Comparison between the Laplace Decomposition Method and Adomian Decomposition in Time-Space Fractional Nonlinear Fractional Differential Equations

The aim of this paper is to discuss application of Laplace Decomposition Method with Adomian Decomposition in time-space Fractional Nonlinear Fractional Differential Equations. The approximate solutions result from Laplace Decomposition Method and Adomian decomposition; those two accessions are comfortable to perform and firm when to PDEs. For caption and further representation of the thought, several examples are tool up.


Introduction
Recently, research has shown that numerous phenomena in fluid mechanics, viscoelasticity, biology, physics, engineering and other field from knowledge mastery are successfully modeled by the use of FPDEs.
Researchers developed numerous methods to resolve FODEs, integral equations and fractional partial differential equations of physical interest.The flowing is an illustration of several of the generality used ones; ADM [3] [4] [5] [6], VIM [7], FDM [2], DTM [9], HPM [8].Among the already mentioned ones, the decomposition method stood up as efficient, easy and accurate in solving a great group of linear and nonlinear ordinary, partial, deterministic or stochastic differential equations.This method is fully appropriate to physically solve problems, some time ago it did not request superfluous linearization, and other bound methods and postulate whom may vary the problem being solved [10] [11].The LDM is a numerical algorithm to solve NODEs and PDEs.Khuri [13] [14] developed this method for the sacrificial solution of a class of NODEs.These numerical technique styles ultimately clarify how the Laplace Transform supposedly is utilized to sacrifice the solutions of the nonlinear differential equations by impacting the Decomposition Method that was initially before Adomian.

( )
F s be the Laplace transform of ( ) defined by the following series representation, valid in the whole complex plane [2] ( ) ( )

The Main Idea of Fractional Laplace Decomposition Method
In this section the class of Nonlinear Fractional Differential According to the Applying of the Laplace transform to Equation (3.1) and the use of linearity Laplace transform, the result is when the property of Laplace transform is used, we get , ,0 ,0 , , , , Stander Laplace decomposition defines the solution ( ) The Nonlinear operator is decomposed as follows: ( ) See n A the Adomian polynomial that are given by when both sides of Equation (3.10) are matched, the following iterative algorithm is yielded: Generally, the recursive relation is given as follows When the inverse Laplace transform Equation (3.13) is applied, the following is obtained where ( ) , g x t is a function that arises from the source term and the prescribed initial condition, the initial solution is important, the choice of Equation (3.15) as the initial solution always leads to noise oscillation during the iteration procedure.

Numerical Results
To show the method of coupled fractional nonlinear partial differential equations, three examples are considered in this section.
The initial condition is subject to: ( ) When the Laplace transform is applied to both sides of Equation (4.1) the following is yielded.
, ,0 , , By using the initial condition, the following recurrence relational is yielded ( ) ( ) ( ) , , Applying the inverse Laplace transform, to both sides of Equation (4.4) yields: The LDM proposes a series solution of the function ( ) , u x t which is given series Equation (3.7) and using Equation (3.7) into Equation (4.5) In above Equation(4.6) is Adomian polynomials that represents nonlinear terms.The few components of Adomian polynomials Then following recurrence relations, we have ) ( ) ) Example 4.2 Consider the nonlinear time-fractional hyperbolic equation [16].
The initial condition is subject to: , , , 2 When the Laplace transform is applied to both sides of Equation (4.8) the following is yielded.
, ,0 ,0 , , By using initial conditions, the following recurrence relational is found , , When the inverse Laplace transform, is applied, to both sides of Equation (4.11) the following is yielded: The LDM the LDM proposes a series solution of the function ( ) , u x t which is given series Equation (3.6) and using Equation (3.6) into Equation (4.12) In above Equation (4.13) Adomian polynomials represent nonlinear terms x N uu = .The few components of the decomposition series are derived as follows: Consider a system of nonlinear coupled with partial differential equation [16].
( ) The initial condition is subject to: ( ) The LADM assumes a series solution of the function ( ) , u x t which is given series Equation (3.6) and through using Equation  , , e , , , , e , , , , e , , where The following recursive relation is obtained: , , e , , , , , x y x y x y y x y x y x y x y y , , , , e e 1 2 1 , , , , e e 1 2 1 x y So the expected solutions are as follows: The solution in a closed form is given as

Conclusion
In treatise, Laplace decomposition method has been successfully used to solve the approximate solution of nonlinear PFDEs (see Table 1 and Table 2).
u u u u u u .4236/am.2018.94032

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The third term approximating a solution to Equation (4.1) gives the following: the Laplace transform is applied the both sides of [Equation (4.14), Equation (4.15) and Equation (4.16)] through using the initial condition, the fol-M.Z. Mohamed, T. M. Elzaki lowing recurrence relational is yielded Laplace transform to both sides of [Equation (4.20), Equation (4.21) and Equation (4.22)] we can yields.
polynomials they represent nonlinearities arising in above system [Equation (4.26), Equation (4.27) and Equation (4.28)] of nonlinear coupled partial differential equations.The components of above Adomian polynomials are given below