Solving Partial Differential Equations of Fractional Order by Using a Novel Double Integral Transform

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Introduction
Integral transformations are seen as the most efficient method of resolving fractional-partial differential equations (FPDEs).FPDEs can mathematically describe a wide variety of phenomena in mathematical physics and in many other scientific fields, making them valuable [1][2][3][4][5].With integral transformations [6][7][8][9], these equations can also be modified to identify precise FPDE solutions.The direct power of transformation techniques has been the inspiration for ongoing research to understand and improve them.Many integral transforms were developed and implemented to solve FPDEs.These transformations allow us to get the exact solutions of the target equations without having to linearize or discretize.They are used to convert FPDEs to ordinary equations when using only one transformation and to algebraic equations when using a double integral transformation.Some examples of these transformations are: the Sumudu transform [10], the natural transform [11], the Elzaki transform [12], the novel transform [13], the Aboodh transform [14], the double Sumudu transform [15,16], the double Elzaki transform [17], the double Shehu transform [18], and the double Laplace-Sumudu transform [19,20].
Diverse partial differential equations have recently been effectively solved using the double Sumudu-Elzaki transform (DSET), a novel double integral transform technique [21].Unfortunately, unlike other integral transforms, this transformation is unable to handle complex mathematical models or nonlinear problems.In order to handle a variety of nonlinear differential equations, some researchers have combined these integral transforms with additional techniques, such as the homotopy perturbation method, the variational iteration method, the differential transform method, and the Adomian decomposition method [22,23].
The primary goal of this research is to broaden the application of DSET by using it to solve FPDEs.We show the effectiveness of the proposed method by applying DSET to a number of interesting applications to get the exact solutions.
The following subjects will be covered in this essay's succeeding sections.We provide some basic definitions and theorems of CFDs in Section 2. Section 3 presents the fundamental DSET definitions, features, and theorems.Section 4 describes the model and process for using the DSET to provide accurate analytical answers to the specified FPDEs.Five exemplary scenarios are utilized in Section 5 to illustrate the recommended approach's liability, convergence, and efficacy.In Section 6, we explain the numerical results and show how the DSET is accurate and efficient.Section 7 also has conclusions.

Preliminaries
In this section, we present basic definitions and notions that will be used in the present work.

Mathematical Problems in Engineering (I) Gives us
The same method can be used to demonstrate the remaining results.
The CFD w.r.t u; for the function ξðu; tÞ can be rewritten as follows [30]: Mathematical Problems in Engineering by applying DSET to Equation (36), we get The same method can be used to demonstrate the remaining result.

Applications of DSET
In this section, we will apply the DSET to a family of FPDEs and get a simple formula for the general solution.
We consider a general nonhomogeneous FPDE of the form: on the ICs: and the BCs: where a; b; and c are constants, Rðξðu; tÞÞ is a linear operator, and hðu; tÞ is the source term.
Applying DSET to Equation (38), we get Using the SST for the conditions Equation (39) and the SET for the conditions Equation (40), to get By substituting Equation (42) into Equation (41), we have Simplifying Equation (43), we obtain 44), we get

Illustrative Examples
In this section, we will construct a few different examples to show how the DSET can be used and how effective it is.
Example 1.Consider the linear fractional heat equation: with the ICs: and the BCs: Solution.Operating the DSET on Equation ( 46) and SST on Equation ( 47) and the SET on Equation (48), we get substituting the SST and SET of initial and boundary conditions in Equation (49), and simplifying, we get In Figure 1, we sketch the approximate solution of Equation ( 52) with different values of the fractional order β when t ¼ 0:03 and u 2 ð0; 6Þ.
Example 2. Consider the linear fractional Klein-Gordon equation: on the ICs: and the BCs: Solution.Operating the DSET on Equation (53) and SST on Equation (54) and the SET on Equation (55), we get substituting the SST and SET of initial and boundary conditions

Mathematical Problems in Engineering
in Equation (56), we get simplifying Equation (58), we obtain In Figure 2, we sketch the approximate solution of Equation ( 60) with different values of the fractional order β when t ¼ 1:5 and u 2 ð1; 6Þ: Example 3. Consider the linear one-dimensional time fractional Burgers equation: on the ICs: and the BCs: Solution.Operating the DSET on Equation (61) and SST on Equation (62) and the SET on Equation (63), we get simplifying Equation (66), we obtain 67), we get In Figure 3, we sketch the approximate solution of Equation ( 68) with different values of the fractional order β when t ¼ 0:8 and u 2 ð1; 6Þ: Example 4. Consider the linear fractional Fokker-Planck equation: on the ICs: and the BCs: Solution.Operating the DSET on Equation (69) and SST on Equation (70) and the SET on Equation (71), we get in Equation (72), we get simplifying Equation (74), we obtain In Figure 4, we sketch the approximate solution of Equation ( 76) with different values of the fractional order β when t ¼ 4 and u 2 ð1; 6Þ:

Mathematical Problems in Engineering
Example 5. Consider the linear fractional telegraph equation: on the ICs: and the BCs: Solution.Operating the DSET on Equation (77) and SST on Equation (78) and the SET on Equation (79), we get in Equation (81), we get simplifying Equation (82), we obtain taking ðS w Þ −1 ðE q Þ −1 of Equation (83), we get In Figure 5, we sketch the approximate solution of Equation ( 84) with different values of the fractional order α when u ¼ 3 and t 2 ð1; 6Þ:

Results and Discussion
In order to show the accuracy and usefulness of the recommended approach, in this section we will look at the numerical evaluation of the results of fractional equations that have been proposed to be solved.Furthermore, we will compare the numerical behavior of the solutions to FPDEs with that of equations with integer derivatives.When β ¼ 1 and α; β ¼ 2; the closed-form solutions for Examples 1-5 is simply calculated.We have chosen to look at the numerical results for different values of fractional-order values α and β.We noticed that the solutions obtained for β ¼ 1; 0:95; 0:85; 0:75; and α; β ¼ 2; 1:95; 1:85; 1:75; are in coordination with the solutions of the closed forms for β ¼ 1 and α; β ¼ 2; as shown in Figures 1-5.It is sufficient to note that when β → 1 and α; β → 2; the solutions resulting from the fractional equations approach these exact solutions.

Conclusion
This article discusses a new double transformation called DSET.First, we applied the DSET to a few particular functions; following that, some theorems and properties connected to the DSET were presented and proved.To demonstrate the applicability and efficacy of the proposed transform, we used DSET to solve a wide range of FPDEs in mathematical physics.Based on the obtained findings, we conclude that the provided transform is efficient, suitable, reliable, and adequate to acquire the accurate solutions of FPDEs according to the taken-intoaccount starting and boundary conditions.Therefore, we may state that a broad class of linear FPDE schemes can be solved using this approach.

TABLE 1 :
DSET for some functions.