Abstract
In this study, we present a new double integral transform called the double formable transform. Several properties and theorems related to existing conditions, partial derivatives, the double convolution theorem, and others are presented. Additionally, we use a convolution kernel to solve linear partial integro-differential equations (PIDE) by using the double formable transform. By solving numerous cases, the double formable transform’s ability to turn the PIDE into an algebraic equation that is simple to solve is demonstrated.
1. Introduction
One of the most significant subjects that might be interesting in life issues is mathematics; it helps to improve mental abilities and foster logical and sound thinking. Additionally, it aids in the understanding and resolution of problems arising from a variety of natural phenomena as well as those pertaining to physical, engineering, biological, and computer science [1–5]. Recent studies have demonstrated the value of integral transforms in the solution of partial equations relating to, for example, sound, heat, static electricity, elasticity, fluid flow, and other physical and engineering phenomena [6, 7].
In addition, the integral transform helped to convert the partial and integral equations into algebraic equations that provide the problems’ precise solutions in the most straightforward manner [8–11]. The double integral transforms have drawn researchers’ attention recently because of their contribution to the solution of equations involving two variables and finding precise solutions in the simplest ways possible. However, with the development of science and the urgent need for mathematics to solve newly emerged problems, it was necessary to pay attention to scientists and researchers to obtain new and advanced methods to keep speed with these problems [12–14]. The most popular double integral transforms that are used to solve partial differential equations are the double Laplace transform [15, 16], the double Laplace-Sumudu transform [17], the double Sumudu transform [18], the ARA-Sumudu transform [19, 20], the double Shehu transform [21], the double Elzaki transform [12], and others [22]. Integral equations of various types played a significant role in explaining many phenomena and coming up with exact solutions when various sciences became complex as a result of their interactions with one another and scientists started studying all physical, chemical, biological, and engineering phenomena. As a result, integral equations are crucial for representing the many disciplines, which prompted scientists and researchers to look for a solution to make it possible for them to solve the integral equations.
There are several types of integral equations, including Volterra integral equations and integro-differential equations; in this paper, we will focus on these two types of integral equations and solve many examples of each [23].
In 2021, with the advent of the formable integral transform (FT) that was presented by Saadah and Ghazal [24], its ability to solve ordinary, partial differential equations, and integral equations, and its properties that could enable us to solve a wide kind of differential and integral problems, we intend to define the new double formable transform. Moreover, it is worth mentioning here that the formable transform might be considered a special case of the new integral transform presented by Jafari in [25], and the idea of double integral transforms is also generalized by the authors in [14]. The main advantage of using the double formable transform is that it preserves the values of constants, which simplifies the calculations during the solving of problems, and it transforms the target problem from two-dimensional space into four-dimensional space. In this study, we focus on the double formable transform and study the main properties and the application of solving integral equations.
Our goal in this paper is to provide a solution to two types of integral equations, the Volterra integral equations, and the partial integro-differential equations, by using the double formable transform (DFT). We have also clarified and explained some theorems and properties with their proofs, and among the most important theorems that will help us solve the examples are the double convolution, partial derivatives, and many more, as we explain in the subsequent sections.
This article is organized as follows: in Section 2, the fundamental facts of formable transform are introduced. In Section 3, we introduce a new double formable transform and present some properties and theorems. In Section 4, we apply the DFT to different types of IDEs and solve some examples. Lastly, in Section 5, the conclusion section is presented.
2. Fundamental Facts of Formable Transform [24]
A new integral transform known as the formable transform of the continuous function on the interval is defined by
The inverse formable transform is defined as
2.1. Formable Transform of Some Basic Functions
2.2. Formable Transform of Derivatives
The above results can be obtained from the definition of FT with simple calculations.
3. Double Formable Transform (DFT)
Definition 1. Let be a continuous function of two variables and . Then, the double formable transform (DFT) of a function is defined aswhich is equivalent toThe inverse of DFT is given byWe denote the single FT as follows:(i)With respect to : (ii)With respect to : Clearly, the DFT is a linear transformation as shown below:where and are constants and are exists.
Property 1. Let . Then,
Proof.
3.1. DFT of Some Basic Functions
(i)Let . Then, From equation(3), we get (ii)Let and are constants. Then, From equation (4), we get (iii)Let and are positive constants. Then,
From equation (5), we get .
Similarly,
Using the property of complex analysis, we have
Using Euler’s formula,
and the formula,
Thus, we find the DFT of the following functions:
Thus, we get
From equations (5) and (6), we get
3.2. Existence Conditions for DFT
If is a function of exponential orders and as and and if there exists a positive constant such that and , we havewe can write as and , and .
Theorem 1. Let be a continuous function on the region of exponential orders and . Then, exists for and provided and .
Proof. Using the definition of DFT, we get
3.3. Some Basic Theorems of DFT
Theorem 2 (Shifting Property). Let be a continuous function, and . Then,
Proof. Using the definition in equation (14),By letting , and and , and in (29), we have
Theorem 3 (Periodic function). Let be exists, where is a periodic function of periods and such that,
Then,
Proof. Using the definition of DFT, we getUsing the property of improper integral, equation (33) can be written asPutting and on the second integral in equation (34). We obtainUsing the periodicity of the function , equation (35) can be written byUsing the definition of DFT, we getThus, equation (37) can be simplified into
Theorem 4 (Heaviside function). Let exists, thenwhere is the Heaviside unit step function defined as
Proof. Using the definition of DFT, we getPutting and in equation (41). We obtainThus, equation (42) can be simplified into
Theorem 5 (Convolution theorem). Let and are exists, thenwhere and the symbol denotes the double convolution with respect to and .
Proof. Using the definition of DFT, we getUsing the Heaviside unit step function, equation (45) can be written asThus, equation (46) can be written as
Theorem 6 (Derivatives properties). Let be a continuous function. Then, we get the following derivatives properties:(a).(b).(c).(d) We only present the proof of (a), (c), and (e), the proof of (b) and (d) are the same.
Proof. (a) Using the integration by part, we obtain. Let , Thus, So, we get,
Proof. (c). Using the integration by part, we obtain. Let Thus, Using equation (49), we have proof of (e) by similar (a) and (c) arguments, one can get Using the integration by part, we obtain Using equations (11) and (49), we haveWe summarize the previous results in the following table below Table 1:
4. Presentation of DFT Method in Solving Integral Equations and Applications
To show the applicability and strength of DFT in solving integral equations, we use it to solve some kinds of integral equations and illustrate the method by solving several examples.
4.1. Volterra Integral Equation
Consider the following Volterra-integral equation:where is the unknown function, and are two known functions and is a constant. The main idea of the method is to apply the DFT to equation (54) as the following:
Now, using the differentiation property of the DFT on equation (55) and Theorem 5, we have
Equation (56) can be simplified as follows:
Operating with the inverse of DFT on both sides of equation (57) giveswhere represents the term arising from the known functions and .
Example 1. Let us consider the following Volterra integral equation:By substituting the values of functions
in the general formula in equation (58), we obtainNow, applying inverse DFT to equation (60), then the solution of equation (59) is given by
Example 2. Let us consider the following Volterra integral equation:Appling the DFT to equation (62), we haveEquation (63) can be simplified as follows:Taking the inverse DFT to equation (64), we get the solution of equation (62) as follows:
Example 3. Let us consider the following Volterra integral equation:Appling the DFT to equation (66), we haveEquation (67) can be simplified as follows:Applying the inverse DFT to equation (68), we get the solution of equation (66), we get
4.2. Volterra Integro-Partial Differential Equations
Consider the following Volterra integro-partial differential equations:with the conditionswhere is the unknown function, and are two known functions, and is a constant.
To get the solution of equation (70), we firstly apply the single formable transform to all conditions in equation (71) and the DFT to equation(70), as follows:
Thus, we have
Using the differentiation property of the DFT to equation (73) and the double convolution theorem, we have
Equation (44) can be simplified as follows:
Operating the inverse of DFT to both sides of equation (75), we getwhere represents the term arising from the known functions , and .
Example 4. Let us consider the following Volterra integro-partial differential equations:subject to the conditionsBy substituting the values of functionsin the general formula in equation (76), we obtainEquation (80) can be simplified as follows:Thus,Taking the inverse of DFT to equation (82), then the solution of equation (77) is
4.3. Integro-Partial Differential Equations
Consider the following integro-partial differential equation:with the conditionswhere is unknown function, and are two known functions.
The main idea of this method is to apply formable transform to all conditions in equation (85) and the DFT to equation (84), as the following:
The single formable transform to conditions in equation (85), we get
The DFT on both sides of equation (84), to get
Using the differentiation property of the DFT of equation (87), above conditions and Theorem 5, we have
Equation (88) can be simplifying as follows:
Operating with the inverse of DFT on both sides of equation (89), giveswhere represents the term arising from the known functions and .
Example 5. Let us consider the following integro-partial differential equation:with the conditionsBy substituting the values of functions in the general formula in equation (89), we obtainTaking the inverse DFT to equation (94), then the solution of equation (91) is
5. Conclusions
Different phenomena are modeled using partial integro-differential equations in sciences, engineering, and social sciences. Solving partial integral equations using the double transform technique is effective. Convolution kernels are used to solve similar problems. In this study, we use the DFT technique to solve integral equations of different kinds. We present the definition of DFT and prove some basic properties and theorems related to its existence and derivatives. Finally, we apply DFT to solve partial integral equations and partial integro-differential equations. The outcomes show the simplicity and applicability of DFT in solving such problems.
Data Availability
No underlying data was collected or produced in this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.