Abstract
This paper deals with the solution of a third kind mixed integro-differential equation (MIDE) in displacement type in space . The singular kernel is modified to take a logarithmic form, while the kernels of time are continuous and positive functions. Using the separation of variables technique, we have a system of Fredholm integral equations (FIEs) that can be transformed into an algebraic system after using orthogonal polynomials. In all the previous researchers’ works, the time periods were divided, and the mixed equation transformed into an algebraic system of FIEs. While when using the separation method, we are able to obtain FIE with time coefficients, and these functions are described as an integral operator in time. Thus, we can study the behavior of the solution with the time dimension in a broader and deeper than the previous one. Some examples are given and discussed to show the performance and efficiency of the proposed methods.
1. Introduction
Various problems in mathematical physics and contact problems in the theory of elasticity lead to an integral equation of the and kinds, (see Papov [1]). In thermoelasticity, Abdou and Basseem in [2] derived a closed form of Gaursat functions for first and second fundamental problems with variant time. Basseem and Alalyani in [3] used Toeplitz and Nystrom methods on the solution of quadratic nonlinear integral equation with different singular kernels. The theory of singular integral equations, developed by the scholar Muskhelishvili in [4], has assumed different techniques and increased important applications in different areas of science. Orthogonal polynomials are widely used in many areas such as airfoil (see [5]), polymer physics (see [6]), fracture mechanics (see [7]), contact radiation, and molecular conduction (see [8]). Abdou et al. in [9] used orthogonal polynomials method to discuss the solution of an MIE with singular kernel in some contact problems. Hafez and Youssri in [10] applied the Legendre-Chebyshev collocation method for solving two-dimensional linear and nonlinear mixed Volterra-Fredholm integral equation (MVFIE). Tohidi and Samadi in [11] used Legendre polynomial in optimal control of nonlinear Volterra integral equation (NLVIE). Nemati et al. in [12] applied Legendre polynomial in a class of two-dimensional nonlinear VIE. Khader et al. in [13, 14] used the Chybyshev polynomial method on the fractional diffusion equation, and they also used Legendre polynomials to solve Ricatii, logistic and delay DE with variable coefficients.
For the third kind of MIDE problem, we will use two orthogonal polynomials for the first time to solve these types of problems. Then, we will compare the numerical results and errors between them.
Consider the bounded differential operator in a general form
Consider MIDE in the form
The functions and are considered as a continuous time functions in the space . are time parameters. The kernel of position is singular, while the unknown function behaves like ; that is, the free term belongs to the class The function determines the kind of Eq. (2). For the displacement and contact problems in the theory of elasticity, we assume the condition where is the external pressure that affects the surface of the elastic material under study, while in the nuclear and mathematical physics problems, we let
Many different and special cases with various applications can be derived from Eq. (2) as seen in some of previous references as an especial cases.
The kernel of Eq. (2) can be written in the form (see [15])
Since and are very small, ,
Hence, Eq. (2) becomes
The importance of Eq. (7) comes from its special cases: (i)When , a third kind FIE is obtained.(ii)When we have an MFVIE of a third kind.(iii)When a third kind VFIE is attained.
The structure of this article is as follows: In Section 2, a separation method is used to obtain a system of FIEs for the third kind of MIDE problem. In Section 3, we represent the unknown function in two different forms, Legendre polynomials and Chebyshev polynomials, and then, we use a technique of orthogonal polynomials to discuss the solution of each system. In Section 4, we illustrate two applications as numerical test examples to show the performance and efficiency of the proposed methods. In Section 5, we present our conclusion.
2. Separation Method
Consider
Eq. (7) becomes where where is a constant.
In order to guarantee the existence of a unique solution of Eq. (9), we assume the following: (i)The singular kernel satisfies the discontinuous condition (ii)While the function and its normality are defined as , the function belongs to the space in which , where is a constant.(iii)For all values of , the given functions of time are bounded, in which and where and are constants.(iv)The unknown function satisfies Hlder condition with respect to time and Lipchitz condition with respect to position.
Let be an operator defined as where
Theorem 1. Under the above assumptions, the solution of the system of IE (9) is unique under the condition in for every .
Proof. Since Using Caushy Schwarz inequality and previous conditions, we get Therefore,
It is obvious that the operator maps the ball into itself where
The inequality (17) involves the boundedness of the operator under the condition .
Let two functions and be two solutions of (9), and then, the formula (12) leads to
Using conditions (i)-(iv) and Cauchy Schwarz inequality, we deduce that
It follows that for is a contraction operator of a system (9). Hence, there exists a unique solution in by a Banach fixed point theorem for every . It is easy to show that the MIE (2) can be changed into the third kind FIE with time coefficients where
The formula (21) leads to discuss the following cases:
At Eq. (2) is called MIE of the first kind, but using the separation technique, we get a system of Fredholm integral equation of the second kind depends on time. This system has a unique solution under the condition
If is a constant function, Eq. (2) is called MIE of the second kind, while is a function of , and we have a system of Fredholm integral equation of the third kind, in which its uniqueness holds under the condition
In all previous research, the time periods in the mixed equation were divided, and this equation transformed into an algebraic system for FIEs (see [16, 17]). Here, we use the separation technique method to get FIEs with time parameters that are described as an integrated operator in time.
3. Method of Solution
3.1. Legendre Polynomial Form
To obtain the solution of Eq. (21), we assume the unknown function in the Legendre polynomial form
Since it is difficult to discuss the numerical solution using formula (24), then it can be truncated to where are constants and are Legendre polynomials that satisfy the orthogonal relation (see [18])
By substituting (25) in Eq. (21), for , we can easily show that
By differentiating Eq. (21) with respect to , for , we get
Eq. (24) through the use of the following relation leads to the following relation
Here, are the associated Legendre polynomials of the first kind that satisfy the following orthogonal relations (see [18])
In sense of Eq. (24) and Eq. (30), the function of Eq. (28) can be represented as (see [18])
The constant coefficients can be determined using Eq. (31) in the following form
Using the Legendre polynomial of the second kind with its relation
Eq. (28) through the use of (30) and (33) yields
Multiplying both sides of Eq. (36) by integrating from to , and through the use of Eq. (31), Eq. (32), and the following relation (see [18]) we get where
3.1.1. Convergence of Algebraic System
The convergence of the algebraic system (38) can be derived from the following:
Since the series behaves like , , where and The solution of this convergent linear algebraic system can be determined for , and then, approximated solution is obtained.
3.2. Chebyshev Polynomial Form
Recall Eq. (21) and consider in Chebyshev form as
To obtain the solution numerically, the formula (41) can be truncated to where are constants and are Chebyshev polynomials that satisfy the following (see [18]): (i)Algebraic formula: (ii)Integral relations: (iii)Orthogonality rule:
In sense of Eq. (41), the two given functions and are written as
By substituting (41), (47), and (48), Eq. (21) through the use of (43) takes the form
Multiplying by , integrating with respect to from -1 to 1, and using (43) and (46), we have the following cases: (i)For we have (ii)For we have (iii)For we have (iv)For , we have where
The formula (53) represents LAS with time coefficients. The unknown constants can be determine after the following cases: (a)When we have where (b)When we have where
The previous algebraic systems have a unique solution which can be obtained after determining the constants,
4. Numerical Results
In this section, some numerical examples are considered to show the accuracy and efficiency of the proposed methods.
4.1. Application 1
Consider the MIE where
In Tables 1 and 2, the solution and its corresponding error are determined by using Legendre polynomials and Chebyshev polynomials .
4.2. Application 2
Let , , and in Eq. (21). In Table 3, the solution and its corresponding error are determined by using Legendre polynomials and Chebyshev polynomials , where .
The following figures represent and its corresponding errors .
5. Conclusion
In this paper, we apply a separation of variable technique in an MIDE to have a system of FIE with time operator coefficients. The solution is then obtained after using the orthogonal polynomials method with two different polynomials (see Figures 1–4). According to numerical results, the Legendre polynomial has a higher efficiency than the Chebyshev polynomial . From all Tables 1–3, corresponding errors show that the Legendre polynomial is a more powerful solver for these kinds of equations. We also observed that the solution behaves like its corresponding given function; that is, is even in the first example; using the Chebyshev orthogonal polynomial method, its solution is symmetric in the domain.
Data Availability
The authors confirm that the data supporting the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest to report regarding the publication of this study.