Hassan ELTAYEB Said MESLOUB

Mathematics Department,College of Science,King Saud University,P.O.Box 2455,Riyadh 11451,Saudi Arabia

E-mail:hgadain@ksu.edu.sa;mesloub@ksu.edu.sa

Abstract In this article,several theorems of fractional conformable derivatives and triple Sumudu transform are given and proved.Based on these theorems,a new conformable triple Sumudu decomposition method(CTSDM)is intrduced for the solution of singular two-dimensional conformable functional Burger’s equation.This method is a combination of the decomposition method(DM)and Conformable triple Sumudu transform.The exact and approximation solutions obtained by using the suggested method in the sense of conformable.Particular examples are given to clarify the possible application of the achieved results and the exact and approximate solution are sketched by using Matlab software.

Key words conformable double Sumudu transform;conformable fractional coupled Burgers’equations;conformable fractional derivative;conformable single Sumudu transform

1 Introduction

Burgers’equation is a famous nonlinear growth equation that was introduced by Burgers in[1].This equation can also be viewed as a particular case of the Navier-Stokes equation.It happens in several areas of applied mathematics,for example,modeling dynamics,heat conduction,and acoustic waves.Analytical solution of Burger equations,founded on Fourier series method,utilizing the suitable conditions was examined in[2].The researchers in[3]explained various exact solutions of Burgers’similar equations and their classi fications.Many strong methods were employed to solve nonlinear partial differential equations,for instance,the fractional Sumudu transform[4,5].The solutions for constant-coefficient and Cauchy-Euler type conformable equations,Cauchy functions,a variation of constants,a self-adjoint equation,and Sturm-Liouville problems have been discussed in[6].The authors in[7],explained the new fractional integration and differentiation operators.The left and right generalized type of fractional derivatives depending on two parameters α and ρ generated by using the(local)proportional derivatives and generalized proportional fractional discussed by applying The Laplace transform for more details see[8].The generalized Lyapunov-type inequality for a conformable boundary value problem has been proved in[9].In[10]the new inequalities of Hermite-Hadamard type for convex functions via conformable fractional integrals are examined.The authors in[11]introduced the initial and final value problems and suggested the basic properties of this transform.Fundamental results of conformable Sturm-Liouville eigenvalue problems have been discussed in[12].The nonlinear fractional predator-prey biological model of two species has been solved by using wavelet and Euler methods for more details see[13,14].The authors in[15,16]examined the dynamical behaviour of the fractional tumor-immune model and obtained results are compared with exiting results by other methods.The possibility for obtaining new chaotic behaviors with the singular fractional operator and shows the chaotic behavior at different values of fractional order has been investigated in[17].Multi-dimensional heat equations of arbitrary order are solved by using an analytical approach homotopy perturbation transform method and residual power series method[18].The new fractional homotopy analysis transform method is used to solve the space-fractional telegraph equation,a new Yang-Abdel-Aty-Cattani fractional diffusion equation and gas dynamics equation[19–21].The author’s in[23]set up the exact solutions for time-fractional Burgers’equations by applying the first integral method.The authors in[24]obtained the solution of the coupled Burgers’equations with space-and time-fractional by using the generalized two-dimensional differential transform method(DTM).In current time the novel concepts and properties of the conformable derivative have been offered for more details we ask the reader to see[25,29,30].Moreover,the authors in[25,26]are solved the fractional differential equations by conformable Laplace transform technique.In[27]the authors provided the conformable double Laplace transform method which is used to solve fractional partial differential equations.The conformable double Laplace decomposition method has been examined to obtain the solutions of one dimensional conformable regular and singular equation of fractional coupled Burgers’for more details see[22].The first integral method has been used to establish the exact solutions of the time-fractional Burgers’equations,see[28].

In this work,we propose a new method which is called the conformable triple sumudu decomposition method(CTSDM)for solving the nonlinear equations.The suggested method is an excellent mixture of the conformable triple Sumudu transform method and decomposition method.This article examines the applicability of the conformable triple Sumudu decomposition method(CTSDM)to solve the regular and singular one-dimensional conformable fractional coupled Burgers’equations.Next,we post some basic concepts and de finitions for the conformable derivatives which are used later in this article.

De finition 1.1([30,32,33]) Given a function g:(0,∞)→R,then the conformable fractional derivative of g of order ζ is de fined by

De finition 1.2([31]) Given a function g(x,t):R×(0,∞)→R.Then,the conformable space fractional partial derivative of order ζ a function f(x,t)is de fined as follows:

De finition 1.3([31]) Given a function g(x,t):R×(0,∞)→R.Then,the time conformable fractional partial derivative of order η a function g(x,t)is determined by:

Conformable fractional derivative of speci fic functions:

Example 1.4([22])

In the following example,we examine the conformable fractional derivative of several functions:

Below,we present some de finition of the conformable Sumudu transform which are useful in this work

De finition 1.5Over the set of function

The double conformable Sumudu transform is denoted by

Example 1.7The triple conformable Sumudu transform for speci fic functions are determined by:

substituting eq.(1.8)into eq.(1.7),we obtain

by taking derivative with respect to u2for eq.(1.9),we have achieve

eq.(1.10),becomes

by arranging the above equation,we get

hence,

The proof is completed.

The conformable triple Sumudu transform of first partial derivative with respect to x,y and t are de fined by

and the conformable triple Sumudu transform of second partial derivative with respect to x,y and t are determined by

respectively.

ProofBy taking partial derivative with respect to u1for eq.(1.2),we have

by taking the partial derivative with respect to u2for eq.(1.18)

by rearranging eq.(1.21),we proof eq.(1.14)

In a similar way,one can prove eq.(1.15).

2 Two-dimensional Fractional Coupled Burgers’Equation

In this part of the paper,regular and irregular two-dimensional conformable fractional coupled Burgers’equations are discussed by employing conformable triple Sumudu decomposition methods(CTSDM).

First problemWe consider the following two-dimensional conformable fractional coupled Burgers’equations:

subject to the conditions

The solution of two-dimensional conformable fractional coupled Burgers’equations is determined by the in finite series as below

The operators An,Bn,Cnand Dnare nonlinear which are de fined by

Some elements of the Adomian polynomials are present as follows

Applying the inverse triple Sumudu transform to eq.(2.3),eq.(2.4)and using eq.(2.6),we get

The components φ0and θ0are given by

Now,we can obtain the following the general form

Now,we stipulate triple inverse Sumudu transform with respect to u1,u2and v exist for eq.(2.14)and eq.(2.15).To clarify the possible application of our method,we examine the following example.

Example 2.1The homogeneous form of two dimensional conformable fractional coupled Burgers’equations is given by

with initial condition

As reported by the above method,the first terms of the Sumudu decomposition sequences are determined as follows

similarly

the remained components becomes zeros

Using eq.(2.5),the sequence solutions are thus given by

By substituting ζ=1,γ=1 and β=1,in the above solution,we get the exact solution.

Figs.(1a),(1b),(1c)shows the approximate solutions of Example 2.1,at t=1,y=0 and ζ=γ=β=1 we get the exact solution of eq.(2.16),also we take different values of ζ,γ,β such as(ζ=0.85,γ=0.90,β=0.95).The surfaces in Figs.(1d),(1e)shows the function ψ(x,y,t)=?(x,y,t)with y=0,x=0,respectively.

Fig.1 a Shows the solutions ψ(x,y,t)=?(x,y,t)for Example 2.1 when ζ=0.95,γ=0.95,β=0.95

Fig.1 b Shows the solutions ψ(x,y,t)=?(x,y,t)for Example 2.1 when ζ=0.85,γ=0.90,β=0.95

Fig.1 c Shows the solutions ψ(x,y,t)=?(x,y,t)for Example 2.1 when ζ=0.95,γ=0.85,β=0.85

Fig.1 d The surface shows the solutions of ψ(x,y,t)=?(x,y,t)at y=0 and ζ=γ=β for Example 2.1

Fig.1 e The surface shows the solutions of ψ(x,y,t)=?(x,y,t)at x=0 and ζ=γ=β for Example 2.1

The second problemThe singular two-dimensional conformable fractional coupled Burgers’equations denoted by:

with conditions

Second,using the conformable triple Sumudu transform on both sides of in eq.(2.20)and after using the differentiation property of conformable Sumudu transform,we get:

On applying Theorems 1.9 and 1.10,we obtain

multiplying both sides of eq.(2.22)byand taking the double integral with respect u1,u2from 0 to u1and 0 to u2,we have

dividing eq.(2.23)by u1u2,and take the inverse triple Sumudu transform,yields:

Consider that triple inverse Sumudu transform with respect u1,u2and v exist for each term on the right-hand side of eqs.(2.28),(2.29)and(2.30).

Example 2.2Consider singular two-dimensional conformable fractional coupled Burgers’equations

Applying the above method,we obtain

where the nonlinear terms An,Bn,Cnand Dnare given in eqs.(2.7),(2.8),(2.10)and(2.9)respectively.In the light of the iterative relations(2.28),(2.29)and(2.30)we obtained other components as follows

hence,the remaining terms become zeros

So the conformable solutions are given in series form

Fig.2 a Shows the solutions ψ(x,y,t)=?(x,y,t)for Example 2.2 when ζ=0.95,γ=0.95,β=0.95

Fig.2 b Shows the solutions ψ(x,y,t)=?(x,y,t)for Example 2.2 when ζ=0.85,γ=0.90,β=0.95

Fig.2 c Shows the solutions ψ(x,y,t)=?(x,y,t)for Example 2.2 when ζ=0.95,γ=0.85,β=0.85

Fig.2 d The surface shows the solutions of ψ(x,y,t)=?(x,y,t)at y=0 and ζ=γ=β for Example 2.2

Fig.2 e The surface shows the solutions of ψ(x,y,t)=?(x,y,t)at x=0 and ζ=γ=β for Example 2.2

Accordingly,the exact solution is denoted by:

and

By substituting ζ=1,γ=1 and β=1,in the above solution,we get the exact solution.

Figs.(2a),(2b),(2c)shows the approximate solutions of Example 2.2,at t=1,y=0 and ζ=γ=β=1 we get the exact solution of eq.(2.31),also we take different values of ζ,γ,β such as(ζ=0.85,γ=0.90,β=0.95).The surfaces in Figs.(2d),(2e)shows the function ψ(x,y,t)=?(x,y,t)with y=0,x=0,respectively.

ConclusionIn this study,we have introduced the conformable triple Sumudu decomposition method.We also proved the conformable triple Sumudu transform of the fractional partial derivatives.Furthermore,the conformable triple Sumudu decomposition method is employed to solve regular and irregular two-dimensional conformable fractional coupled Burgers’equations.Moreover,we provided two examples to explain the efficacy and reliability of the suggested method for both exact and approximation solutions.We applied MATLAB software to sketch the solutions.

AcknowledgementsThe authors would like to extend their sincere appreciation to the Deanship of Scienti fic Research at King Saud University for its funding this Research group No(RG-1440-030).