Yang-Laplace Decomposition Method for Nonlinear System of Local Fractional Partial Differential Equations

The basic motivation of the present study is to extend the application of the local fractional Yang-Laplace decomposition method to solve nonlinear systems of local fractional partial differential equations. The differential operators are taken in the local fractional sense. The local fractional Yang-Laplace decomposition method (LFLDM) can be easily applied to many problems and is capable of reducing the size of computational work to ﬁnd non-differentiable solutions for similar problems. Two illustrative examples are given, revealing the effectiveness and convenience of the method.


Introduction
Fractional partial differential equations is a fundamental tool for the analysis of physical phenomena, such as, electromagnetic, acoustics, viscoelasticity, electrochemistry, and others. These physical and other phenomena are expressed by fractional partial differential equations, which have been solved by several numerical-analytical methods ( [13], [21], [25]). Among them, one of the most popular methods is the so-called Adomian decomposition method (ADM), which has been developed between the 1970s and 1990s by George Adomian ( [1]- [5]).
With the new concepts of fractional derivative and fractional integral, as well as local fractional derivative and local fractional integral, researchers were able to use the ADM method to solve these new types of equations or systems which include, local fractional differential equations, local fractional partial differential equations and local fractional integro-differential equations ( [6]- [10]).
Some more results were obtained by combining fractional differential operators with some known transforms, such as the Young-Laplace transform method and the Sumudu transform method, thus obtaining a solution of fractional equations or systems, even in the nonlinear case. Among these works we find the local fractional Laplace decomposition method [11] and the local fractional Sumudu decomposition method ( [12], [14]). The theory and applications of local fractional derivative and integral operators, has been defined and deeply investigated by Xiao-Jun Yang (see e.g. [15], [16]).
Based on the concept of local fractional operator and on the Yang-Laplace decomposition method, Jassim has recently proposed a method [11] to solve linear local fractional partial differential equations. In this paper, we will generalize the Jassim method and extend it to solve nonlinear systems of local fractional partial differential equations. The Yang-Laplace decomposition will enable us to obtain exact solutions for nonlinear system of local fractional partial differential equations. Two examples are also given to show the effectiveness of this method.
This paper has been organized as follows: In Section 2 some basic definitions and properties of local fractional calculus and local fractional Yang-Laplace transform method are given. In section 3, we present an analysis of the modified method. In section 4 we apply the modified method (LFLDM) to nonlinear systems to obtain the analytical solution. Two applications are given in the same section.

Basic definitions
In this section, we will present the basic concepts of fractional local calculus, and in particular the local fractional derivative, local fractional integral, and local fractional Yang-Laplace transform.

Local fractional derivative
Definition 2.1. The local fractional derivative of Φ(κ) of order σ at κ = κ 0 is defined as ( [15], [16]) where This operator can be extended to the interval (α, β ) so that for any κ ∈ (α, β ), we can define . For the local fractional derivative of high order and local fractional partial derivative of high order

Some properties of the local fractional operators
The local fractional operators fulfill some fundamental equations. In particular, starting from the Mittag-Leffler function we have the following Definition 2.3. The Mittag-Leffler function, the hyperbolic sine and hyperbolic cosine are defined as ( [15], [16], [17]) By using the local fractional derivative (1) and the definitions (2.3) it can be easily shown that ( [15], [16])

Local fractional Yang-Laplace transform
We present here the definition of local fractional Yang-Laplace transform (denoted in this paper by LF L σ [.]) and some properties concerning this transformation.
We will search an analytical solution of this system by the following steps.
Step 1 First we apply the local Yang-Laplace transform to both sides of each equation in system (26), so that: According to the properties of this transform, we have: By taking the inverse transformation on both sides of each system equation (28), there follows: . (29) Step 2 By using the Adomian decomposition method [1], we represent the two unknown functions X and T as infinite series: moreover, the nonlinear terms can be decomposed as: where A n and B n are Adomian polynomials [22]. Substituting (30) and (31) in (29), we get: By comparing both sides of (32a), we have: Step 3 At the last step, we get the solution by a limit of the analytical solution (X, T ) of the system (26):

Applications
In this section, we will implement the proposed method based on local fractional Yang-Laplace decomposition method (LFLDM) [11] and Adomian decomposition for solving two nonlinear systems of local fractional partial differential equations.
For example, The first few components of A n (X), B n (X,t) and C(T ) polynomials [22], are given by: and According to the equations (44)-(45) and formulas (46)-(48), the first terms of local fractional Yang-Laplace decomposition method of the system (36), is given by: and T 0 (κ, τ) = sin σ (κ σ ), Unauthentifiziert | Heruntergeladen 30.12.19 00:49 UTC So that, the local fractional series solution (X, T ), is: and in a closed form, we obtain the non-differentiable solution (X, T ): By letting σ = 1 into (52), we have: It should be noticed that, solution (52) satisfies the initial conditions (37), and in the case σ = 1, we have the same solution obtained in [23] by homotopy perturbation method, and in [24] by the natural decomposition method.
In particular, by letting σ = 1, from (74), we get Let us notice that, our solution (74) fulfills the initial conditions (55), and in the case σ = 1, we re-obtain the same solution already obtained both in [26] by the projected differential transform method and Elzaki transform, and in [24] by the natural decomposition method. Thus showing that the method proposed in this paper is the more general approach to the solution of nonlinear fractional differential system.

Conclusions
The local fractional Yang-Laplace transform decomposition method (LFLDM) has been used to solve nonlinear systems of local fractional partial differential equations. It has been shown that combined with the Adomian decomposiition method, LFLDM enables us to establish an efficient algorithm. This algorithm provides the solution in a series form that converges rapidly to the exact solution, as shown by the results obtained through the two non-trivial examples given in this paper. From the obtained results, it can be concluded that this algorithm is powerful and effective and it can be used to explore some more complicated nonlinear systems with local fractional derivative.

Author contributions
This is an author contribution text. Financial disclosure None reported.

Conflict of interest
The authors declare no potential conflict of interests. lisis non, adipiscing quis, ultrices a, dui.