On Comparison Study between Double Sumudu and Elzaki Linear Transforms Method for Solving Fractional Partial Differential Equations

In this paper, double Sumudu and double Elzaki transforms methods are used to compute the numerical solutions for some types of fractional order partial differential equations with constant coefficients and explaining the efficiently of the method by illustrating some numerical examples that are computed by using Mathcad 15.and graphic in Matlab R2015a.


Introduction:
Fractional differential equations are effective tools to formulate the physical problems. The oldest integral transform is Laplace transform by Laplace in (1780) (1). Watugala introduced the Sumudu transform in (1993), which has some advantages over the Laplace transform (2,3). The Elzaki transform, which was introduced by (4) in (2011), is a revised form of the two previous transforms. For more details and historical review of integral transform (5,6). In (7) double Laplace and double Sumudu transform were used to solve wave equation and Poisson equation. In this study double Sumudu transform and double Elzaki transform are considered to solve some fractional partial differential equations which conclude both the space and time or mixed fractional Caputo derivatives, such as parabolic-hyperbolic, wave and heat fractional equations.

Definition 3: (2)
A function ( ) is said to be of exponent order > 0 if there exist non-negative constants , and such that | ( )| ≤ ≥ .

Definition 4 (3)
The Sumudu transform of the exponent order function ( ) is defined as Table 1 contains Sumudu transform for some famous function.   For the two variables exponent order function ( , ), the double Sumudu transform of the partial fractional integrals are given by the following theorem

Elzaki Transform
A modification to Sumudu transform is a new transform 'Elzaki transform' which is introduced by Tarig Elzaki in 2010, for solving a class of ordinary differential equations with variable coefficients under special initial and boundary conditions, that cannot be done by the Sumudu transform or Laplace transform only (13). Thus, Elzaki transform is a more effective than the Sumudu transform. The researcher (3) designed some conditions for solving such class of differential equations by Elzaki transform method only, these conditions convert the origin equation into other with constant coefficients which can be easy to solve.
But in case of linear ordinary differential equations with constant coefficients, Elzaki and Sumudu transforms usually give the same solution through the duality between the two transforms. If ̃[ ( )] = ( ), then (14):

Definition 6 :(14)
Let the function ( , ) be an exponent order, and , > 0, then the double Elzaki transform for is given by

Some illustrative Examples
The first example is to explain the advantage of the Elzaki transform over the Sumudu transform, through the following simple linear fractional ordinary differential equation with variable coefficients which is generalized for that in (15). Example 1: Consider the following variable coefficients fractional ordinary differential equation

Solution by Sumudu Transform Method
Operating Sumudu transform for both sides of (1) and using Theorem (6) to get Substitute the initial condition and simplify to get Then for all ̃∈ (0,1], again this equation is the first order ordinary fractional differential equation with variable coefficients, so cannot solve it by Sumudu transform method only.

Solution by Elzaki Transform Method
Operating Elzaki transform for both sides of (1) and using Theorem (7) to get So, by letting = 2 , for example " = 1' The last equation is converted to ́( ) = 2 which has the solution ( ) = 3 3 = 3 with = 3, taking inverse Elzaki transform, will have ( ) = Other cases ≠ 1, in order to get an equation with constant coefficients, coefficient of ( ) must be vanish, i. e (1 + ) 1−̃= . So, Elzaki transform alone failed to solve this fractional ordinary differential equation with variable coefficients. Now some of fractional partial differential equations with Caputo derivatives are given to solve it by both double Sumudu and Elzaki transforms methods to get an exact solution for these equations. In the following example, will generalize example 1 in (16)  For ̃= 1,̃= 2, then the standard equation (2) has the exact solution as ( , ) = (1 − − ) Which is the same as in (16). The absolute error of some of 10 -order approximate solutions for equation (2) for different values of ̃,̃, are included in Table 2. Fig.1 illustrates the solution.   When ̃= 1,̃=̃= 2, then have the standard telegraph equation (3) which has ( , ) = − as an exact solution. Table 3 contain the absolute error for some approximated solutions for equation (2) and some of solutions with different values of ̃,̃ ̃ are illustrated graphically in Fig. 2  The exact solution of equation (2) is ( , ) = − when ̃= 1,̃=̃= 2.

Solution by Double Elzaki Transform
That agrees with that for Sumudu transform method. Fig. 2 contains3D-plotted of some numerical solutions of Eq. (3) with different fractional orders ̃−̃ and ̃. Also, Table 3 shows the results of absolutely error for some 10-order approximate solutions.

Solution by Double Elzaki Transform
Which agrees with that for double Sumudu transform method. Fig. 3 contains3D-plotted of some numerical solutions of Eq. (3) with different fractional orders ̃−̃ and ̃. Also, Table 4 shows the results of absolutely error for some 10-order approximate solutions.    Fig.4. Also, Table 5 shows the results of absolutely error for some 10-order approximate solutions.

Discussion:
In this paper used the double Sumudu and Elzaki transformations to compute some of fractional partial differential equations which are proposed as a new approach in this research for these equations to show that the two transformations efficient and accurate by applying these methods and showing perfect results in numerical tables and graphics of the illustrative numerical examples from the increasing of the fractional order convergent to the numerical solution in positive integer through the absolute error goes to zero.
The Sumudu transformation may be used to solve problem without resorting to a new frequency domain and having scale and unit-preserving properties. The Sumudu transform is used to solve PDE by transforming to algebraic equation and after using some mathematical algebra technique using inverse Sumudu transformation to get up the solution. Also the Elzaki transformation is used when the Sumudu transformation cannot solve some of fractional ordinary differential equation when the coefficients are variable, on the other hand the Elzaki transform only can solve such of such equations with changes of the initial conditions of the problem, this is the only advantage of Elzaki transform over the Sumudu one.
Usually, both Sumudu and Elzaki transforms are used to solve some of FPDE with constant coefficients to get the analytic solution. So in this case, need not to study the stability or convergence of the solution and the tables and figures explain the numerical values of the solution with different values of ̃ and ̃.

Conclusion:
In this paper, introduced the double Sumudu and double Elzaki transforms to find the solution for several types of linear partial differential equations with constant coefficients that include space, time and space-time or mixed fractional Caputo derivatives, in this case both transforms give the same solution. All problems which are argued proved the efficiency of this methods throughout the plotting and numerical calculus of approximated solutions.