An Application of the Weighted Mean Value Method to Fredholm integral equations with Toeplitz plus Hankel kernels

In the present work, we study Fredholm integral equations of the second kind with Toeplitz, Hankel, and Toeplitz plus Hankel kernels. The weighted mean-value method is employed and extended to obtain numerical solutions for one-and multidimensional Fredholm integral equations of the second kind. By a successful application of this method, we first convert the integral equation into a system of algebraic equations, then solve this system and obtain promising results. Finally, numerical examples are provided to show the simplicity and applicability of the method.


Introduction
Integral equations appear in many different forms and classification of them depends mostly on their kernels.Before proceeding to solve an integral equation analytically or numerically, it is vital to analyze its kernel first.Integral equations with Toeplitz, Hankel, or Toeplitz plus Hankel (TPH) kernels and their solutions have many useful applications in many diverse areas such as fluid dynamics, quantum mechanics, and scattering theory.The literature is extensive and we list some of them here for interested readers [5,8,4].Before progressing further, it is important to emphasize some important works on the integral equations with TPH kernels.In [15], the authors investigated the relation between the solution and partial differential equations.[2,11,12] proposed solutions in closed-form expression for some particular cases of TPH kernels by the generalized convolution.In [13,14], a new polyconvolution was introduced and applied to solve a system of integral equations with TPH kernels.In [10], integral mean value theorem was used to numerically solve the N-th order Fredholm integro-differential equations.[9] applied mean value theorems for integrals to solve Fredholm integral equations with TPH kernels.In [1], the weighted mean value method (WMVM) was introduced and applied to various classes of one-dimensional Fredholm integral equations including a system of Fredholm integral equations.The method is very simple and the results are promising.In this article, our main aim is to extend the applications of WMVM and to show that it also works for multidimensional Fredholm integral equations with TPH kernels.We consider one-and multidimensional Fredholm Integral equations with TPH kernels which, respectively, have the following form: The rest of the article is organized as follows: Section 2 deals with basic assumptions and notations that will be needed for subsequent sections.In section 3, the main section, algorithms for finding solutions for one-and multidimensional Fredholm integral equations with TPH kernels are introduced.In section 4, some elaborate examples are given to show the simplicity and efficiency of the proposed algorithm.To be able compare our results with those in the literature, we mostly use the same examples as those in [9].

Basic assumptions and notations
In this section, fundamental theorems and basic notations that will be used in the later sections are stated.Also, some key assumptions are made here, once and for all, as to make the subsequent sections more comprehensible and clear from unnecessary complicated notations.Let us start with stating the weighted mean value theorem for integrals.
In recent studies conducted by [6,7], it was shown that an application of the mean value theorem to Fredholm integral equations leaded to significant results.Despite some shortcomings, the simplicity of the algorithm and the promising results obtained from running it make the method worthwhile to investigate further.Another recent article [1] concluded that an application of the weighted mean value theorem to Fredholm integral equations achieved promising results.The main aim of this article is to apply the weighted mean value theorem for integrals to Fredholm integral equations with special kernels.In the foregoing argument, it is mentioned that there are some drawbacks of an application of the mean value theorem to Fredholm integral equations.To be precise, let us consider the following Fredholm integral equation: To implement the algorithm, it is assumed that applying the mean value theorem to Eq. (2.3) results in for some c ∈ [a, b].Indeed, since the kernel K depends both on t, and s, application of the mean value theorem should produce c(t) rather than a constant c.However, explicitly determining this function c(t) is an open problem to the best of our knowledge.On the other hand, implementing the algorithm by taking c(t) as a constant works for many practical cases.A similar argument can be carried out for multidimensional Fredholm integral equations.

Assumptions :
We list all assumptions here and use them when they are needed without stating them again throught the article.
never changes sign in its domain of definition.This assumption will allow us to apply the weighted mean value theorem to the integral equation in hand.
A2 : For multidimensional case, we apply the weighted mean value theorem for one dimensional integrals successively until all the integrals appearing in the equation is removed.Notice that A1 allows us to carry out this step since the integral preserves the sign of the kernel function.
A3 : To implement the algorithm, we assume that an application of the weighted mean value theorem produces a constant c rather a function c(t).This is assumed for multidimensional case too.

Main section
In this section, algorithms for different cases are given in detail.We first investigate one-dimensional case which is followed by two-dimensional case.The last subsection is reserved for multidimensional case.

One-dimensional Case:
Consider the one-dimensional Fredholm integral equation where λ is a constant parameter, f , T, H, F are known continuous functions and ϕ is the desired function.
Apply the WMVM to Eq. (3.5) and get where Substituting c for t in Eq. (3.6) results in which is the first algebraic equation that will be used to find the approximate solution.Since there are two unknowns, c and ϕ (c), one more equation is needed to form a 2 × 2 algebraic system of equations.To obtain the second equation, replace ϕ appearing in Eq. (3.5) under the integral sign by Eq. (3.6).That is, Substituting c for t in Eq. (3.8), we obtain the second desired equation Combining Eq. (3.7) and Eq.(3.9) and solving the resulting system of algebraic equations, the approximate values for c and ϕ (c) are obtained.Using these values in Eq. (3.6), the approximate solution for Eq.(3.5) is obtained.

High-Dimensional Case:
Let us take into consideration the following high-dimensional Fredholm integral equation  For the i th step, we have

.20)
Let that c = (c 1 , c 2 , . . ., c n ).To form an (n + 1) × (n + 1) system of algebraic equations, we first apply WMVM to Eq. (3.18) n−times and get We then replace c with t in Eq. (3.20) and get This is the first equation that will be used to form the (n + 1) × (n + 1) system of algebraic equations.The remaning n equations will be obtained by substituting Eq. (3.21) into the right hand side of Eq. (3.20) and then replacing t by c as follows By solving this system of algebraic equations, c and ϕ (c) will be obtained.Substitution of these values into Eq.(3.21) reveals an approximate solution to Eq. (3.18).

Numerical examples
In this section, we work out some examples to show the simplicity and applicability of the algorithms which are provided in the previous section.The exact solution for this equation is ϕ (t) = e 2t .
An application of the WMVM produces First, substitute c for t in Eq. (4.24) to get Then, if ϕ apearing under the integral sign in Eq. (4.23) is replaced with Eq. ( 4.24), one gets Substituting these values into Eq.( 4.24), it is easy to see that ϕ (t) = e 2t , which is the exact solution.
We apply the WMVM and get ϕ (t) = − sin(t) + 2ϕ 2 (c)(cos(t) + sin(t)).(4.31) and follow the procedure given in the previous section to obtain the following 2 × 2 algebraic system of equations: Solving this sytem of equation, one solution is that Substituting these values into Eq.(4.31), we get ϕ (t) = cos(t), which is the exact solution.

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We apply the WMVM and get and follow the procedure given in the previous section to obtain the following 3 × 3 algebraic system of equations: Solving this sytem of equation, we get
We apply the WMVM and get ) .
Solving this sytem of equation, one gets Substituting these values into Eq.(4.35), the exact solution ϕ (t 1 ,t 2 ) = e t 1 +t 2 is obtained.

Conclusion
This study employed the WMVM for solving one-and multidimensional Fredholm integral equations with Toeplitz, Hankel, and TPH Kernels.Application of the WMVM under some mild assumptions reduced the integral equation to a set of algebraic equations.By solving the obtained system of equations, an approximate solution was obtained.Numerical examples were provided to show the simplicity and efficiency of the algorithm.All of the calculations were carried out in Matlab software.

Theorem 2 . 1 .
[3] Assume f and g are continuous on [a, b].If g never changes sign in [a, b] then, for some c ∈ [a, b], we have
Example 4.5.Consider the following two-dimensional Fredholm integral equation with Toeplitz plus Hankel kernel.