New Applications and Improvements of Sinc Functions for Solving a System of Fredholm Integral Equations

Abstract

This study introduces two novel methodologies for solving systems of Fredholm integral equations, with particular emphasis on second-kind equations. The first method integrates the Sinc-collocation technique with a newly developed singular exponential transformation, enhancing convergence behavior and numerical stability. A comprehensive convergence analysis is conducted to support this approach. The second method employs a double exponential transformation, leading to a pair of linear equations whose solvability is established using the double projection method. Rigorous theoretical analysis is presented, including convergence theorems and newly derived error bounds. A system of two Fredholm integral equations is treated as a practical case study. Numerical examples are provided to illustrate the effectiveness and accuracy of the proposed methods, substantiating the theoretical results.

1. Introduction

The study of numerical solutions for integral equations has recently gained significant attention due to their wide application in modeling complex phenomena across various scientific and engineering fields. These equations are frequently encountered in viscoelasticity, fluid dynamics, electrodynamics, unstable aerodynamics, and thermoplastic processes. Among the numerous numerical techniques available, the Sinc method has emerged as a compelling and versatile approach to addressing these problems. Its remarkable accuracy and exponential convergence, particularly for issues involving singularities or defined on unbounded domains, make it highly effective for solving a diverse range of integral and differential equations. In particular, Sinc-based collocation and Galerkin methods have been successfully applied to manage high-order partial integro-differential equations.

The work presented in [1] aims to enhance certain aspects of the Sinc-Nyström method for solving Volterra integro-differential equations. The authors offer two main theoretical contributions: an analysis of the regularity of the solution and an examination of the convergence rate of the method. In [2], a comprehensive overview of various Sinc-based numerical methods is provided, emphasizing their applicability to computational problems. More recently, Rashidinia and Zarebnia introduced a Sinc-collocation method specifically designed for a certain class of problems [3]. Unlike traditional approaches, this method does not require smoothness at the endpoints and is based on the Sinc approximation framework. The authors assert that this technique can achieve exponential convergence, a claim that is supported by numerical experiments under specific conditions. Additionally, the study in [4] presents two improved variants of the Sinc-collocation method for solving Fredholm integral equations of the second kind, which can be expressed as:

$$u\left(x\right)-{\int }_a^{b}k\left(x,t\right)u\left(t\right)dt=g\left(x\right),a\le x\le b.$$

The authors discussed enhancements to the Sinc-collocation method initially proposed by Rashidinia and Zarebnia in 2005. They introduced two refined versions of this method.

First Improvement: This version builds upon the original scheme to enhance its practicality and theoretical soundness. The authors provided rigorous proof that the convergence rate of this modified approach is exponential, consistent with previous findings in [3].

Second Improvement: In this version, the tanh transformation used in the original scheme is replaced with the double exponential transformation. The authors demonstrated that this change significantly improves the convergence rate.

The paper [5] explores the origins and development of the double exponential transformation in numerical integration. This transformation was initially introduced by H. Takahashi and Mori in 1974. In [6], the authors present a novel numerical approach for solving linear integral equations. They employ the Sinc collocation method, enhanced by the double exponential (DE) transformation. This method is applied to Volterra integral equations of both the first and second kinds, as well as to Fredholm integral equations of the second kind. For the Volterra equations, the authors utilize a numerical indefinite integration formula developed by Muhammad and Mori, which incorporates the DE transformation into the Sinc expansion of the integrand.

In [7], a Sinc-Galerkin method is examined and analyzed for solving a fourth-order partial integro-differential equation with a weakly singular kernel. The time derivative and Riemann–Liouville fractional integral terms are approximated using the Crank–Nicolson method and the trapezoidal convolution quadrature rule. A fully discrete scheme is then constructed via the Sinc-Galerkin spatial approximation. The process is shown to achieve exponential convergence in the appropriate function space. Similarly, in [8], the authors develop a Sinc-Galerkin method for a fourth-order partial integro-differential equation with a weakly singular kernel. They employ the Crank–Nicolson method and the trapezoidal convolution quadrature rule to discretize the temporal terms and formulate a fully discrete scheme using the Sinc-Galerkin approach. Study [9] provides a comprehensive treatment of the Sinc-Galerkin method for solving time-dependent partial differential equations. In [10], two numerical methods are proposed for solving nonlinear Fredholm integral equations of the second kind. These approaches combine the Sinc approximation with single exponential (SE) and double exponential (DE) transformations. To solve the resulting nonlinear systems, the authors apply a Sinc collocation method in conjunction with a Newton iterative process and present a detailed error analysis for both techniques.

Systems of linear integral equations are fundamental tools in applied mathematics and physics, as they frequently arise in the modeling and analysis of complex phenomena. These systems commonly appear when reformulating boundary value problems, particularly in domains comprising multiple subregions or interfaces (see [11]). In such contexts, solving the integral system provides valuable insights into the behavior of the original physical problem. For example, ref. [12] introduces several numerical methods based on piecewise polynomials for solving systems of linear Fredholm integral equations of the second kind.

Despite this progress, systems involving multiple coupled unknown functions appearing both inside and outside the integral terms pose distinct analytical and computational challenges that are not sufficiently addressed in the current literature. In particular, existing Sinc-based methods often exhibit the following limitations:

• A lack of rigorous convergence analysis when extended to coupled systems;
• The absence of optimized transformation strategies (e.g., SE or DE mappings) to improve computational efficiency,
• Unclear definitions of function spaces and operator frameworks, which limit the theoretical robustness and general applicability of their results.

Motivated by these applications, this work aims to address the aforementioned limitations by focusing on the numerical solution of coupled systems of Fredholm integral equations of the second kind:

$$\left\{\begin{array}{c}{\displaystyle x_1\left(s\right)+\sum _{q=1}^r\underset{a}{\overset{b}{\int }}{\Psi }_{1q}\left(s,\eta \right)x_q\left(\eta \right)d\eta =y_1\left(s\right),}\hfill \\ {\displaystyle x_2\left(s\right)+\sum _{q=1}^r\underset{a}{\overset{b}{\int }}{\Psi }_{2q}\left(s,\eta \right)x_q\left(\eta \right)d\eta =y_2\left(s\right),}\hfill \\ \dots \dots \hfill \\ {\displaystyle x_r\left(s\right)+\sum _{q=1}^r\underset{a}{\overset{b}{\int }}{\Psi }_{rq}\left(s,\eta \right)x_q\left(\eta \right)d\eta =y_r\left(s\right).}\hfill \end{array}\right.$$

(1)

Specifically, we concentrate on the practical analysis of the following system comprising two coupled Fredholm integral equations:

$$\left\{\begin{array}{c}{\displaystyle v\left(s\right)+{\int }_a^b\Psi \left(s,\eta \right)u\left(\eta \right)d\eta =f\left(s\right),-\infty <a\le s\le b<\infty ,}\hfill \\ {\displaystyle u\left(s\right)+\underset{a}{\overset{b}{\int }}\Psi \left(s,\eta \right)v\left(\eta \right)d\eta =g\left(s\right),-\infty <a\le s\le b<\infty ,}\hfill \end{array}\right.$$

(2)

which arises in various applied contexts, including composite media modeling and interface problems in mathematical physics. In the system of Fredholm integral equations of the second kind, the unknown functions $u(.)$ and $v(.)$ appear both inside and outside the integral sign. The existing literature on systems of Fredholm integral equations covers a wider range of cases than the type discussed in this paper. However, the type examined here is particularly relevant for practical applications. By focusing on this form, we can conduct a detailed analysis and explore its real-world applications.

Sinc-collocation methods have proven to be effective tools for numerically solving differential and integral equations. In this paper, we introduce two enhanced variants of the Sinc-collocation method, each aimed at improving convergence and computational performance. The first method utilizes a single exponential (SE) transformation, while the second employs a double exponential (DE) transformation. The novelty of our contribution lies in the development of two enhanced Sinc-collocation methods, based on the single exponential (SE) and double exponential (DE) transformations, designed specifically for this class of coupled problems.

Our key theoretical contributions are as follows:

• We derive rigorous error bounds and establish improved convergence rates for both SE- and DE-based Sinc methods, demonstrating sharper asymptotic behavior compared to classical results.
• We develop a comprehensive functional-analytic framework that includes the precise definition of vector function spaces, operators, and the conditions necessary for ensuring convergence.
• We validate our theoretical results through extensive numerical experiments, which confirm the improved accuracy and computational efficiency of the proposed methods.

• A significant contribution of this work lies in the enhanced convergence behavior of the proposed techniques. Classical SE and DE Sinc-collocation methods typically achieve convergence rates of $\mathcal{O}\left(e^{-\sqrt{\pi d\alpha N}}\right)$ and $\mathcal{O}\left(exp\left({\displaystyle \frac{-\pi dN}{log\left(\frac{2dN}{\alpha }\right)}}\right)\right)$, respectively. In contrast, the methods presented in this study have improved convergence rates of $\mathcal{O}\left(N{e}^{-2\sqrt{\pi d\alpha N}}\right)$ and $\mathcal{O}(exp\left({\displaystyle \frac{-2\pi dN}{log(2dN/\alpha )}}\right))$, which are better than the classical methods. Our improvements seek to enhance these rates, leading to better computational efficiency and reliability.

The organization of this paper is as follows: in Section 2, we establish some basic definitions and preliminaries. In Section 3, we introduce the collocation methods and analyze their convergence. We will also prove the associated theorems, offering new and improved estimates for the rates of convergence. Section 4 discusses the numerical results. Finally, we present our conclusions in Section 5.

2. Basic Definitions and Preliminaries

For the reader’s convenience, we collect some known results and useful formulas related to the space of continuous functions, compactness, modes of convergence, and Sinc functions, which will be used throughout this work.

2.1. Space of Continuous Functions and Modes of Convergence

The space $\mathcal{X}:=C\left(\left[a,b\right],\mathbb{R}\right)$ consists of continuous functions, a well-known and well-behaved class of functions that avoids many of the technical challenges associated with more irregular functions. Since continuous functions defined on a compact interval, such as $\left[a,b\right]$, are inherently bounded, it follows naturally that the supremum norm

$${∥\phi ∥}_{\infty }:=\underset{s\in \left[a,b\right]}{sup}\left|\phi \left(s\right)\right|$$

is the most appropriate choice for this space. Here, $\left|.\right|$ is the standard Euclidean norm on $\mathbb{R}$.

A linear operator T from the normed space $\mathcal{X}$ into itself is called bounded if there exists a positive number ${\alpha }_0$ such that

$${∥T\phi ∥}_{\infty }\le {\alpha }_0{∥\phi ∥}_{\infty } \mathrm{for} \mathrm{all} \phi \in \mathcal{X},$$

and

$$∥T∥:=\underset{{∥x∥}_{\infty }\le 1}{sup}{∥Tx∥}_{\infty },$$

is the norm of T. We will denote by $\mathcal{B}\left(\mathcal{X}\right)$ the space of bounded linear operators from $\mathcal{X}$ into itself.

We recall that a subset $\Omega$ of a normed space $\mathcal{X}$ is called compact if every open covering of $\Omega$ contains a finite subcovering.

The operator T is compact if it is bounded, and the set $\left\{Tx:{∥x∥}_{\infty }\le 1\right\}$ has compact closure in $\mathcal{X}$. That is, T is called compact if it maps each bounded set in $\mathcal{X}$ into a relatively compact set in $\mathcal{X}$.

Let $\left(T_n\right)\in \mathcal{B}\left(\mathcal{X}\right)$ be a sequence of bounded linear operators on the space $\mathcal{X}$. Let $T\in \mathcal{B}\left(\mathcal{X}\right)$.

Consider the following two well-established modes of convergence:

The pointwise convergence, expressed as $T_n\stackrel{p}{\to }T$, is defined as

$${∥T_n\phi -T\phi ∥}_{\infty }\to 0 \mathrm{for} \mathrm{every} \phi \in \mathcal{X}.$$

The norm convergence, expressed as $T_n\stackrel{n}{\to }T$, is given by

$$∥T_n-T∥\to 0.$$

$T_n\stackrel{n}{\to }T$ implies $T_n\stackrel{p}{\to }T$. The converse fails in general.

2.2. Sinc-Functions

The Sinc-function is defined on the whole real line as follows:

$$\mathrm{Sinc}\left(s\right):=\left\{\begin{array}{cc}{\displaystyle \frac{sin\left(\pi s\right)}{\pi s}},\hfill & s\ne 0,\hfill \\ 1,\hfill & s=0.\hfill \end{array}\right.$$

The shifted Sinc-function with evenly spaced nodes is defined by

$$S(j,h)\left(s\right):=\left\{\begin{array}{cc}{\displaystyle \frac{sin\left(\pi \right(s-jh)/h)}{\pi (s-jh)/h}},\hfill & s\ne jh,\hfill \\ 1,\hfill & s=jh.\hfill \end{array}\right.$$

To approximate the solution of (2), we consider two types of variable transformation functions, single and double exponential transformation, respectively. The authors of [13] proposed the double exponential transformation that improves the first one.

Let us consider the following single exponential transformation

$${\mathrm{S}}_{\mathrm{e}}\left(s\right)={\displaystyle \frac{b-a}{2}}tanh\left({\displaystyle \frac{s}{2}}\right)+{\displaystyle \frac{b+a}{2}},s\in \mathbb{R},$$

and its inverse is described as follows:

$${\left\{{\mathrm{S}}_{\mathrm{e}}\right\}}^{-1}\left(t\right)=log\left({\displaystyle \frac{t-a}{b-t}}\right),t\in (a,b).$$

Moreover, the double exponential transformation is given by

$${\mathrm{D}}_{\mathrm{e}}\left(s\right)={\displaystyle \frac{b-a}{2}}tanh\left({\displaystyle \frac{\pi }{2}}sinh\left(s\right)\right)+{\displaystyle \frac{b+a}{2}},s\in \mathbb{R},$$

also,

$${\left\{{\mathrm{D}}_{\mathrm{e}}\right\}}^{-1}\left(t\right)=log\left[{\displaystyle \frac{1}{\pi }}log\left({\displaystyle \frac{t-a}{b-t}}\right)+\sqrt{1+{\left\{{\displaystyle \frac{1}{\pi }}log\left({\displaystyle \frac{t-a}{b-t}}\right)\right\}}^2}\right],t\in (a,b).$$

• Let ${\mathcal{X}}_N$ be the space spanned by the first $2N+1$ of Sinc-functions.
• Let ${\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}$ and ${\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}$ be interpolatory projection operators defined from $\mathcal{X}$ onto ${\mathcal{X}}_N$ as follows:

  $$\left({\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}\phi \right)\left(s\right):=\phi \left(a\right){\displaystyle \frac{b-s}{b-a}}+\sum _{j=-N}^N\left(\mathcal{L}\phi \right)\left(t_j^{{\mathrm{S}}_{\mathrm{e}}}\right)S(j,h)\left({\left\{{\mathrm{S}}_{\mathrm{e}}\right\}}^{-1}\left(s\right)\right)+\phi \left(b\right){\displaystyle \frac{s-a}{b-a}},a\le s\le b,$$

  and

  $$\left({\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}\phi \right)\left(s\right):=\phi \left(a\right){\displaystyle \frac{b-s}{b-a}}+\sum _{j=-N}^N\left(\mathcal{L}\phi \right)\left(t_j^{{\mathrm{D}}_{\mathrm{e}}}\right)S(j,h)\left({\left\{{\mathrm{D}}_{\mathrm{e}}\right\}}^{-1}\left(s\right)\right)+\phi \left(b\right){\displaystyle \frac{s-a}{b-a}},a\le s\le b,$$

  respectively, where

  $$\left(\mathcal{L}\phi \right)\left(s\right):=\phi \left(s\right)-{\displaystyle \frac{(b-s)\phi \left(a\right)+(s-a)\phi \left(b\right)}{b-a}}.$$

  Through the paper, assume that $\underset{N}{sup}∥{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}∥<\infty$ and $\underset{N}{sup}∥{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}∥<\infty$. The collocation points $t_j^{{\mathrm{S}}_{\mathrm{e}}}$ and $t_j^{{\mathrm{D}}_{\mathrm{e}}}$ are defined as follows:

  $$\begin{array}{cc}\hfill t_j^{{\mathrm{S}}_{\mathrm{e}}}& =\left\{\begin{array}{cc}a\hfill & j=-N-1,\hfill \\ {\mathrm{S}}_{\mathrm{e}}\left(jh\right)\hfill & j=-N\dots N,\hfill \\ b\hfill & j=N+1,\hfill \end{array}\right.\hfill \end{array}$$

  and

  $$\begin{array}{cc}\hfill t_j^{{\mathrm{D}}_{\mathrm{e}}}& =\left\{\begin{array}{cc}a\hfill & j=-N-1,\hfill \\ {\mathrm{D}}_{\mathrm{e}}\left(jh\right)\hfill & j=-N\dots N,\hfill \\ b\hfill & j=N+1.\hfill \end{array}\right.\hfill \end{array}$$

  Following [4], there exist two constants $C_1,C_2$ such that

  $${∥u-{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}u∥}_{\infty }\le C_1\sqrt{N}{e}^{-\sqrt{\pi d\alpha N}},\mathrm{for} \mathrm{some} \mathrm{positive} \mathrm{constants}d\mathrm{and}\alpha ,$$

  (3)

  and

  $${∥u-{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}u∥}_{\infty }\le C_{2}exp\left({\displaystyle \frac{-\pi dN}{log\left(\frac{2dN}{\alpha }\right)}}\right).$$

  (4)

3. System of Linear Fredholm Integral Equations of the Second Kind

3.1. Generalized Case

We examine the system of linear Fredholm integral equations of the second kind, given by the following form:

$$\left\{\begin{array}{c}{\displaystyle x_1\left(s\right)+\sum _{q=1}^r\underset{a}{\overset{b}{\int }}{\Psi }_{1q}\left(s,\eta \right)x_q\left(\eta \right)d\eta =y_1\left(s\right),}\hfill \\ {\displaystyle x_2\left(s\right)+\sum _{q=1}^r\underset{a}{\overset{b}{\int }}{\Psi }_{2q}\left(s,\eta \right)x_q\left(\eta \right)d\eta =y_2\left(s\right),}\hfill \\ \dots \dots \hfill \\ {\displaystyle x_r\left(s\right)+\sum _{q=1}^r\underset{a}{\overset{b}{\int }}{\Psi }_{rq}\left(s,\eta \right)x_q\left(\eta \right)d\eta =y_r\left(s\right).}\hfill \end{array}\right.$$

(5)

Set

$${\mathcal{T}}_{pq}{x}_q\left(s\right)=\underset{a}{\overset{b}{\int }}{\Psi }_{pq}\left(s,\eta \right)x_q\left(\eta \right)d\eta ,p,q=1,2,\dots ,r,$$

and

$${\mathbf{I}}_r={\left[\begin{array}{cccc}I& O& \dots & O\\ O& I& \dots & O\\ \dots & & \dots & \\ O& O& \dots & I\end{array}\right]}_{r\times r},\mathbf{T}={\left[\begin{array}{cccc}{\mathcal{T}}_{11}& {\mathcal{T}}_{12}& \dots & {\mathcal{T}}_{1r}\\ {\mathcal{T}}_{21}& {\mathcal{T}}_{22}& \dots & {\mathcal{T}}_{2r}\\ \dots & & \dots & \\ {\mathcal{T}}_{r1}& {\mathcal{T}}_{r2}& \dots & {\mathcal{T}}_{rr}\end{array}\right]}_{r\times r}.$$

The system (5) can be represented as an operator equation in the following manner:

$$x_p\left(s\right)+\sum _{q=1}^r\left({\mathcal{T}}_{pq}{x}_q\right)\left(s\right)=y_p\left(s\right),1\le p\le r.$$

Or equivalently,

$$\left({\mathbf{I}}_r+\mathbf{T}\right)\mathbf{X}=\mathbf{Y},$$

which is the matrix form, where

$$\mathbf{X}=\left[\begin{array}{c}{x}_1\\ x_2\\ \dots \\ x_r\end{array}\right]\mathrm{and}\mathbf{Y}=\left[\begin{array}{c}{y}_1\\ y_2\\ \dots \\ y_r\end{array}\right].$$

3.2. Development of the Method

In this section, we consider the approximate operators

$${\mathbf{T}}_N^{{\mathrm{S}}_{\mathrm{e}}}:=\mathbf{T}{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}},\mathrm{and}{\mathbf{T}}_N^{{\mathrm{D}}_{\mathrm{e}}}:=\mathbf{T}{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}.$$

We note that

$$\begin{array}{cc}\hfill \left({\mathcal{T}}_{pq,N}^{{\mathrm{S}}_{\mathrm{e}}}{x}_q\right)\left(s\right)& ={\int }_a^b{\Psi }_{pq}\left(s,\eta \right)\left({\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}{x}_q\right)\left(\eta \right)d\eta ,a<s<b,\hfill \\ \hfill \left({\mathcal{T}}_{pq,N}^{{\mathrm{D}}_{\mathrm{e}}}{x}_q\right)\left(s\right)& ={\int }_a^b{\Psi }_{pq}\left(s,\eta \right)\left({\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}{x}_q\right)\left(\eta \right)d\eta ,a<s<b.\hfill \end{array}$$

Let ${\mathbf{X}}_N^{{\mathrm{S}}_{\mathrm{e}}}={\left(x_{1,N}^{{\mathrm{S}}_{\mathrm{e}}},x_{2,N}^{{\mathrm{S}}_{\mathrm{e}}},\dots x_{r,N}^{{\mathrm{S}}_{\mathrm{e}}}\right)}^T$ be the solution of the system

$$x_{p,N}^{{\mathrm{S}}_{\mathrm{e}}}\left(s\right)+\sum _{q=1}^r\left({\mathcal{T}}_{pq}{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}{x}_{q,N}^{{\mathrm{S}}_{\mathrm{e}}}\right)\left(s\right)=y_p\left(s\right),1\le p\le r.$$

Or equivalently,

$$\left({\mathbf{I}}_r+{\mathbf{T}}_{\mathbf{N}}^{{\mathrm{S}}_{\mathrm{e}}}\right){\mathbf{X}}_{\mathbf{N}}^{{\mathrm{S}}_{\mathrm{e}}}=\mathbf{Y},$$

where

$${\mathbf{T}}_{\mathbf{N}}^{{\mathrm{S}}_{\mathrm{e}}}={\left[\begin{array}{cccc}{\mathcal{T}}_{11}{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}& {\mathcal{T}}_{12}{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}& \dots & {\mathcal{T}}_{1r}{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}\\ {\mathcal{T}}_{21}{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}& {\mathcal{T}}_{22}{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}& \dots & {\mathcal{T}}_{2r}{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}\\ \dots & & \dots & \\ {\mathcal{T}}_{r1}{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}& {\mathcal{T}}_{r2}{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}& \dots & {\mathcal{T}}_{rr}{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}\end{array}\right]}_{r\times r}.$$

Also, let ${\mathbf{X}}_{\mathbf{N}}^{{\mathrm{D}}_{\mathrm{e}}}={\left(x_{1,N}^{{\mathrm{D}}_{\mathrm{e}}},x_{2,N}^{{\mathrm{D}}_{\mathrm{e}}},\dots x_{r,N}^{{\mathrm{D}}_{\mathrm{e}}}\right)}^T$ be the solution of the system

$$x_{p,N}^{{\mathrm{D}}_{\mathrm{e}}}\left(s\right)+\sum _{q=1}^r\left({\mathcal{T}}_{pq}{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}{x}_{q,N}^{{\mathrm{D}}_{\mathrm{e}}}\right)\left(s\right)=y_p\left(s\right),1\le p\le r.$$

That is to say,

$$\left({\mathbf{I}}_r+{\mathbf{T}}_{\mathbf{N}}^{{\mathrm{D}}_{\mathrm{e}}}\right){\mathbf{X}}_{\mathbf{N}}^{{\mathrm{D}}_{\mathrm{e}}}=\mathbf{Y},$$

with

$${\mathbf{T}}_{\mathbf{N}}^{{\mathrm{D}}_{\mathrm{e}}}={\left[\begin{array}{cccc}{\mathcal{T}}_{11}{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}& {\mathcal{T}}_{12}{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}& \dots & {\mathcal{T}}_{1r}{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}\\ {\mathcal{T}}_{21}{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}& {\mathcal{T}}_{22}{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}& \dots & {\mathcal{T}}_{2r}{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}\\ \dots & & \dots & \\ {\mathcal{T}}_{r1}{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}& {\mathcal{T}}_{r2}{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}& \dots & {\mathcal{T}}_{rr}{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}\end{array}\right]}_{r\times r}.$$

In order to prove the existence of the operators ${\left(I+{\mathbf{T}}_{\mathbf{N}}^{{\mathrm{S}}_{\mathrm{e}}}\right)}^{-1}$ and ${\left(I+{\mathbf{T}}_{\mathbf{N}}^{{\mathrm{D}}_{\mathrm{e}}}\right)}^{-1}$, we will use the $\nu$-convergence introduced in [14].

We recall that the approximate operator ${\mathbf{T}}_{\mathbf{N}}^{{\mathrm{S}}_{\mathrm{e}}}$ is $\nu$-convergent to the operator $\mathbf{T}$ if and only if

S1. 

$∥{\mathbf{T}}_{\mathbf{N}}^{{\mathrm{S}}_{\mathrm{e}}}∥<{\gamma }_1<\infty ,\mathrm{for} \mathrm{the} \mathrm{positive} \mathrm{constant}{\gamma }_1$;

S2. 

$∥\left({\mathbf{T}}_{\mathbf{N}}^{{\mathrm{S}}_{\mathrm{e}}}-\mathbf{T}\right){\mathbf{T}}_{\mathbf{N}}^{{\mathrm{S}}_{\mathrm{e}}}∥\to 0\mathrm{and}∥\left({\mathbf{T}}_{\mathbf{N}}^{{\mathrm{S}}_{\mathrm{e}}}-\mathbf{T}\right)\mathbf{T}∥\to 0\mathrm{as}N\to \infty$.

Also, the approximate operator ${\mathbf{T}}_{\mathbf{N}}^{{\mathrm{D}}_{\mathrm{e}}}$ is $\nu$-convergent to the operator $\mathbf{T}$ if and only if

D1. 

The sequence ${\left({\mathbf{T}}_{\mathbf{N}}^{{\mathrm{D}}_{\mathrm{e}}}\right)}_{N\ge 1}$ is bounded;

D2. 

$∥\left({\mathbf{T}}_{\mathbf{N}}^{{\mathrm{D}}_{\mathrm{e}}}-\mathbf{T}\right){\mathbf{T}}_{\mathbf{N}}^{{\mathrm{D}}_{\mathrm{e}}}∥\to 0\mathrm{and}∥\left({\mathbf{T}}_{\mathbf{N}}^{{\mathrm{D}}_{\mathrm{e}}}-\mathbf{T}\right)\mathbf{T}∥\to 0\mathrm{as}N\to \infty .$

Lemma 1 and 2 establish that the conditions S1 and D1 ensuring $\nu$-convergence of the approximate operators ${\mathbf{T}}_N^{{\mathrm{S}}_{\mathrm{e}}}$ and ${\mathbf{T}}_N^{{\mathrm{D}}_{\mathrm{e}}}$ to $\mathbf{T}$ are satisfied.

Lemma 1.

The following estimate holds:

$$∥{\mathbf{T}}_N^{{\mathrm{S}}_{\mathrm{e}}}∥\le \sqrt{b-a}\underset{N}{sup}{∥{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}∥}_{\mathcal{B}(\mathcal{X},\mathcal{X})}\underset{1\le p\le r}{max}\sum _{q=1}^r\left\{\underset{a\le s\le b}{max}\sqrt{{\int }_a^b{\left|{\Psi }_{pq,s}\left(\eta \right)\right|}^{2}d\eta }\right\}.$$

Proof. 

In fact, we have

$$\begin{array}{cc}\hfill \left|\left({\mathcal{T}}_{pq,N}^{{\mathrm{S}}_{\mathrm{e}}}\phi \right)\left(s\right)\right|& =\left|{\int }_a^b{\Psi }_{pq}\left(s,\eta \right)\left({\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}\phi \right)\left(\eta \right)d\eta \right|,\mathrm{for} \mathrm{all}\phi \in \mathcal{X}\hfill \\ \\ \hfill & \le \sqrt{{\int }_a^b{\left|{\Psi }_{pq,s}\left(\eta \right)\right|}^{2}d\eta }\sqrt{{\int }_a^b{\left|\left({\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}\phi \right)\left(\eta \right)\right|}^{2}d\eta },a\le s\le b\hfill \\ \hfill & \le \sqrt{b-a}{∥{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}∥}_{\mathcal{B}(\mathcal{X},\mathcal{X})}\underset{a\le s\le b}{max}\sqrt{{\int }_a^b{\left|{\Psi }_{pq,s}\left(\eta \right)\right|}^{2}d\eta }{∥\phi ∥}_{\infty }.\hfill \end{array}$$

Hence,

$$\begin{array}{cc}\hfill {∥{\mathcal{T}}_{pq,N}^{{\mathrm{S}}_{\mathrm{e}}}\phi ∥}_{\infty }& \le \sqrt{b-a}\underset{N}{sup}{∥{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}∥}_{\mathcal{B}(\mathcal{X},\mathcal{X})}\underset{a\le s\le b}{max}\sqrt{{\int }_a^b{\left|{\Psi }_{pq,s}\left(\eta \right)\right|}^{2}d\eta }{∥\phi ∥}_{\infty }<\infty ,\hfill \end{array}$$

and hence

$$\begin{array}{cc}\hfill ∥{\mathcal{T}}_{pq,N}^{{\mathrm{S}}_{\mathrm{e}}}∥& \le \sqrt{b-a}\underset{N}{sup}{∥{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}∥}_{\mathcal{B}(\mathcal{X},\mathcal{X})}\underset{a\le s\le b}{max}\sqrt{{\int }_a^b{\left|{\Psi }_{pq,s}\left(\eta \right)\right|}^{2}d\eta }.\hfill \end{array}$$

But

$$\begin{array}{cc}\hfill ∥{\mathbf{T}}_N^{{\mathrm{S}}_{\mathrm{e}}}∥& =\underset{1\le p\le r}{max}\sum _{q=1}^r\left\{∥{\mathcal{T}}_{pq,N}^{{\mathrm{S}}_{\mathrm{e}}}∥\right\}.\hfill \end{array}$$

Consequently, the desired result is achieved through above estimation.    □

Lemma 2.

The following estimate holds:

$$∥{\mathbf{T}}_N^{{\mathrm{D}}_{\mathrm{e}}}∥\le \sqrt{b-a}\underset{N}{sup}{∥{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}∥}_{\mathcal{B}(\mathcal{X},\mathcal{X})}\underset{1\le p\le r}{max}\sum _{q=1}^r\left\{\underset{a\le s\le b}{max}\sqrt{{\int }_a^b{\left|{\Psi }_{pq,s}\left(\eta \right)\right|}^{2}d\eta }\right\}.$$

Proof. 

Analogously to the proof of Lemma 1, we obtain

$$\begin{array}{cc}\hfill \left|\left({\mathcal{T}}_{pq,N}^{{\mathrm{D}}_{\mathrm{e}}}\phi \right)\left(s\right)\right|& \le \sqrt{b-a}{∥{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}∥}_{\mathcal{B}(\mathcal{X},\mathcal{X})}\underset{a\le s\le b}{max}\sqrt{{\int }_a^b{\left|{\Psi }_{pq,s}\left(\eta \right)\right|}^{2}d\eta }{∥\phi ∥}_{\infty }.\hfill \end{array}$$

Consequently,

$$\begin{array}{cc}\hfill {∥{\mathcal{T}}_{pq,N}^{{\mathrm{D}}_{\mathrm{e}}}\phi ∥}_{\infty }& \le \sqrt{b-a}\underset{N}{sup}{∥{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}∥}_{\mathcal{B}(\mathcal{X},\mathcal{X})}\underset{a\le s\le b}{max}\sqrt{{\int }_a^b{\left|{\Psi }_{pq,s}\left(\eta \right)\right|}^{2}d\eta }{∥\phi ∥}_{\infty }<\infty .\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill ∥{\mathcal{T}}_{pq,N}^{{\mathrm{D}}_{\mathrm{e}}}∥& \le \sqrt{b-a}\underset{N}{sup}{∥{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}∥}_{\mathcal{B}(\mathcal{X},\mathcal{X})}\underset{a\le s\le b}{max}\sqrt{{\int }_a^b{\left|{\Psi }_{pq,s}\left(\eta \right)\right|}^{2}d\eta }.\hfill \end{array}$$

But

$$\begin{array}{cc}\hfill ∥{\mathbf{T}}_N^{{\mathrm{D}}_{\mathrm{e}}}∥& =\underset{1\le p\le r}{max}\sum _{q=1}^r\left\{∥{\mathcal{T}}_{pq,N}^{{\mathrm{D}}_{\mathrm{e}}}∥\right\}.\hfill \end{array}$$

Hence, the result follows.    □

To prove that the conditions S2 and D2 for the $\nu$-convergence of the approximate operators ${\mathbf{T}}_N^{{\mathrm{S}}_{\mathrm{e}}}$ and ${\mathbf{T}}_N^{{\mathrm{D}}_{\mathrm{e}}}$ to $\mathbf{T}$ are satisfied, we need the following results:

Theorem 1.

The following estimate holds:

$${∥{\mathcal{T}}_{pq}(I-{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}})\phi ∥}_{\infty }\le {\alpha }_0(b-a)N{e}^{-2\sqrt{\pi d\alpha N}},$$

for the positive constant ${\alpha }_0$.

Proof. 

For all $\phi \in \mathcal{X}$,

$$\begin{array}{cc}\hfill \left|{\mathcal{T}}_{pq}(I-{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}})\phi \left(s\right)\right|& =\left|{\int }_a^b{\Psi }_{pq}\left(s,\eta \right)\left((I-{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}})\phi \right)\left(\eta \right)d\eta \right|,a\le s\le b\hfill \\ \hfill & =\left|〈(I-{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}})\phi ,{\Psi }_{pq,s}〉\right|\hfill \\ \hfill & =\left|〈(I-{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}})\phi ,(I-{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}){\Psi }_{pq,s}〉\right|\hfill \\ \hfill & \le \sqrt{{\int }_a^b{\left|(I-{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}})\phi \left(\eta \right)\right|}^{2}d\eta }\sqrt{{\int }_a^b{\left|(I-{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}){\Psi }_{pq,s}\left(\eta \right)\right|}^{2}d\eta }\hfill \\ \hfill & \le {∥(I-{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}})\phi ∥}_2{∥(I-{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}){\Psi }_{pq,s}∥}_2.\hfill \end{array}$$

So,

$$\begin{array}{cc}\hfill {∥{\mathcal{T}}_{pq}(I-{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}})\phi ∥}_{\infty }& \le {∥(I-{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}})\phi ∥}_2{∥(I-{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}){\Psi }_{pq,s}∥}_2\hfill \\ \hfill & \le (b-a){∥(I-{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}})\phi ∥}_{\infty }{∥(I-{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}){\Psi }_{pq,s}∥}_{\infty }.\hfill \end{array}$$

Thus, by (3), we obtain the desired result.    □

Theorem 2.

The following estimate holds:

$${∥{\mathcal{T}}_{pq}(I-{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}})\phi ∥}_{\infty }\le {\alpha }_1(b-a)exp\left({\displaystyle \frac{-2\pi dN}{log(2dN/\alpha )}}\right),$$

for the positive constant ${\alpha }_1$.

Proof. 

Similarly to the proof of Theorem 1, we obtain

$$\begin{array}{cc}\hfill {∥{\mathcal{T}}_{pq}(I-{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}})\phi ∥}_{\infty }& \le (b-a){∥(I-{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}})\phi ∥}_{\infty }{∥(I-{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}){\Psi }_{pq,s}∥}_{\infty }.\hfill \end{array}$$

Consequently, by (4), we obtain the desired result.    □

Corollary 1.

The following estimate holds:

$${∥{\mathcal{T}}_{pq}(I-{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}){\mathcal{T}}_{pq}\phi ∥}_{\infty }\le {\alpha }_2(b-a)N{e}^{-2\sqrt{\pi d\alpha N}},$$

for the positive constant ${\alpha }_2$.

Proof. 

Theorem 1 gives

$$\begin{array}{cc}\hfill {∥{\mathcal{T}}_{pq}(I-{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}){\mathcal{T}}_{pq}\phi ∥}_{\infty }& \le (b-a){∥(I-{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}){\mathcal{T}}_{pq}\phi ∥}_{\infty }{∥(I-{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}){\Psi }_{pq,s}∥}_{\infty }.\hfill \end{array}$$

Consequently, by (3), we obtain the desired result.    □

Corollary 2.

The following estimate holds:

$${∥{\mathcal{T}}_{pq}(I-{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}){\mathcal{T}}_{pq}\phi ∥}_{\infty }\le {\alpha }_3(b-a)exp\left({\displaystyle \frac{-2\pi dN}{log(2dN/\alpha )}}\right),$$

for the positive constant ${\alpha }_3$.

Proof. 

Theorem 2 gives

$$\begin{array}{cc}\hfill {∥{\mathcal{T}}_{pq}(I-{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}){\mathcal{T}}_{pq}\phi ∥}_{\infty }& \le (b-a){∥(I-{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}){\mathcal{T}}_{pq}\phi ∥}_{\infty }{∥(I-{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}){\Psi }_{pq,s}∥}_{\infty }.\hfill \end{array}$$

Consequently, by (4), we obtain the desired result.    □

Corollary 3.

The following estimate holds:

$${∥{\mathcal{T}}_{pq}(I-{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}){\mathcal{T}}_{pq}{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}\phi ∥}_{\infty }\le {\alpha }_4(b-a)N{e}^{-2\sqrt{\pi d\alpha N}},$$

for the positive constant ${\alpha }_4$.

Proof. 

Theorem 1 gives

$$\begin{array}{cc}\hfill {∥{\mathcal{T}}_{pq}(I-{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}){\mathcal{T}}_{pq}{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}\phi ∥}_{\infty }& \le (b-a){∥(I-{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}){\mathcal{T}}_{pq}{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}\phi ∥}_{\infty }{∥(I-{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}){\Psi }_{pq,s}∥}_{\infty }.\hfill \end{array}$$

Consequently, by (3), we obtain the desired result.    □

Corollary 4.

The following estimate holds:

$${∥{\mathcal{T}}_{pq}(I-{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}){\mathcal{T}}_{pq}{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}\phi ∥}_{\infty }\le {\alpha }_5(b-a)exp\left({\displaystyle \frac{-2\pi dN}{log(2dN/\alpha )}}\right),$$

for the positive constant ${\alpha }_5$.

Proof. 

Theorem 2 gives

$$\begin{array}{cc}\hfill {∥{\mathcal{T}}_{pq}(I-{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}){\mathcal{T}}_{pq}{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}\phi ∥}_{\infty }& \le (b-a){∥(I-{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}){\mathcal{T}}_{pq}{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}\phi ∥}_{\infty }{∥(I-{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}){\Psi }_{pq,s}∥}_{\infty }.\hfill \end{array}$$

Consequently, by (4), we obtain the desired result.    □

From the above analysis, we conclude that conditions S2 and D2 governing the $\nu$-convergence of the approximate operators ${\mathbf{T}}_N^{{\mathrm{S}}_{\mathrm{e}}}$ and ${\mathbf{T}}_N^{{\mathrm{D}}_{\mathrm{e}}}$ to $\mathbf{T}$ are satisfied. This conclusion is substantiated by the results presented below:

Corollary 5.

We have

$$\begin{array}{c}\hfill ∥\left({\mathbf{T}}_{\mathbf{N}}^{{\mathrm{S}}_{\mathrm{e}}}-\mathbf{T}\right){\mathbf{T}}_{\mathbf{N}}^{{\mathrm{S}}_{\mathrm{e}}}∥\to 0,\end{array}$$

and

$$\begin{array}{c}\hfill ∥\left({\mathbf{T}}_{\mathbf{N}}^{{\mathrm{S}}_{\mathrm{e}}}-\mathbf{T}\right)\mathbf{T}∥\to 0.\end{array}$$

Proof. 

Since

$$\begin{array}{cc}\hfill ∥\left({\mathbf{T}}_{\mathbf{N}}^{{\mathrm{S}}_{\mathrm{e}}}-\mathbf{T}\right){\mathbf{T}}_{\mathbf{N}}^{{\mathrm{S}}_{\mathrm{e}}}∥& =\underset{1\le p\le r}{max}\sum _{q=1}^r\left\{∥\left({\mathcal{T}}_{pq,N}^{{\mathrm{S}}_{\mathrm{e}}}-{\mathcal{T}}_{pq}\right){\mathcal{T}}_{pq,N}^{{\mathrm{S}}_{\mathrm{e}}}∥\right\},\hfill \end{array}$$

and since

$$\begin{array}{cc}\hfill ∥\left({\mathbf{T}}_{\mathbf{N}}^{{\mathrm{S}}_{\mathrm{e}}}-\mathbf{T}\right)\mathbf{T}∥& =\underset{1\le p\le r}{max}\sum _{q=1}^r\left\{∥\left({\mathcal{T}}_{pq,N}^{{\mathrm{S}}_{\mathrm{e}}}-{\mathcal{T}}_{pq}\right){\mathcal{T}}_{pq}∥\right\},\hfill \end{array}$$

we obtain the desired result. □

Corollary 6.

We have

$$\begin{array}{c}\hfill ∥\left({\mathbf{T}}_{\mathbf{N}}^{{\mathrm{D}}_{\mathrm{e}}}-\mathbf{T}\right){\mathbf{T}}_{\mathbf{N}}^{{\mathrm{D}}_{\mathrm{e}}}∥\to 0,\end{array}$$

and

$$\begin{array}{c}\hfill ∥\left({\mathbf{T}}_{\mathbf{N}}^{{\mathrm{D}}_{\mathrm{e}}}-\mathbf{T}\right)\mathbf{T}∥\to 0.\end{array}$$

Proof. 

As both conditions

$$\begin{array}{cc}\hfill ∥\left({\mathbf{T}}_{\mathbf{N}}^{{\mathrm{D}}_{\mathrm{e}}}-\mathbf{T}\right){\mathbf{T}}_{\mathbf{N}}^{{\mathrm{D}}_{\mathrm{e}}}∥& =\underset{1\le p\le r}{max}\sum _{q=1}^r\left\{∥\left({\mathcal{T}}_{pq,N}^{{\mathrm{D}}_{\mathrm{e}}}-{\mathcal{T}}_{pq}\right){\mathcal{T}}_{pq,N}^{{\mathrm{D}}_{\mathrm{e}}}∥\right\},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill ∥\left({\mathbf{T}}_{\mathbf{N}}^{{\mathrm{D}}_{\mathrm{e}}}-\mathbf{T}\right)\mathbf{T}∥& =\underset{1\le p\le r}{max}\sum _{q=1}^r\left\{∥\left({\mathcal{T}}_{pq,N}^{{\mathrm{D}}_{\mathrm{e}}}-{\mathcal{T}}_{pq}\right){\mathcal{T}}_{pq}∥\right\},\hfill \end{array}$$

hold, we arrive at the desired result. □

Consequently, we conclude that the sequences ${\mathbf{T}}_N^{{\mathrm{S}}_{\mathrm{e}}}$ and ${\mathbf{T}}_N^{{\mathrm{D}}_{\mathrm{e}}}$ converge to $\mathbf{T}$ in the $\nu$-sense.

Proposition 1.

For N large enough, the operators ${\mathbf{I}}_r+{\mathbf{T}}_{\mathbf{N}}^{{\mathrm{S}}_{\mathrm{e}}}$ and ${\mathbf{I}}_r+{\mathbf{T}}_{\mathbf{N}}^{{\mathrm{D}}_{\mathrm{e}}}$ are invertible, and the constants

$${\beta }^{{\mathrm{S}}_{\mathrm{e}}}:=\underset{N}{sup}\parallel {\left({\mathbf{I}}_r+{\mathbf{T}}_{\mathbf{N}}^{{\mathrm{S}}_{\mathrm{e}}}\right)}^{-1}\parallel$$

and

$${\gamma }^{{\mathrm{D}}_{\mathrm{e}}}:=\underset{N}{sup}\parallel {\left({\mathbf{I}}_r+{\mathbf{T}}_{\mathbf{N}}^{{\mathrm{D}}_{\mathrm{e}}}\right)}^{-1}\parallel$$

are finite.

Proof. 

Since the operators ${\mathbf{I}}_r+\mathbf{T}$ are invertible and the approximate operators ${\mathbf{T}}_{\mathbf{N}}^{{\mathrm{S}}_{\mathrm{e}}}$ and ${\mathbf{T}}_{\mathbf{N}}^{{\mathrm{D}}_{\mathrm{e}}}$ converge to $\mathbf{T}$ in $\nu$-sense, it follows that the inverse operators ${\left({\mathbf{I}}_r+{\mathbf{T}}_{\mathbf{N}}^{{\mathrm{S}}_{\mathrm{e}}}\right)}^{-1}$ and ${\left({\mathbf{I}}_r+{\mathbf{T}}_{\mathbf{N}}^{{\mathrm{D}}_{\mathrm{e}}}\right)}^{-1}$ exist and are uniformly bounded for N large enough, (see [14]). □

The Sinc-collocation method via single exponential transformation leads to the following linear system:

$$\left({\mathbf{I}}_r+{\mathbf{T}}_{\mathbf{N}}^{{\mathrm{S}}_{\mathrm{e}}}\right){\mathbf{X}}_{\mathbf{N}}^{{\mathrm{S}}_{\mathrm{e}}}\left(t_i^{{\mathrm{S}}_{\mathrm{e}}}\right)=\mathbf{Y}\left(t_i^{{\mathrm{S}}_{\mathrm{e}}}\right),i=-N-1,\cdots ,N+1.$$

Once this linear system is solved, the approximate solution is given by

$$\begin{array}{cc}\hfill {\mathbf{X}}_N^{{\mathrm{S}}_{\mathrm{e}}}& =\mathbf{Y}-\mathbf{T}{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}{\mathbf{X}}_N^{{\mathrm{S}}_{\mathrm{e}}}.\hfill \end{array}$$

Also, the Sinc-collocation method via double exponential transformation leads to the following linear system:

$$\left({\mathbf{I}}_r+{\mathbf{T}}_N^{{\mathrm{D}}_{\mathrm{e}}}\right){\mathbf{X}}_N^{{\mathrm{D}}_{\mathrm{e}}}\left(t_i^{{\mathrm{D}}_{\mathrm{e}}}\right)=\mathbf{Y}\left(t_i^{{\mathrm{D}}_{\mathrm{e}}}\right),i=-N-1,\cdots ,N+1.$$

Once this linear system is solved, the approximate solution is given by

$$\begin{array}{cc}\hfill {\mathbf{X}}_N^{{\mathrm{D}}_{\mathrm{e}}}& =\mathbf{Y}-\mathbf{T}{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}{\mathbf{X}}_N^{{\mathrm{D}}_{\mathrm{e}}}.\hfill \end{array}$$

3.3. Convergence Analysis

The convergence order of the proposed method is established in the following Theorem.

Theorem 3.

The following estimates hold:

$$\begin{array}{cc}\hfill \parallel \mathbf{X}-{\mathbf{X}}_{\mathbf{N}}^{{\mathrm{S}}_{\mathrm{e}}}{\parallel }_{\infty }& \le {\beta }_0(b-a)N{e}^{-2\sqrt{\pi d\alpha N}},\hfill \end{array}$$

where ${\beta }_0$ is some finite positive constant.

Proof. 

We have

$$\begin{array}{cc}\hfill {\mathbf{X}}_{\mathbf{N}}^{{\mathrm{S}}_{\mathrm{e}}}-\mathbf{X}& ={({\mathbf{I}}_r+{\mathbf{T}}_N^{{\mathrm{S}}_{\mathrm{e}}})}^{-1}\mathbf{Y}-{({\mathbf{I}}_r+\mathbf{T})}^{-1}\mathbf{Y}\hfill \\ \hfill & ={({\mathbf{I}}_r+{\mathbf{T}}_N)}^{-1}(\mathbf{T}-{\mathbf{T}}_N^{{\mathrm{S}}_{\mathrm{e}}})\mathbf{X}\hfill \\ \hfill & ={({\mathbf{I}}_r+{\mathbf{T}}_N)}^{-1}\mathbf{T}({\mathbf{I}}_r-{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}})\mathbf{X}.\hfill \end{array}$$

Since

$$\underset{N}{sup}\parallel {({\mathbf{I}}_r+{\mathbf{T}}_N^{{\mathrm{S}}_{\mathrm{e}}})}^{-1}\parallel <\infty ,$$

and by using Theorem (1), we obtain the desired result. □

Theorem 4.

The following estimates hold:

$$\begin{array}{cc}\hfill \parallel \mathbf{X}-{\mathbf{X}}_{\mathbf{N}}^{{\mathrm{D}}_{\mathrm{e}}}{\parallel }_{\infty }& \le {\beta }_1(b-a)exp\left({\displaystyle \frac{-2\pi dN}{log(2dN/\alpha )}}\right),\hfill \end{array}$$

where ${\beta }_1$ is some finite positive constant.

Proof. 

As above, we obtain

$$\begin{array}{cc}\hfill {\mathbf{X}}_{\mathbf{N}}^{{\mathrm{D}}_{\mathrm{e}}}-\mathbf{X}& ={({\mathbf{I}}_r+{\mathbf{T}}_N)}^{-1}\mathbf{T}({\mathbf{I}}_r-{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}})\mathbf{X}.\hfill \end{array}$$

Since

$$\underset{N}{sup}\parallel {({\mathbf{I}}_r+{\mathbf{T}}_N^{{\mathrm{D}}_{\mathrm{e}}})}^{-1}\parallel <\infty ,$$

and by using Theorem (2), we obtain the desired result. □

3.4. System of Two Fredholm Integral Equations

For given real functions $\left(f,g\right)$, consider the system of two Fredholm integral Equation (2). Letting

$$\mathcal{K}\phi \left(s\right):={\int }_a^b\Psi \left(s,\eta \right)\phi \left(\eta \right)d\eta ,a<s<b,$$

system (2) reads as

$$\left\{\begin{array}{c}{\displaystyle v+\mathcal{K}u=f,}\hfill \\ u+\mathcal{K}v=g.\hfill \end{array}\right.$$

To simplify the above problem, we can reduce it to a system of two separate equations. To this end, we follow [15] in letting

$$\vartheta :=u-v,F:=g+f;$$

$$\theta :=u+v,G:=g-f.$$

So,

$$u={\displaystyle \frac{\theta +\vartheta }{2}},f={\displaystyle \frac{F-G}{2}};$$

$$v={\displaystyle \frac{\theta -\vartheta }{2}},g={\displaystyle \frac{F+G}{2}}.$$

Substituting these into (2), we obtain

$$\begin{array}{cc}\hfill \left(\theta -\vartheta \right)\left(s\right)+{\int }_a^b\Psi \left(s,\eta \right)\left(\theta +\vartheta \right)\left(\eta \right)d\eta & =\left(F-G\right)\left(s\right),\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill \left(\theta +\vartheta \right)\left(s\right)+{\int }_a^b\Psi \left(s,\eta \right)\left(\theta -\vartheta \right)\left(\eta \right)d\eta & =\left(F+G\right)\left(s\right).\hfill \end{array}$$

(7)

By adding Equations (6) and (7) together, and by subtracting (6) from (7), we obtain the following equivalent system:

$$\left\{\begin{array}{c}{\displaystyle \theta \left(s\right)+{\int }_a^b\Psi \left(s,\eta \right)\theta \left(\eta \right)d\eta =F\left(s\right),a\le s\le b,}\hfill \\ \vartheta \left(s\right)-{\displaystyle {\int }_a^b}\Psi \left(s,\eta \right)\vartheta \left(\eta \right)d\eta =G\left(s\right),a\le s\le b.\hfill \end{array}\right.$$

(8)

System (8) can be rewritten in operator form as follows:

$$\left\{\begin{array}{c}{\displaystyle \theta +\mathcal{K}\theta =F,}\hfill \\ \vartheta -\mathcal{K}\vartheta =G,\hfill \end{array}\right.$$

Recall that for each $\left(f,g\right)\in {\mathcal{X}}^2$, the compactness of the Fredholm operator $\mathcal{K}$ from X into itself confirms that the solution $\left(\vartheta ,\theta \right)\in {\mathcal{X}}^2$ of the problem (8) is unique.

In this section, we consider the approximate operators:

$${\mathcal{K}}_N^{{\mathrm{S}}_{\mathrm{e}}}:=\mathcal{K}{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}},\mathrm{and}{\mathcal{K}}_N^{{\mathrm{D}}_{\mathrm{e}}}:=\mathcal{K}{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}.$$

We note that

$$\begin{array}{cc}\hfill \left({\mathcal{K}}_N^{{\mathrm{S}}_{\mathrm{e}}}\phi \right)\left(s\right)& ={\int }_a^b\Psi \left(s,\eta \right)\left({\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}\phi \right)\left(\eta \right)d\eta ,a<s<b,\hfill \\ \hfill \left({\mathcal{K}}_N^{{\mathrm{D}}_{\mathrm{e}}}\phi \right)\left(s\right)& ={\int }_a^b\Psi \left(s,\eta \right)\left({\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}\phi \right)\left(\eta \right)d\eta ,a<s<b,\hfill \end{array}$$

Let $\left({\theta }_N^{{\mathrm{S}}_{\mathrm{e}}},{\vartheta }_N^{{\mathrm{S}}_{\mathrm{e}}}\right)$ be the solution of the system

$$\left\{\begin{array}{c}{\displaystyle (I+{\mathcal{K}}_N^{{\mathrm{S}}_{\mathrm{e}}}){\theta }_N^{{\mathrm{S}}_{\mathrm{e}}}=F,}\hfill \\ (I-{\mathcal{K}}_N^{{\mathrm{S}}_{\mathrm{e}}}){\vartheta }_N^{{\mathrm{S}}_{\mathrm{e}}}=G,\hfill \end{array}\right.$$

and let $\left({\theta }_N^{{\mathrm{D}}_{\mathrm{e}}},{\vartheta }_N^{{\mathrm{D}}_{\mathrm{e}}}\right)$ be the solution of the system

$$\left\{\begin{array}{c}{\displaystyle (I+{\mathcal{K}}_N^{{\mathrm{D}}_{\mathrm{e}}}){\theta }_N^{{\mathrm{D}}_{\mathrm{e}}}=F,}\hfill \\ (I-{\mathcal{K}}_N^{{\mathrm{D}}_{\mathrm{e}}}){\vartheta }_N^{{\mathrm{D}}_{\mathrm{e}}}=G.\hfill \end{array}\right.$$

We use the same analysis presented in the previous section to prove the existence of the operators ${\left(I-{\mathcal{K}}_N^{{\mathrm{S}}_{\mathrm{e}}}\right)}^{-1}$, ${\left(I+{\mathcal{K}}_N^{{\mathrm{S}}_{\mathrm{e}}}\right)}^{-1}$, ${\left(I-{\mathcal{K}}_N^{{\mathrm{D}}_{\mathrm{e}}}\right)}^{-1}$, and ${\left(I+{\mathcal{K}}_N^{{\mathrm{D}}_{\mathrm{e}}}\right)}^{-1}$. To this end, we employ the concept of $\nu$-convergence.

That is to say,

S1. 

$∥{\mathcal{K}}_N^{{\mathrm{S}}_{\mathrm{e}}}∥<{\gamma }_1<\infty ,\mathrm{for} \mathrm{the} \mathrm{positive} \mathrm{constant}{\gamma }_1$;

S2. 

$∥\left({\mathcal{K}}_N^{{\mathrm{S}}_{\mathrm{e}}}-\mathcal{K}\right){\mathcal{K}}_N^{{\mathrm{S}}_{\mathrm{e}}}∥\to 0\mathrm{and}∥\left({\mathcal{K}}_N^{{\mathrm{S}}_{\mathrm{e}}}-\mathcal{K}\right)\mathcal{K}∥\to 0\mathrm{as}N\to \infty$.

Also, the approximate operator ${\mathcal{K}}_N^{{\mathrm{D}}_{\mathrm{e}}}$ is $\nu$-convergent to the operator $\mathcal{K}$; that is to say,

D1. 

The sequence ${\left({\mathcal{K}}_N^{{\mathrm{D}}_{\mathrm{e}}}\right)}_{N\ge 1}$ is bounded;

D2. 

$∥\left({\mathcal{K}}_N^{{\mathrm{D}}_{\mathrm{e}}}-\mathcal{K}\right){\mathcal{K}}_N^{{\mathrm{D}}_{\mathrm{e}}}∥\to 0\mathrm{and}∥\left({\mathcal{K}}_N^{{\mathrm{D}}_{\mathrm{e}}}-\mathcal{K}\right)\mathcal{K}∥\to 0\mathrm{as}N\to \infty .$

For N large enough, the operators $I-{\mathcal{K}}_N^{{\mathrm{S}}_{\mathrm{e}}}$ and $I+{\mathcal{K}}_N^{{\mathrm{S}}_{\mathrm{e}}}$ are invertible, and the constants

$${\beta }^{{\mathrm{S}}_{\mathrm{e}}}:=\underset{N}{sup}\parallel {\left(I-{\mathcal{K}}_N^{{\mathrm{S}}_{\mathrm{e}}}\right)}^{-1}\parallel$$

and

$${\gamma }^{{\mathrm{S}}_{\mathrm{e}}}:=\underset{N}{sup}\parallel {\left(I+{\mathcal{K}}_N^{{\mathrm{S}}_{\mathrm{e}}}\right)}^{-1}\parallel$$

are finite. That is, the inverse operators ${(I-{\mathcal{K}}_N^{{\mathrm{S}}_{\mathrm{e}}})}^{-1}$ and ${(I+{\mathcal{K}}_N^{{\mathrm{S}}_{\mathrm{e}}})}^{-1}$ exist and are uniformly bounded for N large enough.

Also, for N large enough, the operators $I-{\mathcal{K}}_N^{{\mathrm{D}}_{\mathrm{e}}}$ and $I+{\mathcal{K}}_N^{{\mathrm{D}}_{\mathrm{e}}}$ are invertible, and the constants

$${\beta }^{{\mathrm{D}}_{\mathrm{e}}}:=\underset{N}{sup}\parallel {\left(I-{\mathcal{K}}_N^{{\mathrm{D}}_{\mathrm{e}}}\right)}^{-1}\parallel$$

and

$${\gamma }^{{\mathrm{D}}_{\mathrm{e}}}:=\underset{N}{sup}\parallel {\left(I+{\mathcal{K}}_N^{{\mathrm{D}}_{\mathrm{e}}}\right)}^{-1}\parallel$$

are finite. That is, the inverse operators ${(I-{\mathcal{K}}_N^{{\mathrm{D}}_{\mathrm{e}}})}^{-1}$ and ${(I+{\mathcal{K}}_N^{{\mathrm{D}}_{\mathrm{e}}})}^{-1}$ exist and are uniformly bounded for N large enough.

The Sinc-collocation method via single exponential transformation leads to the following linear systems:

$$(I+{\mathcal{K}}_N^{{\mathrm{S}}_{\mathrm{e}}}){\theta }_N\left(t_i^{{\mathrm{S}}_{\mathrm{e}}}\right)=F\left(t_i^{{\mathrm{S}}_{\mathrm{e}}}\right),i=-N-1,\cdots ,N+1$$

and

$$(I-{\mathcal{K}}_N^{{\mathrm{S}}_{\mathrm{e}}}){\vartheta }_N\left(t_i^{{\mathrm{S}}_{\mathrm{e}}}\right)=G\left(t_i^{{\mathrm{S}}_{\mathrm{e}}}\right),i=-N-1,\cdots ,N+1.$$

Once these linear systems are solved, the approximate solutions are given by

$$\begin{array}{cc}\hfill {\theta }_N^{{\mathrm{S}}_{\mathrm{e}}}& =F-\mathcal{K}{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}{\theta }_N^{{\mathrm{S}}_{\mathrm{e}}},\hfill \\ \hfill {\vartheta }_N^{{\mathrm{S}}_{\mathrm{e}}}& =G+\mathcal{K}{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}{\vartheta }_N^{{\mathrm{S}}_{\mathrm{e}}}.\hfill \end{array}$$

Hence

$${\theta }_N^{{\mathrm{S}}_{\mathrm{e}}}\left(s\right)={\theta }_N\left(a\right){\displaystyle \frac{b-s}{b-a}}+\sum _{j=-N}^N\left(\mathcal{L}\left[\Psi \left(s,t_j^{{\mathrm{S}}_{\mathrm{e}}}\right)\theta \left(t_j^{{\mathrm{S}}_{\mathrm{e}}}\right)\right]\right)S(j,h)\left({\left\{{\mathrm{S}}_{\mathrm{e}}\right\}}^{-1}\left(s\right)\right)+{\theta }_N\left(b\right){\displaystyle \frac{s-a}{b-a}},$$

and

$${\vartheta }_N^{{\mathrm{S}}_{\mathrm{e}}}\left(s\right)={\vartheta }_N\left(a\right){\displaystyle \frac{b-s}{b-a}}+\sum _{j=-N}^N\left(\mathcal{L}\left[\Psi \left(s,t_j^{{\mathrm{S}}_{\mathrm{e}}}\right)\vartheta \left(t_j^{{\mathrm{S}}_{\mathrm{e}}}\right)\right]\right)S(j,h)\left({\left\{{\mathrm{S}}_{\mathrm{e}}\right\}}^{-1}\left(s\right)\right)+{\vartheta }_N\left(b\right){\displaystyle \frac{s-a}{b-a}}.$$

Also, the Sinc-collocation method via double exponential transformation leads to the following linear systems:

$$(I+{\mathcal{K}}_N^{{\mathrm{D}}_{\mathrm{e}}}){\theta }_N\left(t_i^{{\mathrm{D}}_{\mathrm{e}}}\right)=F\left(t_i^{{\mathrm{D}}_{\mathrm{e}}}\right),i=-N-1,\cdots ,N+1$$

and

$$(I-{\mathcal{K}}_N^{{\mathrm{D}}_{\mathrm{e}}}){\vartheta }_N\left(t_i^{{\mathrm{D}}_{\mathrm{e}}}\right)=G\left(t_i^{{\mathrm{D}}_{\mathrm{e}}}\right),i=-N-1,\cdots ,N+1.$$

Once these linear systems are solved, the approximate solutions are given by

$$\begin{array}{cc}\hfill {\theta }_N^{{\mathrm{D}}_{\mathrm{e}}}& =F-\mathcal{K}{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}{\theta }_N^{{\mathrm{D}}_{\mathrm{e}}}\hfill \\ \hfill {\vartheta }_N^{{\mathrm{D}}_{\mathrm{e}}}& =G+\mathcal{K}{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}{\vartheta }_N^{{\mathrm{D}}_{\mathrm{e}}}.\hfill \end{array}$$

Hence

$${\theta }_N^{{\mathrm{D}}_{\mathrm{e}}}\left(s\right)={\theta }_N\left(a\right){\displaystyle \frac{b-s}{b-a}}+\sum _{j=-N}^N\left(\mathcal{L}\left[\Psi \left(s,t_j^{{\mathrm{S}}_{\mathrm{e}}}\right)\theta \left(t_j^{{\mathrm{D}}_{\mathrm{e}}}\right)\right]\right)S(j,h)\left({\left\{{\mathrm{D}}_{\mathrm{e}}\right\}}^{-1}\left(s\right)\right)+{\theta }_N\left(b\right){\displaystyle \frac{s-a}{b-a}},$$

and

$${\vartheta }_N^{{\mathrm{D}}_{\mathrm{e}}}\left(s\right)={\vartheta }_N\left(a\right){\displaystyle \frac{b-s}{b-a}}+\sum _{j=-N}^N\left(\mathcal{L}\left[\Psi \left(s,t_j^{{\mathrm{D}}_{\mathrm{e}}}\right)\vartheta \left(t_j^{{\mathrm{D}}_{\mathrm{e}}}\right)\right]\right)S(j,h)\left({\left\{{\mathrm{D}}_{\mathrm{e}}\right\}}^{-1}\left(s\right)\right)+{\vartheta }_N\left(b\right){\displaystyle \frac{s-a}{b-a}}.$$

Here, the convergence analysis of the proposed method for this practical case is presented in the following theorems, using the same approach as in the previous section.

Theorem 5.

Assume that $F,G\in \mathcal{X}$. The following estimates hold:

$$\begin{array}{cc}\hfill \parallel \theta -{\theta }_N^{{\mathrm{S}}_{\mathrm{e}}}{\parallel }_{\infty }& \le {\beta }_2(b-a)N{e}^{-2\sqrt{\pi d\alpha N}},\hfill \\ \hfill \parallel \vartheta -{\vartheta }_N^{{\mathrm{S}}_{\mathrm{e}}}{\parallel }_{\infty }& \le {\beta }_3(b-a)N{e}^{-2\sqrt{\pi d\alpha N}},\hfill \end{array}$$

where ${\beta }_0$ and ${\beta }_1$ are some finite positive constants.

Theorem 6.

Assume that $F,G\in \mathcal{X}$. The following estimates hold:

$$\begin{array}{cc}\hfill \parallel \theta -{\theta }_N^{{\mathrm{D}}_{\mathrm{e}}}{\parallel }_{\infty }& \le {\beta }_4(b-a)exp\left({\displaystyle \frac{-2\pi dN}{log(2dN/\alpha )}}\right),\hfill \\ \hfill \parallel \vartheta -{\vartheta }_N^{{\mathrm{D}}_{\mathrm{e}}}{\parallel }_{\infty }& \le {\beta }_5(b-a)exp\left({\displaystyle \frac{-2\pi dN}{log(2dN/\alpha )}}\right),\hfill \end{array}$$

where ${\beta }_4$ and ${\beta }_5$ are some finite positive constants.

3.5. Appendix: Application to a Fredholm Integro-Differential Equation

To enhance the originality of our method, we examine in this appendix an application to the following Fredholm integro-differential equation:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle {\phi }^{\left(m\right)}\left(s\right)+{\int }_a^b\Psi (s,\eta )\phi \left(\eta \right)d\eta =g\left(s\right),a\le s\le b,}\hfill \\ \phi \left(a\right)=a_0,{\phi }^{\prime }\left(a\right)=a_1,\cdots ,{\phi }^{(m-1)}\left(a\right)=a_{m-1}.\hfill \end{array}\right.\end{array}$$

(9)

Define ${\mathbf{D}}^{\mathbf{m}}:\mathcal{D}\to \mathcal{X}$ as follows:

$${\mathbf{D}}^m\phi ={\phi }^{\left(m\right)}.$$

$$\mathcal{D}=\{\phi \in \mathcal{X}:{\phi }^{\left(m\right)}\in \mathcal{X},\phi \left(a\right)=a_0,{\phi }^{\prime }\left(a\right)=a_1,\cdots ,{\phi }^{(m-1)}\left(a\right)=a_{m-1}\}.$$

Equation (9) reads as

$$\begin{array}{c}\hfill {\mathbf{D}}^m\phi +\mathcal{K}\phi =g,\end{array}$$

(10)

or equivalently,

$$\begin{array}{c}\hfill \phi +{\mathbf{V}}^m\mathcal{K}\phi =g_m,\end{array}$$

(11)

where

$$\begin{array}{ccc}\hfill {\mathbf{V}}^m\phi \left(s\right)& :=& {\left({\mathbf{D}}^{\mathbf{m}}\right)}^{-1}\phi \left(s\right)\hfill \\ & =& {\int }_a^s{\int }_a^s\cdots {\int }_a^s\phi \left({\tau }_1\right)d{\tau }_1\cdots d{\tau }_m\hfill \\ & =& {\displaystyle \frac{1}{(m-1)!}}{\int }_a^s{(s-\tau )}^{m-1}\phi \left(\tau \right)d\tau ,\hfill \end{array}$$

and

$$g_m:={\mathbf{V}}^{m}g-\sum _{j=0}^{m-1}{\displaystyle \frac{a_j}{j!}}{(s-a)}^j.$$

In this section, we consider the approximate operators:

$${\mathbf{V}}_N^{m,{\mathrm{S}}_{\mathrm{e}}}:={\mathbf{V}}^m\mathcal{K}{\pi }_N^{{\mathrm{S}}_{\mathrm{e}}},\mathrm{and}{\mathbf{V}}_N^{m,{\mathrm{D}}_{\mathrm{e}}}:={\mathbf{V}}^m\mathcal{K}{\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}.$$

We note that

$$\begin{array}{cc}\hfill \left({\mathbf{V}}_N^{m,{\mathrm{S}}_{\mathrm{e}}}\phi \right)\left(s\right)& ={\displaystyle \frac{1}{(m-1)!}}{\int }_a^s{(s-\tau )}^{m-1}\left({\pi }_N^{{\mathrm{S}}_{\mathrm{e}}}\phi \right)\left(\tau \right)d\tau ,\hfill \\ \hfill \left({\mathbf{V}}_N^{m,{\mathrm{D}}_{\mathrm{e}}}\phi \right)\left(s\right)& ={\displaystyle \frac{1}{(m-1)!}}{\int }_a^s{(s-\tau )}^{m-1}\left({\pi }_N^{{\mathrm{D}}_{\mathrm{e}}}\phi \right)\left(\tau \right)d\tau .\hfill \end{array}$$

Let ${\phi }_N^{m,{\mathrm{S}}_{\mathrm{e}}}$ be the solution of the equation

$$(I+{\mathbf{V}}_N^{m,{\mathrm{S}}_{\mathrm{e}}}){\phi }_N^{m,{\mathrm{S}}_{\mathrm{e}}}=g_m,$$

and let ${\phi }_N^{m,{\mathrm{D}}_{\mathrm{e}}}$ be the solution of the equation

$$(I+{\mathbf{V}}_N^{m,{\mathrm{D}}_{\mathrm{e}}}){\phi }_N^{m,{\mathrm{D}}_{\mathrm{e}}}=g_m.$$

To establish the existence of the operators ${\left(I+{\mathbf{V}}_N^{m,{\mathrm{S}}_{\mathrm{e}}}\right)}^{-1}$ and ${\left(I+{\mathbf{V}}_N^{m,{\mathrm{D}}_{\mathrm{e}}}\right)}^{-1}$, we rely on the same analytical framework presented in the previous section, employing the notion of $\nu$-convergence.

In other words,

S1. 

$∥{\mathbf{V}}_N^{m,{\mathrm{S}}_{\mathrm{e}}}∥<{\gamma }_2<\infty ,\mathrm{for} \mathrm{the} \mathrm{positive} \mathrm{constant}{\gamma }_2$;

S2. 

$∥\left({\mathbf{V}}_N^{m,{\mathrm{S}}_{\mathrm{e}}}-{\mathbf{V}}^m\mathcal{K}\right){\mathbf{V}}_N^{m,{\mathrm{S}}_{\mathrm{e}}}∥\to 0\mathrm{and}∥\left({\mathbf{V}}_N^{m,{\mathrm{S}}_{\mathrm{e}}}-{\mathbf{V}}^m\mathcal{K}\right){\mathbf{V}}^m\mathcal{K}∥\to 0\mathrm{as}N\to \infty$.

Also, the approximate operator ${\mathbf{V}}_N^{m,{\mathrm{D}}_{\mathrm{e}}}$ is $\nu$-convergent to the operator ${\mathbf{V}}^m\mathcal{K}$; that is to say,

D1. 

The sequence ${\left({\mathbf{V}}_N^{m,{\mathrm{D}}_{\mathrm{e}}}\right)}_{N\ge 1}$ is bounded;

D2. 

$∥\left({\mathbf{V}}_N^{m,{\mathrm{D}}_{\mathrm{e}}}-{\mathbf{V}}^m\mathcal{K}\right){\mathbf{V}}_N^{m,{\mathrm{D}}_{\mathrm{e}}}∥\to 0\mathrm{and}∥\left({\mathbf{V}}_N^{m,{\mathrm{D}}_{\mathrm{e}}}-{\mathbf{V}}^m\mathcal{K}\right){\mathbf{V}}^m\mathcal{K}∥\to 0\mathrm{as}N\to \infty .$

For sufficiently large N, the operator $I+{\mathbf{V}}_N^{m,{\mathrm{S}}_{\mathrm{e}}}$ is invertible, and the constant

$${\gamma }^{m,{\mathrm{S}}_{\mathrm{e}}}:=\underset{N}{sup}\parallel {\left(I+{\mathbf{V}}_N^{m,{\mathrm{S}}_{\mathrm{e}}}\right)}^{-1}\parallel$$

is finite. That is, the inverse operator ${\left(I+{\mathbf{V}}_N^{m,{\mathrm{S}}_{\mathrm{e}}}\right)}^{-1}$ exists and is uniformly bounded for large N. Similarly, for sufficiently large N the operator ${\left(I+{\mathbf{V}}_N^{m,{\mathrm{D}}_{\mathrm{e}}}\right)}^{-1}$ is invertible, and the constant is also finite. Hence, the inverse operator ${\left(I+{\mathbf{V}}_N^{m,{\mathrm{D}}_{\mathrm{e}}}\right)}^{-1}$ exists and is uniformly bounded for large N.

The Sinc-collocation method, employing the single exponential transformation, leads to the following linear system:

$$(I+{\mathbf{V}}_N^{m,{\mathrm{S}}_{\mathrm{e}}}){\phi }_N^{m,{\mathrm{S}}_{\mathrm{e}}}\left(t_i^{{\mathrm{S}}_{\mathrm{e}}}\right)=g_m\left(t_i^{{\mathrm{S}}_{\mathrm{e}}}\right),i=-N-1,\cdots ,N+1,$$

After solving this system, the approximate solution is constructed as follows:

$$\begin{array}{cc}\hfill {\phi }_N^{m,{\mathrm{S}}_{\mathrm{e}}}& =g_m-{\mathbf{V}}_N^{m,{\mathrm{S}}_{\mathrm{e}}}{\phi }_N^{m,{\mathrm{S}}_{\mathrm{e}}}.\hfill \end{array}$$

Similarly, the Sinc-collocation method based on the double exponential transformation yields the following linear system:

$$(I+{\mathbf{V}}_N^{m,{\mathrm{D}}_{\mathrm{e}}}){\phi }_N^{m,{\mathrm{D}}_{\mathrm{e}}}\left(t_i^{{\mathrm{D}}_{\mathrm{e}}}\right)=g_m\left(t_i^{{\mathrm{D}}_{\mathrm{e}}}\right),i=-N-1,\cdots ,N+1.$$

Upon solving this linear system, the approximate solution can be expressed as:

$$\begin{array}{cc}\hfill {\phi }_N^{m,{\mathrm{D}}_{\mathrm{e}}}& =g_m-{\mathbf{V}}_N^{m,{\mathrm{D}}_{\mathrm{e}}}{\phi }_N^{m,{\mathrm{D}}_{\mathrm{e}}}.\hfill \end{array}$$

The convergence analysis of the proposed method for this practical case is carried out using the same approach as in the previous section.

4. Discussion of Numerical Results

Fredholm systems are not only of theoretical interest but also play a crucial role in various applied sciences. In electromagnetic theory, for example, boundary integral formulations used to analyze wave scattering phenomena often reduce to Fredholm integral equations of the first kind. Likewise, in the study of viscoelastic materials, models incorporating memory effects typically involve integral operators that fit within the Fredholm framework. These real-world occurrences highlight the importance of developing robust analytical and numerical tools for such systems, reinforcing the practical motivation behind the present study.

In this section, numerical comparisons between the SE-Sinc-collocation and DE-Sinc-collocation methods and the two provided methods are presented. The computation was performed on Sony PC (Sony, Tokyo, Japan) with 2.13GHz Intel(R) Core(TM)2 Duo with 4GB memory, running Microsoft Windows 7 version 6.1.7601. The Maple 17 programming language was used to implement the computational programs. We conduct numerical tests to clarify the theoretical results obtained in the previous section.

Letting

$$\begin{array}{cc}\hfill {\mathcal{E}}_{1,N}& :=\mathcal{O}\left(\sqrt{N}{e}^{-\sqrt{\pi d\alpha N}}\right),\hfill \\ \hfill {\mathcal{E}}_{2,N}& :=\mathcal{O}(exp\left({\displaystyle \frac{-\pi dN}{log\left(\frac{2dN}{\alpha }\right)}}\right)),\hfill \\ \hfill {\mathcal{E}}_{3,N}& :=\mathcal{O}\left(N{e}^{-2\sqrt{\pi d\alpha N}}\right),\hfill \\ \hfill {\mathcal{E}}_{4,N}& :=\mathcal{O}(exp\left({\displaystyle \frac{-2\pi dN}{log(2dN/\alpha }}\right)).\hfill \end{array}$$

We note that ${\mathcal{E}}_{1,N}$ and ${\mathcal{E}}_{2,N}$ represent the rates of convergence obtained in [3], while ${\mathcal{E}}_{3,N}$ and ${\mathcal{E}}_{4,N}$ represent the rates of convergence achieved by our methods. We compare these rates in the following table for various values of $\rho$, $\alpha$, d, and N. The numerical results demonstrate that ${\mathcal{E}}_{3,N}$ is better than ${\mathcal{E}}_{1,N}$, and ${\mathcal{E}}_{4,N}$ is better than ${\mathcal{E}}_{2,N}$. Comparison between rates of convergence is given in

Table 1. Comparison between rates of convergence.

N	${\mathcal{E}}_{1,\mathit{N}}$	${\mathcal{E}}_{3,\mathit{N}}$	${\mathcal{E}}_{2,\mathit{N}}$	${\mathcal{E}}_{4,\mathit{N}}$
5	1.55871 × 10−2	2.42956 × 10−4	1.28975 × 10−4	1.66343 × 10−8
10	2.81803 × 10−3	1.55871 × 10−6	6.10004 × 10−7	3.72142 × 10−13
15	7.12040 × 10−4	5.07003 × 10−7	4.56248 × 10−9	2.081201 × 10−17
20	2.17310 × 10−4	4.72286 × 10−8	4.48499 × 10−11	2.01151 × 10−21
25	7.52300 × 10−5	5.65952 × 10−9	5.32390 × 10−13	2.83411 × 10−25
30	2.85572 × 10−5	8.15433 × 10−10	7.28106 × 10−15	5.30191 × 10−29

for $d={\displaystyle \frac{3.14}{2}}$ and $\alpha =1$.

In Figure 1, we present convergence plots illustrating the rates reported in Table 1. Figure 2 shows a comparison between the numerical solutions ${\vartheta }_N^{{\mathrm{S}}_{\mathrm{e}}}$ and ${\theta }_N^{{\mathrm{S}}_{\mathrm{e}}}$ and their exact counterparts $\vartheta \left(s\right)=e^{-2s}$ and $\theta \left(s\right)=e^{-s}$ for $N=10$. Convergence plots corresponding to the results in

Table 2. Improved single exponential approximation for Example 1.

N	${∥\mathit{\vartheta }-{\mathit{\vartheta }}_{\mathit{N}}^{{\mathbf{S}}_{\mathbf{e}}}∥}_{\mathit{\infty }}$	${∥\mathit{\theta }-{\mathit{\theta }}_{\mathit{N}}^{{\mathbf{S}}_{\mathbf{e}}}∥}_{\mathit{\infty }}$
5	9.365 × 10−4	8.759 × 10−4
10	5.365 × 10−6	7.254 × 10−6
15	7.265 × 10−7	5.687 × 10−7
20	5.982 × 10−8	4.325 × 10−8

and

Table 3. Improved double exponential approximation for Example 1.

N	${∥\mathit{\vartheta }-{\mathit{\vartheta }}_{\mathit{N}}^{{\mathbf{D}}_{\mathbf{e}}}∥}_{\mathit{\infty }}$	${∥\mathit{\theta }-{\mathit{\theta }}_{\mathit{N}}^{{\mathbf{D}}_{\mathbf{e}}}∥}_{\mathit{\infty }}$
5	8.365 × 10−8	9.789 × 10−8
10	4.125 × 10−13	4.265 × 10−13
15	6.351 × 10−17	6.354 × 10−17
20	8.298 × 10−21	6.187 × 10−21

are displayed in Figure 3. In Figure 4, we compare the convergence behavior in Example 1 for the single-exponential and double-exponential approximations. Figure 5 and Figure 6 present convergence plots corresponding to the rates reported in

Table 4. Comparison results for Example 2.

N	${\mathcal{E}}_{\mathit{N}}^{{\mathbf{S}}_{\mathbf{e}}}$	${\mathcal{E}}_{\mathit{N}}^{{\mathbf{D}}_{\mathbf{e}}}$	${\tilde{\mathcal{E}}}_{\mathit{N}}^{{\mathbf{S}}_{\mathbf{e}}}$	${\tilde{\mathcal{E}}}_{\mathit{N}}^{{\mathbf{D}}_{\mathbf{e}}}$
5	6.21548 × 10−4	1.37362 × 10−3	2.38777 × 10−4	9.89875 × 10−7
10	5.07910 × 10−6	4.92750 × 10−7	1.92754 × 10−7	8.62578 × 10−12
15	1.06313 × 10−7	1.59098 × 10−10	6.38425 × 10−8	6.25487 × 10−15
20	3.77483 × 10−9	5.55627 × 10−14	7.62548 × 10−10	6.32565 × 10−20
25	1.90739 × 10−10	1.72236 × 10−16	5.75261 × 10−11	4.25142 × 10−24
30	1.24724 × 10−11	1.77636 × 10−16	6.23687 × 10−12	5.32154 × 10−28

and

Table 5. Comparison results for Example 3.

s	$\left|\mathit{u}\left(\mathit{s}\right)-{\mathit{u}}_{\mathit{N}}\left(\mathit{s}\right)\right|$	$\left|\mathit{u}\left(\mathit{s}\right)-{\mathit{u}}_{\mathit{N}}^{{\mathbf{S}}_{\mathbf{e}}}\left(\mathit{s}\right)\right|$	$\left|\mathit{u}\left(\mathit{s}\right)-{\mathit{u}}_{\mathit{N}}^{{\mathbf{D}}_{\mathbf{e}}}\left(\mathit{s}\right)\right|$
0.00	1.788 × 10−3	2.125 × 10−4	6.859 × 10−7
0.25	2.224 × 10−3	3.254 × 10−4	5.476 × 10−7
0.50	2.711 × 10−3	1.847 × 10−4	7.162 × 10−7
0.75	3.261 × 10−3	4.298 × 10−4	6.354 × 10−7
1.00	3.882 × 10−3	1.651 × 10−4	6.235 × 10−7

, respectively. Finally, Figure 7 visualizes the convergence behavior of the single and double exponential methods based on the results in

Table 6. Improved single exponential results for Example 4.

s	$\left|\mathit{\vartheta }\left(\mathit{s}\right)-{\mathit{\vartheta }}_{\mathit{N}}^{{\mathbf{S}}_{\mathbf{e}}}\left(\mathit{s}\right)\right|$	$\left|\mathit{\theta }\left(\mathit{s}\right)-{\mathit{\theta }}_{\mathit{N}}^{{\mathbf{S}}_{\mathbf{e}}}\left(\mathit{s}\right)\right|$
0	0	0
0.25	6.72663 × 10−6	8.62145 × 10−6
0.5	8.40647 × 10−6	9.16232 × 10−6
0.75	3.00578 × 10−6	5.36214 × 10−6

and

Table 7. Improved double exponential results for Example 4.

s	$\left|\mathit{\vartheta }\left(\mathit{s}\right)-{\mathit{\vartheta }}_{\mathit{N}}^{{\mathbf{S}}_{\mathbf{e}}}\left(\mathit{s}\right)\right|$	$\left|\mathit{\theta }\left(\mathit{s}\right)-{\mathit{\theta }}_{\mathit{N}}^{{\mathbf{S}}_{\mathbf{e}}}\left(\mathit{s}\right)\right|$
0	0	0
0.25	8.26541 × 10−11	4.26514 × 10−11
0.5	7.62547 × 10−11	7.26547 × 10−11
0.75	2.32564 × 10−11	5.62545 × 10−11

.

Example 1.

Table 2 and Table 3 displayed the absolute errors as a function of N, respectively.

First, let us investigate the system of integral Equation (2), where $(f,g)$ is such that the precise solution is

$$\begin{array}{c}\hfill u\left(s\right)={\displaystyle \frac{e^{-s}+e^{-2s}}{2}},v\left(s\right)={\displaystyle \frac{e^{-s}-e^{-2s}}{2}}.\end{array}$$

The kernel is given by

$$\Psi \left(s,\eta \right)=s\eta .$$

The method’s rate of convergence is presented in Table 2 and Table 3. The outcomes validated the convergence characteristics that had been previously shown.

To compare our current approaches with those published in the literature for solving Fredholm integral equations of the second kind, specifically the methods described in [4,16], we note that our approaches are particularly valuable for certain cases of these equations.

$$u\left(s\right)+\lambda {\int }_a^b\Psi \left(s,\eta \right)u\left(\eta \right)d\eta =f\left(s\right),a\le s\le b,$$

(12)

Example 2

([4]). In this Example, we consider the following Fredholm integral equation:

$$u\left(s\right)-{\int }_0^1\left(3\eta -6s^2\right)u\left(\eta \right)d\eta ={\displaystyle \frac{1}{4}}-s,0\le s\le 1,$$

Let ${\mathcal{E}}_N^{{\mathrm{S}}_{\mathrm{e}}}$ and ${\mathcal{E}}_N^{{\mathrm{D}}_{\mathrm{e}}}$ denote the maximum absolute errors introduced at the respective collocation points as presented in [4]. Meanwhile, ${\tilde{\mathcal{E}}}_N^{{\mathrm{S}}_{\mathrm{e}}}$ and ${\tilde{\mathcal{E}}}_N^{{\mathrm{D}}_{\mathrm{e}}}$ indicate the maximum absolute errors generated by our methods. We compare these rates for various values of ρ, α, d, and N in the following table. The numerical results demonstrate that ${\tilde{\mathcal{E}}}_N^{{\mathrm{S}}_{\mathrm{e}}}$ outperforms ${\mathcal{E}}_N^{{\mathrm{S}}_{\mathrm{e}}}$, and ${\tilde{\mathcal{E}}}_N^{{\mathrm{D}}_{\mathrm{e}}}$ surpasses ${\mathcal{E}}_N^{{\mathrm{D}}_{\mathrm{e}}}$. A comparison of the rates of convergence is provided in Table 5 for $d={\displaystyle \frac{3.14}{2}}$ and $\alpha =1$.

Example 3

([16]). In [16], the author employed the composite Trapezoidal rule to approximate the solution of the Fredholm integral equation of the second kind of the form:

$$\begin{array}{c}\hfill u\left(s\right)-{\displaystyle \frac{1}{2}}{\int }_0^1(s-\eta )e^{-s\eta }u\left(\eta \right)d\eta =e^{-s}-{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}{e}^{-(s-1)}.\end{array}$$

with the exact solution

$$\begin{array}{c}\hfill u\left(s\right)=e^{-s}.\end{array}$$

In this example, we will solve the above model problem that has a smooth exact solution.

Table 5 presents the comparison results for Example 4, where we evaluate the errors $\left|u\left(s\right)-u_N\left(s\right)\right|$, $\left|u\left(s\right)-u_N^{{\mathrm{S}}_{\mathrm{e}}}\left(s\right)\right|$, and $\left|u\left(s\right)-u_N^{{\mathrm{D}}_{\mathrm{e}}}\left(s\right)\right|$ at selected points s. Specifically, we report the errors between the exact solution $u\left(s\right)$ and the approximations $u_N\left(s\right)$, $u_N^{{\mathrm{S}}_{\mathrm{e}}}\left(s\right)$, and $u_N^{{\mathrm{D}}_{\mathrm{e}}}\left(s\right)$ at the points $s=0,0.25,0.5,0.75,and1$ for $n=5$.

Example 4.

Let us investigate the system of two integral Equation (2), where $(f,g)$ is such that $[a,b]=[0,{\displaystyle \frac{\pi }{3}}]$ and the exact solution is

$$\begin{array}{c}\hfill u\left(s\right)=\sqrt{s},v\left(s\right)=\sqrt{2s}.\end{array}$$

The kernel is given by

$$\Psi \left(s,\eta \right)={\left(s{\eta }^2\right)}^{(4/5)}.$$

The numerical results for this example are presented in Table 6 and Table 7.

Example 5

([17]). The authors of [17] employed a collocation-based Chebyshev method to approximate the solution of the Fredholm integral equation of the second kind of the form:

$$\phi \left(s\right)-\underset{-1}{\overset{1}{\int }}{e}^{\left(2s-{\displaystyle \frac{5}{3}}\eta \right)}\phi \left(\eta \right)d\eta =e^{2s}\left[1-3e^{{\displaystyle \frac{1}{3}}}+3e^{{\displaystyle \frac{-1}{3}}}\right],$$

with the exact solution

$$\begin{array}{c}\hfill \phi \left(s\right)=e^{2s}.\end{array}$$

The numerical results for this Example are presented in

Table 8. Comparison results for Example 5.

s	$\left|\mathit{\phi }\left(\mathit{s}\right)-{\mathit{\phi }}_{\mathit{N}}\left(\mathit{s}\right)\right|$	$\left|\mathit{\phi }\left(\mathit{s}\right)-{\mathit{\phi }}_{\mathit{N}}^{{\mathbf{S}}_{\mathbf{e}}}\left(\mathit{s}\right)\right|$	$\left|\mathit{\phi }\left(\mathit{s}\right)-{\mathit{\phi }}_{\mathit{N}}^{{\mathbf{D}}_{\mathbf{e}}}\left(\mathit{s}\right)\right|$
0.0	1.0000 × 10−5	3.546 × 10−7	9.612 × 10−9
0.2	5.0000 × 10−6	7.241 × 10−7	7.143 × 10−9
0.4	1.0000 × 10−6	4.292 × 10−7	5.214 × 10−9
0.6	1.0000 × 10−6	2.137 × 10−7	1.714 × 10−9
0.8	1.0000 × 10−6	3.825 × 10−7	6.251 × 10−9

.

5. Conclusions

This study analyzes the convergence of Sinc-collocation methods, which utilize a double projection scheme with a novel process, to approximate solutions for a system of second-kind Fredholm integral equations. We assess the convergence of the Sinc-collocation method through a single exponential transformation. Additionally, our analysis includes a demonstration of the convergence of the Sinc-collocation method using a double exponential transformation. A system of linear equations is considered, and we present solutions based on a twofold projection scheme. This study features a comprehensive convergence analysis and discusses relevant theorems, along with the provision of additional error bounds. Numerical examples are included to illustrate the theoretical results and to demonstrate the effectiveness of our methods. We can extend this work to integral equations with nonsmooth exact solutions; Volterra integro-differential equations; and the nonlinear case.