the solution of the system of first-order singular nonlinear IVP of Eq.(3.1)is obtained as follows:

and

Example 3.2.We consider the following system of first-order(in t)non-homogeneous singular nonlinear PDEs with homogeneous boundary conditions:

Using the infinite series (2.7) and the MDM rules (10) in Eq. (3.6) results in:

The Am’s and Bm’s are obtained as [11]:

Also the expressions for um’s and vm’s become:

Therefore the solution of the system of first-order singular nonlinear PDEs of Eq.(3.5)is:

3.2 Examples of systems of second-order singular nonlinear PDEs

Example 3.3.We consider the following system of second order singular nonlinear IVP:

By using the infinte series in Eq. (2.7) and the MDM in Eq. (2.17), Eq. (3.12) can be rewritten as:

The Am’s and Bm’s take the following forms[11]:

Therefore,by summing up the solution terms(3.15),the solution of IVP(3.11)is obtained as follows:

which is the exact solution of the system of second-order singular IVP of Eq.(3.11).

Example 3.4.We consider the following system of second-order non-homogeneous singular PDEs with the homogeneous boundary conditions:

The inverse operators

from Eq.(2.14)with n1=-1/3,n2=-1/4,are applied on the PDEs of Eq.(3.17)which give:

Using the infinite series in Eq. (2.7) and MDM definition in Eq. (2.17), Eq. (3.18) isrewritten as:

The Adomian polynomials Am’s and Bm’s are obtained as[11,12]:

and the expressions for um’s and vm’s are obtained as:

The solution of the systemof second-ordersingular initial-value problem(3.17)becomes:

which is exact solution of the system of second-ordernon-homogeneoussingular nonlinear initial-value problem of Eq.(3.17).

4 Conclusions

The improved modified decomposition method(IMDM),i.e.the modified decomposition method(MDM)in conjunction with new developed inverse differential operators, is used to solve the systems of singular nonlinear initial-value problems(IVPs)in the first and second-order partial differential equations(PDEs).Four case study problems, two systems of first-order singular nonlinear initial-value problems and two systems of second-order singular nonlinear initial-value problems are solved using the IMDM,and the results of the solutions are compared with existing exact solutions.The comparisons show that the new inverse differential operators incorporated into the MDM are capable of predicting the exact solution of the systems of singular nonlinear PDEs effectively.

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[2]Wazwaz A.M.,The modified decomposition method applied to unsteady flow of gas through a porous medium.Appl.Math.Comput.,118(2001),123-132.

[3]Wazwaz A.M.,Analytic treatment for variable-coefficient fourth-order parabolic partial differential equations.Appl.Math.Comput.,123(2001),219-227.

[4]Wazwaz A.M.,A new algorithm for solving differential equations of Lane-Emden type. Appl.Math.Comput.,118(2001),287-310.

[5]Wazwaz A.M.,A reliable treatment for mixed Volterra-Fredholm integral equations.Appl. Math.Comput.,127(2002),405-414.

[6]Wazwaz A.M.,A domian decomposition method for a reliable treatment of the Bratu-type equations.Appl.Math.Comput.,166(2005),652-663.

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[11]Khan Y.,Vazquez-Leal H.,Faraz N.,An auxiliary parameter method using Adomian polynomials and Laplace transformation for nonlinear differential equations.Appl.Math.Model.,37(2013),2702-2708.

[12]Motsa S.S.,SibandaP.,A linearization method for non-linear singular boundary value problems.Comput.Math.Appl.,63(2012),1197-1203.

[13]Khan M.,Gondal M.A.,A reliable treatment of Abel’s second kind singular integral equations.Appl.Math.Lett.,25(2012),1666-1670.

[14]Kumar M.,Singh N.,Phase range analysis and traveling wave solution of third order nonlinear singular problems arising in thin film evolution.Comput.Math.Appl.,64(2012),2886-2895.

?Corresponding author.Email addresses:icp@aut.ac.ir(S.S.Nourazar),dalir@aut.ac.ir(N.Dalir)

10.4208/jpde.v27.n4.6 December 2014

AMS Subject Classifications:35F25,35G25,35C10,35A20,35Q35