An Efficient Method for Solving Coupled Time Fractional Nonlinear Evolution Equations with Conformable Fractional Derivatives

In this article, an efficient reliable method, which is the residual power series method (RPSM), is used in order to investigate the approximate solutions of conformable time fractional nonlinear evolution equations with conformable derivatives under initial conditions. In particular, two types of equations are considered, which are time coupled diffusion-reaction equations (CD-REs) and MKdv equations coupled with conformable fractional time derivative of order α. The attitude of RPSM and the influence of different values of α are shown graphically.

3083 homotopy perturbation method [12], and variational iteration method [13], were discovered to provide the approximate solutions of FDEs. The RPSM was used successfully to produce a series of solutions for tumor models [14]. RPSM was used to investigate a numerical solution for the fractional Burger equation [15]. The exact analytical solution of the time-fractional Schrodinger equation was found in another work [16]. The main aim of this paper is to employ RPSM for two models of nonlinear FDEs of special interest physically, in terms of the convergent fractional power series. The rest of this article is arranged as follows: In section 2 some preliminaries are given. In section 3 we describe the RPSM. The models of the proposed study are described in sections 4 and 5. Numerical simulations are drawn in section 6. Finally, the conclusions are presented in section 7.

2.
Preliminaries Definition 2.1 [8]: Given a function :[0, ) y   , then the conformable fractional derivative of order α of y is defined by And the conformable integral of order α is defined by

An Overview of CRPSM
We consider the following fundamental concept of RPSM operator: (4) Where N(u) and R(u) are nonlinear and linear terms, respectively, with an initial condition (IC): The RPSM suggests the solution for Eqs.(4) as a fractional power series (FPS) about the (IC)t=0 as: Truncating the infinite series (6) after k th terms implies: For the convergence of the FBS, refer for instance to a previous work [17]. Using Equation (5), then Eq. (7) may be expressed as: The residual function ( RF ) for Eq.(4) is defined by: However, finding 1 needs to solve the algebraic equations: ( 1) ,

4.
RPSM for solving CD-REs The performing of the RPSM for finding the solution of the CD-REs in terms of the FPS is represented in this section. First consider that: , With 0. 5 2 ( , 0) , [1 ] kx kx e ux e   and the exact solution when α=1 is given by a previous work [19] as: where, z x ct,  k is constant, and The k th truncated series of u(x,t) and v(x,t) is defined by: It is obvious that 0 () fxand 0 g ( ) x can be obtained directly from the initial conditions given by Eqs.  Let us define the RF of Eqs. (19) and (20) as follows: (28) Then the k th residual function becomes: The coefficients of f ( ) and g ( ), n 1, 2, .k, nn xx   respectively, can be computed by solving the following system:

RPSM for solving coupled MKdv equations with a conformable fractional order derivative (CMKdv)
In this segment, the RPSM will also be considered in order to get the solution of the CD-R as FBS, as follows:

7.
Conclusions In this study, RPSM was implemented to find the solutions of CD-REs and CMKdv. The approximate solution was given as an infinite FPS. The suggested method introduced an easy manner to find the coefficients of the solution, which converges quickly to the closed form. The numerical results demonstrate the significant feature, efficiency, and reliability of the proposed method for solving CD-REs and CMKdv.