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# Results Involving Partial Differential Equations and Their Solution by Certain Integral Transform

1 Department of Mathematics, Faculty of Science, Zarqa University, Zarqa, 13110, Jordan

2 Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Amman, 11134, Jordan

3 Department of Mathematics, Faculty of Science and Letters, Agri Ibrahim Çeçen University, Agri, 04100, Turkey

* Corresponding Author: Shrideh Al-Omari. Email:

(This article belongs to the Special Issue: On Innovative Ideas in Pure and Applied Mathematics with Applications)

*Computer Modeling in Engineering & Sciences* **2024**, *138*(2), 1593-1616. https://doi.org/10.32604/cmes.2023.029180

**Received** 06 February 2023; **Accepted** 25 April 2023; **Issue published** 17 November 2023

## Abstract

In this study, we aim to investigate certain triple integral transform and its application to a class of partial differential equations. We discuss various properties of the new transform including inversion, linearity, existence, scaling and shifting, etc. Then, we derive several results enfolding partial derivatives and establish a multi-convolution theorem. Further, we apply the aforementioned transform to some classical functions and many types of partial differential equations involving heat equations, wave equations, Laplace equations, and Poisson equations as well. Moreover, we draw some figures to illustrate 3-D contour plots for exact solutions of some selected examples involving different values in their variables.## Keywords

Mathematical modeling is one of the main aspects of mathematics that deals with the real-life problems involved in various fields of science. Mathematical modeling transforms life problems into mathematical models and analyzes models mathematically. It is generally used in the computational theory of the field of natural sciences, computer sciences, engineering and social sciences as well. Mathematical models are the cornerstone of the modern scientific understanding of sound, heat, diffusion, static electricity, thermodynamics, fluid dynamics, elasticity, quantum mechanics and general relativity. They can be generated from various mathematical considerations comprising differential geometry and calculus of variation. Partial differential equations (PDEs) occupy a large sector of oriented scientific fields in physics and engineering. Various methods have been developed over the years to find solutions of partial differential equations, some of which reduce partial differential equations to one or more than one ordinary differential equation (ODE). However, a number of approaches have been proposed to deal with partial differential equations, such as the method of group static solutions [1,2], the non-classical method of group static [3], the method of partially invariant solutions [1] and the general method of variable separation for both linear systems. In the non-linear equations we recall the Hamilton-Jacobi equation, residual power series [4], reproducing kernel reproduction [5,6], Laplace residual power series and others [7–9].

Partial differential equations can also be handled using integral transformations to determine the exact solution to the target equations without the need for linearization or discretization. The transformation techniques are powerful and straightforward. Therefore, scientists have put a lot of efforts into understanding and improving techniques. Many integral transforms have been developed such as the Laplace transform [10], Novel transform [11], Sumudu transform [12], Elzaki transform [13], Natural transform [14], ARA transform [15], formable transform [16] and others [17–19].

In the sequel, extensions of double transformations are extensively developed for solving PDEs and obtaining results by performing solutions based on numerical approaches [20]. Double transforms such as the double Laplace transform [21–23], double Shehu transform [24], double Kamal transform [25], double Sumudu transform [26–31], double Elzaki transform [32], double Laplace-Sumudu transform [33–35] and ARA–Sumudu transform [36,37] have been considered in the literature. The ARA transform [15] was used for solving ordinary fractional differential equations [38] and some nonlinear PDEs [39]. The double ARA transform was employed for some kinds of partial and integral equations in [40–47].

In this work, we extend the idea of the single and double ARA transforms to a triple ARA transform which can be used in solving larger classes of PDEs. In this context, we define the new transform and present its main properties. More results regarding the existence, partial derivatives, and convolution theorems are also presented. However, the novelty of this study fulfills the generalization of the double ARA transform and its application to wider classes of PDEs.

Motivation in this work emerges from introducing a new approach that solves partial differential equations of multi-variables. We present characteristics of the transform and compute its values for some basic functions. Further, we show its applicability to various types of PDEs of great importance in physics. However, we organize this paper as follows. In Sections 2 and 3, we discuss the single and double ARA transforms and obtain their main properties. In Section 4, we introduce the triple ARA transform and prove some basic results on existence, elementary functions, convolutions, and partial derivatives. In Section 5, we apply the TARAT in solving PDEs and consider some examples and their 3D graphs. In Section 6, we give a conclusion section.

2 Basic Definitions and Preliminary Results

In this section, we recall some definitions and notations from [15] and establish several basic properties of the ARA transform.

Definition 2.1. The

provided the integral exists. In particular, for

when the integral exists. For simplicity, we make use of the notation

The inverse ARA transform is defined by [15]

Following is a result which is very needful in the sequel.

Theorem 2.1. Let the function

where

Proof. By employing the definition of the ARA transform we obtain

which reveals

It is clear from the above equation that the improper integral converges for all

This completes the proof of the theorem.

Note that, the existence condition of the ARA transform fulfils when the function

Let

•

•

•

•

•

•

This section discusses the new double ARA transform DARAT. It provides fundamental properties and characteristics of the transform including existence conditions, linearity and inversion formula for the proposed new double transform. Moreover, it provides some important properties and results and computes the double ARA transform of some elementary functions.

Definition 3.1. Let

provided the integral exists. Clearly, the double ARA transform is linear, which can be shown from the equation

provided

Now, we state without proof the following theorems [40,41].

Theorem 3.1 [40] Let

Theorem 3.2 (Periodic Function) [40] Let

Theorem 3.3 (Heaviside Function) [40] Let

where

Theorem 3.4 (Convolution Theorem). [40] Let

where

In the following, we present Table 1 to illustrate the values of the double ARA transform of some elementary functions and give main properties of the transform.

In this section, we introduce a new triple transform, called the triple ARA transform (TARAT), and derive some basic properties and establish certain results involving partial derivatives.

Definition 4.1. Let

where

Note that, the triple ARA transform is a linear transform, due to the fact that if

To simplify the application of the triple ARA transform, we need to recall the following notations:

By a similarity in the variables

The existence conditions of the triple ARA transform are presented in the following theorem.

Theorem 4.1. Let

where

Proof. Assume that

Also, the inverse triple ARA transform implies that

This finishes the proof of the theorem.

In Table 2, we provide some values of the triple ARA transform for some basic functions.

Now, we introduce some properties of the new approach.

Property 4.1. (Change of scale property) Let

where

Proof. Based on the definition of the triple ARA transform, we have

Putting

This finishes the proof of our result.

Property 4.2. (First shifting property) Let

where

Proof. Based on the definition of the triple ARA transform, we obtain

This finishes the proof of our result.

The following result is very useful in solving some kinds of partial differential equations.

Theorem 4.2. If

where

Proof: The definition of the triple ARA transform implies

From the properties of the single ARA, we get

Again, using some properties of the ARA transform to both variables

This finishes the proof of the theorem.

The application of triple ARA on partial derivatives of different types is illustrated on the following theorem.

Theorem 4.3. Let

i)

ii)

iii)

iv)

Proof (i). From the definition of the triple ARA transform, we write

To solve the integration inside the brackets, we use the idea of integration by parts with the assumptions that

(ii). By using part (i) of the theorem, it follows that:

where,

Similarly, one can follow the proof of (iii) and (iv).

This proves the theorem.

In view of the similarity between

The following property is a basic result from Theorem 4.3, and the proof can be obtained by similar arguments to the above theorem.

Corollary 4.1. Let

•

•

•

•

•

The following corollary presents a new application of the triple ARA transform, that is useful in solving some kinds of integral equations.

Corollary 4.2. If

Proof. Suppose that

and

Applying the triple ARA transform to both sides of Eq. (4), and using Theorem 4.3 suggest to write

Thus, we write

which implies that

This ends the proof of our corollary.

Now we study the effect of the triple ARA on the Heaviside function.

Theorem 4.4. Let

where

Proof. From the definition of the triple ARA transform, we have

Putting

Thus, we infer

This ends the proof of our theorem.

The following theorem presents the application of the triple ARA on the multi convolution property.

Theorem 4.5.(Convolution Theorem) If

where

Proof. Following the definition of the triple ARA transform, we write

The definition of the Heaviside function implies

Thus, we get

Using Theorem 4.4 reveals

This completes the proof of the theorem.

In this section, we present an algorithm for using the triple ARA transform in solving partial differential equations and provide solutions of some examples including partial differential equations.

5.1 Algorithm of the TARAT Method

To illustrate the method of using the triple ARA transform in solving partial differential equations, let us consider the following partial differential equation:

where

which yields

where

where

To obtain the solution in the original space, we apply the inverse triple ARA transform to Eq. (14) to have

Hence, the desired result is therefore established.

In this section, we apply the triple ARA transform operator to solve the wave equation, heat equation, Poisson equation and some few others.

Example 1: Consider the following wave equation

where

Solution: Operating the triple ARA transform to both sides of Eq. (5) yields

Hence, using the derivative properties implies

To simplify Eq. (17), we find the transformed values of the conditions (16) as follows:

By substituting the above values and using simple calculations, we can derive

Applying the inverse triple ARA transform

The following Fig. 1 illustrates the contours 3D plot of the exact solution to Example 1, with different values of the variables as follows.

Example 2: Consider the heat partial differential equation

where

Solution: Operating the triple ARA transform to both sides of Eq. (19) gives rise to

Employing properties of the derivative infer

To simplify Eq. (22), we compute the transformed values of the conditions as follows:

Substituting the above values included in Eq. (22) and applying some calculations we reach the conclusion

Applying the inverse

Our example has, therefore, been solved.

The following Fig. 2 illustrates the contours 3D plot of the exact solution to Example 2, with different values of the variables as follows.

Example 3: Consider the following Poisson partial differential equation:

where

Solution: By arguments alike to those employed for Examples 1 and 2, we apply the TARAT to Eq. (24) and compute the transformed values of the initial conditions to infer

Thus, we have the solution, in the triple ARA space, as

Now, we reach the solution to Example 3 in the original space, by applying the inverse TARAT

Our example has been solved.

The following Fig. 3 illustrates the contours 3D plot of the exact solution to Example 3, with different values of the variables as follows.

Example 4: Consider the following Laplace partial differential equation:

where

Solution: Following steps alike to those used in the previous examples, then by applying the TARAT to the conditions in (28) and operating the triple ARA transform to Eq. (27) we establish that

Thus, we get the solution in the triple ARA space in the form

Following that, we get a solution to Example 4 in the original space after applying the inverse triple ARA transform

Our example has been solved.

The following Fig. 4 illustrates the contours 3D plot of the exact solution to Example 4, with different values of the variables as follows.

Example 5: Consider the following partial differential equation

where

Solution: Applying the triple ARA transform to the condition (31) infers

Now, allowing the triple ARA transform to act on both sides of Eq. (31) and using the transformed values of the conditions with simple calculations reveal

This indeed implies

Operating the inverse TARAT

Our example has then been solved.

The following Fig. 5 illustrates the contours 3D plot of the exact solution to Example 5, with different values of the variables as follows.

In this paper, a new triple transform called the triple ARA transform is presented and several properties and theorems including linearity, existence, partial derivatives, and the multiple convolution theorem are introduced. Tables that summarize the new results are established to illustrate the new approach. The methodology of using the triple ARA transform in solving partial differential equations is given and further examples of several types of partial differential equations involving heat, wave, and Poisson equations are discussed. The outcomes have shown the simplicity of using the proposed transform in different aspects. In future work, we aim to solve integral equations and nonlinear partial differential equations based on the triple ARA transform.

Acknowledgement: Not applicable.

Funding Statement: This research received no external funding.

Author Contributions: All authors have read and agreed to the published version of the manuscript.

Availability of Data and Materials: Not applicable.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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*Computer Modeling in Engineering & Sciences*,

*138*

*(2)*, 1593-1616. https://doi.org/10.32604/cmes.2023.029180

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*Comput. Model. Eng. Sci.*, vol. 138, no. 2, pp. 1593-1616, 2024. https://doi.org/10.32604/cmes.2023.029180

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