Numerical Solutions for the Nonlinear PDEs of Fractional Order by Using a New Double Integral Transform with Variational Iteration Method

This paper considers a new Double Integral transform called Double Sumudu-Elzaki transform DSET. The combining of the DSET with a semi-analytical method, namely the variational iteration method DSETVIM, to arrive numerical solution of nonlinear PDEs of Fractional Order derivatives. The proposed dual method property decreases the number of calculations required, so combining these two methods leads to calculating the solution's speed. The suggested technique is tested on four problems. The results demonstrated that solving these types of equations using the DSETVIM was more advantageous and efficient.


Introduction
Based on the idea of fractional calculus, which originated more than three decades ago. The study and use of arbitrary order integrals and derivatives using real or complex number powers of the differential and integral operators are the subjects of the mathematical analysis branch known as fractional calculus. Models of real-world problems may be more accurately represented using fractional derivatives than integer-order derivatives [1][2][3] .
Integral transform methods are essential for the solution of many different varieties of problems. Multiple integral transforms, including the Laplace, Sumudu, Fourier, Natural, Mellin, and Elzaki, have been used for the solution of PDEs [4][5][6][7] , as a result of the rapid developments in research and engineering. Therefore, notice that several academics are attempting to create new methods that allow us to solve this form of problem. These attempts, which are still continuing, have resulted in the promotion of these studies in numerous ways, including the Homotopy analysis method (HAM), Adomian decomposition method (ADM), and Variational iteration method (VIM) [8][9][10] , which have become well-known among a significant number of researchers in this field. A new approach has just been developed, which combines the Laplace transform, Sumudu transform, Natural transform, or Elzaki transform, with these techniques [11][12][13][14] .
The properties and theories of double integrals, such as [15][16][17] , are novel. Some authors have used these transforms in conjunction with other mathematical techniques, such as the HAM, ADM, and VIM [18][19][20][21] , to solve linear and nonlinear fractional differential equations.
In all applied science and engineering. Partial Differential Equations (PDEs) of fractional order are utilized to explain various situations. Finding exact or approximate solutions to these kinds of equations has received a lot of attention in recent research [22][23][24] .
Many nonlinear phenomena are major parts of applied research and engineering [25][26][27][28][29][30][31][32][33][34][35][36][37] . Nonlinear equations of fractional order have been found in a variety of real-world problems. Different phenomena may be described with the help of nonlinear PDEs of fractional order. Nonlinear PDEs of fractional-order derivatives computed with unknown functions of two variables are challenging to solve, such equations are more difficult to solve than linear PDEs. The fact that these equations are so widely used has made mathematicians aware of them. Nonetheless, solving these mathematical problems is neither numerically nor conceptually simple.
In this paper, the DSETVIM has been used to solve nonlinear time-fractional derivatives NT-FDPDEs. Our current article has been structured as follows: Definitions of the Sumudu transform and the ELzaki transforms in the context of fractional calculus are presented in Section 2. Our proposed analysis of the revised approach with the convergence theorem will be presented in Section 3. There are four examples of how this method was employed were provided in Section 4. The last part is the conclusion.

Basic definitions:
With the use of the Sumudu and ELzaki Transform, the fundamental ideas and features of the fractional calculus theory are given in this part. Definition 1: 1 A real function Φ( ), > 0, is said to be in the space ϑ , ϑ ∈ R, if there exists a real number , ( > ), such that Φ( ) = Φ 1 ( ), where Φ 1 ( ) ∈ C[0, ∞), and it is said to be in the space if Φ ( ) ∈ ϑ , ∈ . Definition 2: 2 The Riemann-Liouville fractional integral of order ( ≥ 0) of a function Φ( ) is defined as: Additionally, the Riemann-Liouville fractional integral has the following property:  The operator has the following fundamental characteristics: Definition 4: 38 The Sumudu Transform ST of the function Φ( ) for all ≥ 0 is defined as: ( 1 , 2 ), where the operator S z is called the Sumudu transform operator.
Definition 5: 39 The Elzaki Transform ET of the function Φ( ) for all ≥ 0 is defined as: ( 1 , 2 ), where the operator E t is called the Elzaki transform operator.
These functions are of exponential order, and they take into consideration functions in the set G described by: Basic derivative properties of the DSET 40 : .
For the Existence condition and the properties of DSET see 40 .

Principle of the DSETVIM:
This paragraph will use the suggested method DSETVIM for solving NT-FDPDEs , ( − 1 < ≤ , = 1,2, . . . ). where ( , ) is the source term, denotes the linear differential operator, stands for the generic nonlinear differential operator, and = ( , ) is the Caputo fractional derivative. Applying the DSET on both sides of Eq.1, Depending on the derivative properties of DSET, the Eq.3 becomes The results of this calculation, which uses the IDSET on both sides of Eq.4, are as follows: By applying the variational iteration technique 8 , which can then be used to create the correct functional, as shown below: Or alternately Recall that ( , ) = lim →∞ ( , ).
The limit stated above will determine whether the equation under consideration has an exact solution ES or an approximate solution AS.

The Convergence Theorem
The convergence theorem of DSET is shown in this section.
, converges at = 0 , then the integral converges for < 0 . Proof: for the proof see 41 .
converges for < 0 , and by using Theorem 9 the integral 1 ∫

Applications:
The technique mentioned in the preceding paragraph will be used to solve the following NT-FDPDEs in the following cases: Recall that the ES of Eq.12 is calculated by ( , ) = lim →∞ ( , ).
Then, ( , ) = 1 + 1+ , | | < 1.  The following result is obtained by employing the differentiation property and applying DSET, including both sides of Eq.13:   The AS of some of 4-order approximate solutions for Eq.13 for different values of are included in Table 1.   This can be assumed to be the m th AS of Eq.22. The ES when = 1 of Eq.18 is given by: The comparison between the suggested method with the method that combines Yang transform with the variational iteration method described in reference 44 to some of the 4-order approximate solutions for Eq.18 for various values of and various values of , , as well as the absolute error between the ES and AS when = 1 are included in Table 2   The formula shown below may be created using Eq.7: The comparison between the suggested method with the method that combines Elzaki transform with Adomian decomposition method described in reference 45 to some of the 4-order approximate solutions for Eq.23 for various values of and = 1.1, along with the absolute error between the ES and AS when = 2 are included in Table 3,

Conclusion
Combining Sumudu-Elzaki Transforms and the Variational Iteration Method is an effective strategy for solving NT-FDPDEs. The suggested method is very effective and appropriate for these types of problems. The results demonstrate that the DSETVIM produces very accurate approximations with just a few iterations. The numerical results demonstrate how effective, simple, and speedy this new analytical approach is, producing a series solution that rapidly converges to the right solution.

Acknowledgment
We would like to express our gratitude to Al-Esraa University College for supporting this work.