Solving System of Conformable Fractional Differential Equations by Conformable Double Laplace Decomposition Method

ALFAQEIH Sulimanand KAYIJUKA Idrissa

Ege University,Faculty of Science,Department of Mathematics,35100 Bornova Izmir,Turkey.

Abstract. Herein,an approach known as conformable double Laplace decomposition method(CDLDM)is suggested for solving system of non-linear conformable fractional differential equations. The devised scheme is the combination of the conformable double Laplace transform method(CDLTM)and,the Adomian decomposition method(ADM). Obtained results from mathematical experiments are in full agreement with the results obtained by other methods. Furthermore,according to the results obtained we can conclude that the proposed method is efficient,reliable and easy to be implemented on related many problems in real-life science and engineering.

Key Words: Fractional differential equation;double Laplace transform;Adomian decomposition method;conformable fractional derivative.

1 Introduction

Recently,many researchers have been attracted by the fractional partial differential equations,and the topic has been received more attention during the last decades due to their significant role in engineering and real-life science [1–3]. Consequently, many authors introduced several classes of fractional derivatives such as, Caputo, Riemann-Liouville,Hadamard, Caputo-Hadamard and so on [4,5]. Unfortunately, these fractional derivatives do not obey a lot of the usual properties such as the product rule, the chain rule,Quotient Rule for two functions, due to that, these fractional derivatives have a lot of difficulties in applications, to overcome these difficulties,Khalil et.al.[6]introduced the conformable fractional derivative that satisfying all the classical properties of the normal derivatives. Consequently, some techniques for solving ordinary differential equations are used to solve conformable fractional differential equations. Recently, many authors have developed various analytical and approximate methods to obtain the solution of partial differential equations,such as Homotopy perturbation method(HPM)[7,8],Adomian decomposition method (ADM) [9,10], variational iteration method (VIM) [11], differential transform method (DTM) [12], transform methods[13–19], and many others.Among all the previous methods, the double Laplace method (DLM). In the last few years there was no work or very little work available on(DLT),therefore,recently many authors have been paying a significant attention towards the applying of double Laplace transform to solve integral,partial differential equations including ordinary and fractional[20–22]. Recently,Ozan Ozkan and Ali kurt in(2018)[23]introduced the conformable double Laplace transform (CDLT) and they implemented it to solve conformable fractional heat equation and Conformable fractional Telegraph equation. For more about(CDLTM)see[24,25].

The main target of this article is to solve systems of nonlinear fractional differential equations involving conformable fractional derivatives by a new method called conformable double Laplace decomposition method,this method is a combination between two methods,which are the(CDLTM)and(ADM).

2 Preliminaries

In this section,properties and some basic definitions of the conformable fractional derivative(CFD)are presented.

Definition 2.1.([26])The(CFD)of order ρ of a function ?:(0,∞)→Ris given by:is definedby:

Definition 2.2.([27])The(CFD)of order ρ of a function

Definition 2.3.([27])The(CFD)of order β of a functionisdefinedby:

2.1 Conformable fractional derivative of certain functions:

3 Conformable Laplace transform(CLT)

In this part,we present some notes on conformable Laplace transforms(CLTs).

Definition 3.1.([28])The(CLT)of a real valued function ?:[0,∞)→Ris given by:

Definition 3.2.([23])The(CDLT)of a piecewise continuous function ?:[0,∞)×[0,∞)→Ris given by:

Theorem 3.1.([23])The(CDLT)of the,n,m∈Ntimes conformable partial fractional derivativesof the function ? is given by:

Theorem 3.2.The(CDLT)ofare give by:

Proof.See[23].

4 Analysis of(CDLDM)

Herein,we demonstrate the purpose of the approach by considering the general system of nonlinear conformable fractional differential equations of the form;

Subject to the initial conditions:

where,are given functions,R1(v,w),R2(v,w) are the linear terms,andχ1(v,w),χ2(v,w)are the nonlinear terms.

Applying the(CDLT)to both sides of(4.1),we get:

whereF(p),G(p) are the conformable single Laplace transforms ofrespectively,andH1(p,q),H2(p,q),are the(CDLT)ofrespectively.Now,applying the inverse of(CDLT)to(4.3),we get:

The solution is represented by the infinite series

And,we represent the nonlinear terms with

whereAm,Bmare the Adomian polynomials, which can be easily computed by the following formulas:

By substituting equations(4.5),(4.6)in equation(4.4),we get,

Eq.(4.7)is the coupling of(CDLT)and the(ADM).Comparing both sides of Eq.(4.7),we obtian the following general recursive relation formulas:

Consequently, our solutions of the system(4.1) subject to the initial conditions (4.2) are given by:

5 Applications

Example 5.1.Consider the system of conformable fractional differential equation

With initial conditions:

Applying the(CDLDM)to Eq.(5.1)and initial conditions given in Eq.(5.2),we obtain the recurrence relations given by:

whereAn,Bnare the Adomian polynomials to be determined from the nonlinear terms:

Consequently,the first few components are given by:

Similarly,we find

Finally,the series solution of the unknown functionsare given by:

Ifρ,β=1,then the series solutions are:

which are in full agreement with the results obtained by[29].

Example 5.2.Consider the the following system of the nonlinear Kdv conformable frac-tional differential equations:

Subject to the initial conditions:

Depending on(4.8)and(4.9),we obtain

where,An,Bn,Cnare Adomian polynomials,which represent the nonlinear terms.

Consequently,the remaining components of the solution can be computed as follows:

Finally,the series solution of the unknown functionsare given by:

Forρ,β=1,the exact solution of(5.3)is given by:

which are in full agreement with the results obtained by[28].

Example 5.3.Consider the following system of nonlinear Dispersive Long Wave equations of conformable fractional derivatives

Subject to initial conditions

Applying the aforementioned steps, it is straight forward to generate the following recursive relations:

Eventually,the remaining terms of the solution can be computed as follows:

Thus,the solution of system(5.5)is given by

Forρ,β=1,the exact solutions are given by:

which are in full agreement with the results obtained by[28].

Example 5.4.Consider the following nonlinear system of conformable fractional partial differential equations:

With initial conditions

From Eqs.(4.8)and(4.9),we generate the recursive relations as follows:

where

Consequently,the first solution terms of the system(5.7),are given by:

Proceeding in a similar manner,we get:

Finally,the series solution of the unknown functionsandare given by:

ifρ=γ=β=1, then we have:

which are in full agreement with the results obtained by[29].

6 Conclusions

In this paper,the(CDLDM) was successfully implemented for solving a system of nonlinear conformable partial differential equations. this new approach is a merge of the(CDLT)and the(ADM).The presented experiments are in full agreement with the results obtained in the literature. Consequently,the results obtained confirm that the(CDLDM)is more efficient and reliable approach in solving systems of fractional partial differential equations involving conformable fractional derivatives that arise in different areas of science and engineering. Finally,it is worthwhile to mention that the(CDLT)can be applied to solve linear conformable partial differential equations and conformable integral equations that arising in applied mathematics,applied physics and engineering,without combination with other methods,which will be discussed in subsequent papers.

Acknowledgement

The authors thank the referees for their useful comments and suggestions that substantially helped improving the quality of the paper.