Numerical Solutions of Volterra Integral Equations of Third Kind and Its Convergence Analysis

Abstract

The current work suggests a method for the numerical solution of the third type of Volterra integral equations (VIEs), based on Lagrange polynomial, modified Lagrange polynomial, and barycentric Lagrange polynomial approximations. To do this, the interpolation of the unknown function is considered in terms of the above polynomials with unknown coefficients. By substituting this approximation into the considered equation, a system of linear algebraic equations is obtained. Then, we demonstrate the method’s convergence and error estimations. The proposed approaches retain the possible singularity of the solution. To the best of the authors’ knowledge, the singularity case has not been addressed by researchers yet. To illustrate the applicability, effectiveness, and correctness of new methods for the proposed integral equation, examples with both types of kernels, symmetric as well as non-symmetric, are provided at the end.

1. Introduction

At the beginning of the 20th century, Vito Volterra developed new forms of equations, which he called integral equations, for their study of the phenomena of population expansion. More specifically, the equation with the unknown function under the integral sign is known as an integral equation. Integral equations have been the topic of extensive research in recent years, and they may help model and analyze a wide range of problems in mechanics, physics, astronomy, engineering, physics, biology, chemistry, potential theory, electrostatics, and economics, [1,2,3,4,5,6,7]. Furthermore, as in the natural world, many symmetrical objects exist. From the symmetrical point of view, many problems are modeled by integral equations. The extension of the symmetrical concept to the Volterra integral equations is detailed in the study [8].

There are three types of integral equations; however, due to its unique characteristics, we focus on the third type in this study. In the literature, the third type of Volterra integral Equation (VIE) is presented as follows:

$${\mu }^{\beta }\varphi (\mu )=y(\mu )+{\int }_0^{\mu }{(\mu -t)}^{-\gamma }W(\mu ,t)\varphi (t)dt,\mu \in I=[0,T],$$

(1)

where $\gamma \in [0,1),\beta \in {\mathbb{R}}^{+}=(0,\infty ),\gamma +\beta >0$, and $y(\mu )$ represents continuous function on I. $\varphi (\mu )$ is an unknown function that is to be determined.

Moreover, $W(\mu ,t)$ is continuous on $\mathsf{\Delta }=\{W(\mu ,t):0\le t\le x\le T\}$ and defined as

$$\begin{array}{c}\hfill W(\mu ,t)={\mu }^{\gamma +\beta -1}{k}_1(\mu ,t),\end{array}$$

wherein $k_1$∈$C(\mathsf{\Delta })$. This kind of equation has garnered the interest of researchers in the last few years. Likewise the authors of [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26] used different techniques to solve the different types of VIEs of third kind. The existence, uniqueness, and regularity of theorems as well as the computational solutions to the third type of VIE, comprising of equations related to both weakly singular kernels $(0<\gamma <1)$ and smooth kernel $(\gamma =0)$ have been presented in Allaei et al. [27] for $\beta$ > 1 and $\beta$ $\in (0,1)$. Furthermore, the authors of the same study, have deduced the prerequisites for converting Equation (1) into a so-called cordial VIE [28,29]. However, the situation of $\gamma +\beta$ > 1 has been especially intriguing since integral operators associated with Equation (1) would not be compact, and it would not be possible to solve equations using classical numerical methods if $k_1(0,0)\ne 0$. To address the problem of solvability, researchers [30] proposed a modified graded mesh. Furthermore, Gabbasov [31,32] investigated these equations using a unique direct approach as well as a special correlation technique. Shulaia [33] had also examined equations using the spectral expansion solution approach. Additionally, a spectral collocation method based on extended Jacobi wavelets and the Gauss–Jacobi numerical integration formula has been published [34]. Furthermore, these types of equations were examined by Ritz approximations discussed in [35]. Furthermore, the Bernstein approach were used by the authors of [36] for solving such types of equations. The authors of [37] used the regularizing operator created by Lavrentiev’s approach to solve the third kind VIEs. The new class of the third kind of integral equations discussed in [38] wherein the internal point of integration interval, the external known function reduces to zero. Furthermore, the solutions of systems of linear Fredholm integral equations of the third kind with multi-point singularities were discussed in [39]. An operational matrix was introduced [40] as a different method for resolving third kind of VIEs. These studies collectively show that the third kind of VIEs is a topic worth exploring, and those efforts will continue to rise day by day. Given that the authors approximated the solution of the weakly singular Volterra- Fredholm integral equations presented in [41] by the use of Bernstein polynomials. The Bernstein multi-scaling polynomials (BMSPs) were employed by the authors of [42] to solve stochastic Itô-Volterra integral equations.

Furthermore, polynomials are one of the most often used mathematical tools due to their simplicity in expression, ease of definition, and speedy computation on current computers. They thus contributed significantly to approximation theory and numerical analysis for many years. Weierstrass’s demonstration that it is feasible to approximate continuous functions using polynomials. As a result, the current work provided a new approach for numerically solving third kind of VIEs by Lagrange polynomials. The beauty and analytic utility of Lagrange interpolation are both acknowledged. In most circumstances, dealing with polynomial interpolation, the Lagrange technique is the method of choice [43]. This technique has the benefit of not requiring the numbers ${\mu }_1$, ${\mu }_2$, ${\mu }_3$, …, ${\mu }_n$ to be evenly spaced or in sequential sequence.

The proposed method to solve Equation (1) will retain the possible singularity of the solution. To the best of the authors’ knowledge, the singularity case in the solution has remained unaddressed by researchers by employing techniques, some of which are briefed below:

• In 2021, Nemati et al. [34], examined Equation (1) based on Jacobi wavelets and the Gauss -Jacobi quadrature method with the variable transformation by fixing the integral interval into $[-1,1]$ to avoid the singularity problem.
• In 2021, Usta [36] proposed the Bernstein approximation to examine the numerical solution of Equation (1). Based on their technique, splitting the integral interval with $x_i=i/n+ϵ$, $i=0,1,2,\cdots n-1$ and $x_n=1-ϵ$, $ϵ>0$, to bypass the singularity.

The following is an outline of the paper. Theorems and definitions are discussed in Section 2. The main goal of Section 3 is to develop the methodology and numerical system. Section 4 discusses convergence and error analysis. Section 5 is devoted to presenting a few numerical examples, tables, and figures. Finally, Section 6 establishes the work’s conclusion.

2. Preliminaries

In this section, definitions, theorems, and auxiliary facts are mentioned which we need throughout the paper.

Definition 1 

(Lagrange formula [43]). For a collection of $n+1$ data points $\{({\mu }_0,t_0),({\mu }_1,t_1),$$({\mu }_2,t_2),({\mu }_3,t_3),\dots ,({\mu }_n,t_n)\}$ the Lagrange formula can be defined as:

$$P_n(\mu )=\sum _{m=0}^n{\varphi }_m{L}_{n,m}(\mu ),$$

(2)

where

$$L_{n,m}(\mu )=\prod _{k=0,k\ne m}^n\frac{(\mu -{\mu }_k)}{({\mu }_m-{\mu }_k)},$$

(3)

and

$$\begin{array}{c}\hfill L_{n,m}({\mu }_k)=\left\{\begin{array}{c}0,\mathit{for}m\ne k\hfill \\ 1,\mathit{for}m=k.\hfill \end{array}\right.\end{array}$$

Definition 2 

(Barycentric Lagrange formula [44]). For a collection of $n+1$ data points $\{({\mu }_0,t_0),({\mu }_1,t_1),({\mu }_2,t_2),({\mu }_3,t_3),\cdots ,({\mu }_n,t_n)\}$ the barycentric Lagrange formula can be defined as:

$$P_n(\mu )=\frac{{\displaystyle \sum _{j=0}^n}\frac{w_j}{(\mu -{\mu }_j)}{\varphi }_j}{{\displaystyle \sum _{j=0}^n}\frac{w_j}{(\mu -{\mu }_j)}},$$

(4)

where

$$w_j=\frac{1}{\prod _{j\ne k}({\mu }_j-{\mu }_k)},k,j=0,1,\cdots ,n.$$

(5)

Definition 3 

(Modified Lagrange formula [44]). For a collection of $n+1$ data points $\{({\mu }_0,t_0),$$({\mu }_1,t_1),({\mu }_2,t_2),({\mu }_3,t_3),\dots ,({\mu }_n,t_n)\}$ the modified Lagrange formula can be defined as:

$$P_n(\mu )=l(\mu )\sum _{j=0}^n\frac{W_j}{(\mu -{\mu }_j)}{\varphi }_j,$$

(6)

where

$$l(\mu )=\prod _{j=0}^n(\mu -{\mu }_j),$$

(7)

and

$$W_j=\frac{1}{\prod _{k\ne j}({\mu }_j-{\mu }_k)}k,j=0,1,2,\cdots ,n.$$

(8)

Definition 4 

(Absolute error [43]). If p is an approximation to ϕ, the absolute error is given by $|\varphi -p|$.

Definition 5 

Let $\varphi :[0,T]\to \mathbb{R}$ be a function, then $C^{n+1}[0,T]$ is the set of all continuous functions whose ${(n+1)}^{th}$ derivatives are also continuous on $[0,T]$, where $\mathbb{R}$ denotes the set of real numbers.

Definition 6 

The kernel $W(\mu ,t)$ is of the form

$$\begin{array}{c}\hfill W(\mu ,t)=\overline{W}(t,\mu ),\end{array}$$

is called the complex symmetric or Hermitian, the real kernel $W(\mu ,t)$ is symmetric if

$$\begin{array}{c}\hfill W(\mu ,t)=W(t,\mu ).\end{array}$$

Theorem 1 

(Polynomial interpolation theorem [43]). For a set of given $n+1$ points, there exists a unique polynomial of degree less or equal to n that interpolates the points.

Theorem 2 

(Weierstrass approximation theorem [43]). Suppose $\varphi (\mu )$ is a continuous function on the closed interval $[a,b]$ then for a given ϵ > 0 there exists a polynomial $p_n(\mu )$, where the value of n depends on the value of ϵ, such that

$$|\varphi (\mu )-p_n(\mu )|<ϵ,\forall \mu \in [a,b].$$

Therefore, any continuous function of high enough degree can be approximated to any accuracy by a polynomial.

3. Construction of the Numerical Scheme

The main goal of this section is to construct the numerical scheme to solve the third kind of integral equation

$${\mu }^{\beta }\varphi (\mu )=y(\mu )+{\int }_0^{\mu }{(\mu -t)}^{-\gamma }W(\mu ,t)\varphi (t)dt,\mu \in [0,T],$$

where $y(\mu )$ and $W(\mu ,t)$ are continuous functions and $\varphi (\mu )$ is an unknown function that has to be determined.

For solving Equation (1) numerically, the derivation of the methods is presented as follows:

3.1. Lagrange Polynomial Method

Using Lagrange’s polynomial approach to determine the approximate solutions of our considered integral Equation (1) on $[0,T]$, the unknown function $\varphi$ is approximated by relation (2). As a result, the following equations would be provided:

$$\begin{array}{cc}\hfill & {\mu }^{\beta }[{\phi }_0\frac{(\mu -{\mu }_0)(\mu -{\mu }_1)\dots (\mu -{\mu }_n)}{({\mu }_0-{\mu }_1)({\mu }_0-{\mu }_2)\dots ({\mu }_0-{\mu }_n)}+{\varphi }_1\frac{(\mu -{\mu }_0)(\mu -{\mu }_2)\dots (\mu -{\mu }_n)}{({\mu }_1-{\mu }_0)({\mu }_1-{\mu }_2)\dots ({\mu }_1-{\mu }_n)}+\cdots \hfill \\ \hfill & +{\varphi }_n\frac{(\mu -{\mu }_0)(\mu -{\mu }_1)\dots (\mu -{\mu }_{n-1})}{({\mu }_n-{\mu }_0)({\mu }_n-{\mu }_2)\dots ({\mu }_n-{\mu }_{n-1})}]\hfill \\ \hfill & =y(x)+{\int }_0^{\mu }{(\mu -t)}^{-\gamma }W(\mu ,t)[{\varphi }_0\frac{(t-{\mu }_1)(t-{\mu }_2)\dots (\mu -{\mu }_n)}{({\mu }_0-{\mu }_1)({\mu }_0-{\mu }_2)\dots ({\mu }_0-{\mu }_n)}\hfill \\ \hfill & +{\varphi }_1\frac{(t-{\mu }_0)(\mu -{\mu }_2)\dots (t-{\mu }_n)}{({\mu }_1-{\mu }_0)({\mu }_1-{\mu }_2)\dots ({\mu }_1-{\mu }_n)}+\dots +{\varphi }_n\frac{(t-{\mu }_0)(t-{\mu }_1)\dots (t-{\mu }_{n-1})}{({\mu }_n-{\mu }_0)({\mu }_n-{\mu }_2)\dots ({\mu }_n-{\mu }_{n-1})}]dt,\hfill \\ \hfill & \frac{{\varphi }_0}{({\mu }_0-{\mu }_1)({\mu }_0-{\mu }_2)\dots ({\mu }_0-{\mu }_n)}[{\mu }^{\beta }(\mu -{\mu }_1)(\mu -{\mu }_2)\cdots (\mu -{\mu }_n)\hfill \\ \hfill & -{\int }_0^{\mu }{(\mu -t)}^{-\gamma }W(\mu ,t)(t-{\mu }_1)(t-{\mu }_2)\cdots (t-{\mu }_n)dt]\hfill \\ \hfill & +\frac{{\phi }_1}{({\mu }_1-{\mu }_0)({\mu }_1-{\mu }_2)\cdots ({\mu }_1-{\mu }_n)}[{\mu }^{\beta }(\mu -{\mu }_0)(\mu -{\mu }_2)\cdots (\mu -{\mu }_n)\hfill \\ \hfill & -{\int }_0^{\mu }{(\mu -t)}^{-\gamma }W(\mu ,t)(t-{\mu }_0)(t-{\mu }_2)\cdots (t-{\mu }_n)dt]+\cdots \hfill \\ \hfill & +\frac{{\varphi }_n}{({\mu }_n-v_0)({\mu }_n-v_1)\cdots ({\mu }_n-{\mu }_{n-1})}[{\mu }^{\beta }(\mu -{\mu }_0)(\mu -{\mu }_1)\cdots (\mu -{\mu }_n)\hfill \\ \hfill & -{\int }_0^{\mu }{(\mu -t)}^{-\gamma }W(\mu ,t)(t-{\mu }_0)(t-{\mu }_1)\cdots (t-{\mu }_{n-1})dt]=y(\mu ).\hfill \end{array}$$

To obtain the system of $n+1$ equations, put $\mu ={\mu }_i$ for $i=0,1,2,\dots ,n$, we obtain

$$A_1\overrightarrow{v}=\overrightarrow{B_1},$$

(9)

where $A_1$ = $\left[a_{ij}\right]$, $B_1$ = $b_i$,

$$b_i=y_i,i=0,1,2,3,\dots ,n,$$

(10)

and

$$\begin{array}{c}\hfill a_{ij}=\left\{\begin{array}{c}{\mu }_i^{\beta }-\frac{1}{q}\times \left[{\int }_0^{{\mu }_i}{({\mu }_i-t)}^{-\gamma }W({\mu }_i,t)\times q_{1}dt\right],\mathrm{if}i=j\hfill \\ -\frac{1}{q}\times \left[{\int }_0^{{\mu }_i}{({\mu }_i-t)}^{-\gamma }W({\mu }_i,t)\times q_{1}dt\right],\mathrm{if}i\ne j\hfill \end{array}\right.\end{array}$$

(11)

where

$$q=\prod _{k=0,j\ne k}^n({\mu }_j-{\mu }_k),$$

(12)

and

$$q_1=\prod _{k=0,}^n(t-{\mu }_k),$$

(13)

for all $i,j=0,1,2,\cdots n$.

Methodology:

Using the Lagrange polynomial, the numerical solution of third kind of VIEs is described in the following steps:

Step 1: 

Put $h=\frac{b-a}{n}$,   $n\in \mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers.

Step 2: 

Set ${\mu }_i=a+ih$, with ${\mu }_0=a$ and ${\mu }_n=b$,   $i=0,1,\cdots ,n.$

Step 3: 

Using Step 1, Step 2, and Equation (11) to find $a_{ij}$ (note that in Equation (11) the exact value of integral is to be considered).

Step 4: 

Using Equation (10) for determining $b_i^{\prime }s$.

Step 5: 

To solve the system (9) use Steps 3 and 4 and the Biconjugate Gradient Stabilized Method (BiCGSTAB) [45].

3.2. Modified Lagrange Polynomial Method

With the same procedure, the unknown function $\varphi$ is approximated by relation (6), we obtain the following equations:

$$\begin{array}{cc}\hfill & {\mu }^{\beta }(\mu -{\mu }_1(\mu -{\mu }_2)\cdots (\mu -{\mu }_n)w_0{\phi }_0+{\mu }^{\beta }(\mu -{\mu }_0)(\mu -{\mu }_2)\cdots (\mu -{\mu }_n)w_1{\varphi }_1\hfill \\ \hfill & +\cdots +{\mu }^{\beta }(\mu -{\mu }_0)(\mu -{\mu }_1)\cdots (\mu -{\mu }_{n-1})w_n{\varphi }_n=y(\mu )\hfill \\ \hfill & +{\int }_0^{\mu }{(\mu -t)}^{-\gamma }W(\mu ,t)[(t-{\mu }_1)(t-{\mu }_2)\cdots (t-{\mu }_n)w_0{\varphi }_0\hfill \\ \hfill & +(t-{\mu }_0)(t-{\mu }_2)\cdots (t-{\mu }_n)w_1{\varphi }_1+\cdots +(t-{\mu }_0)\cdots (t-{\mu }_1)(t-{\mu }_{n-1})w_n{\varphi }_n]dt,\hfill \\ \hfill & w_0{\varphi }_0\left[{\mu }^{\beta }(\mu -x_1)(\mu -{\mu }_2)\cdots (\mu -{\mu }_n)\right.\hfill \\ \hfill & \left.-{\int }_0^{\mu }{(\mu -t)}^{-\gamma }W(\mu ,t)(t-{\mu }_1)(t-{\mu }_2)\cdots (t-{\mu }_n)dt\right]\hfill \\ \hfill & +w_1{\varphi }_1[{\mu }^{\beta }(\mu -{\mu }_0)(\mu -{\mu }_2)\cdots (\mu -{\mu }_n)\hfill \\ \hfill & -{\int }_0^{\mu }{(\mu -t)}^{-\gamma }W(\mu ,t)(t-{\mu }_0)(t-{\mu }_2)\cdots (t-{\mu }_n)dt]+\cdots \hfill \\ \hfill & +w_n{\varphi }_n[{\mu }^{\beta }(\mu -{\mu }_0)(\mu -{\mu }_2)\cdots (\mu -{\mu }_{n-1})\hfill \\ \hfill & -{\int }_0^{\mu }{(\mu -t)}^{-\gamma }W(\mu ,t)(t-{\mu }_0)(t-{\mu }_2)\cdots (t-{\mu }_{n-1})dt]=y(\mu ).\hfill \end{array}$$

for $i=0,1,2,\cdots ,n$, put $\mu ={\mu }_i$, the system of $n+1$ equations obtained

$$A_1\overrightarrow{v}=\overrightarrow{B_1},$$

(14)

where    $A_1$ = $\left[a_{ij}\right]$, $B_1$ = $b_i$,

$$b_i=y_i,i=0,1,2,3,\dots ,n$$

(15)

and

$$\begin{array}{c}\hfill a_{ij}=\left\{\begin{array}{c}{\mu }_i^{\beta }-\left[{\int }_0^{{\mu }_i}{({\mu }_i-t)}^{-\gamma }W({\mu }_i,t)\times qdt\right],\mathrm{if}i=j\hfill \\ -1\times \left[{\int }_0^{{\mu }_i}{({\mu }_i-t)}^{-\gamma }W({\mu }_i,t)\times qdt\right],\mathrm{if}i\ne j\hfill \end{array}\right.\end{array}$$

(16)

where

$$q=\prod _{k=0,j\ne k}^n\frac{(t-{\mu }_k)}{({\mu }_j-{\mu }_k)},$$

(17)

for all $i,j=0,1,2,\cdots n$.

Methodology:

Using the modified Lagrange polynomial, the numerical solution of third kind of VIEs is described in the following steps:

Step 1: 

Put $h=\frac{b-a}{n}$,   $n\in \mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers.

Step 2: 

Set ${\mu }_i=a+ih$, with ${\mu }_0=a$ and ${\mu }_n=b$,   $i=0,1,\cdots ,n.$

Step 3: 

Using Step 1, Step 2, and Equation (16) to find $a_{ij}$ (note that in Equation  (16) the exact value of integral is to be considered).

Step 4: 

Using Equation  (15) for determining $b_i^{\prime }s$.

Step 5: 

To solve the system (14) use Steps 3 and 4 and the Biconjugate Gradient Stabilized Method (BiCGSTAB) [45].

3.3. Barycentric Lagrange Polynomial Method

Using the barycentric Lagrange’s polynomial method, we can find the approximate solutions of our considered integral Equation  (1) on $[0,T]$.

To numerically solve Equation  (1), the unknown function $\varphi$ is approximated by relation (4). As a result the following equations would be provided:

$$\begin{array}{cc}\hfill & \frac{(\mu -{\mu }_1)(\mu -{\mu }_2)\cdots (\mu -{\mu }_n){\mu }^{\beta }{w}_0{\varphi }_0}{{\displaystyle \sum _{k\ne j=0}^n}({\displaystyle \prod _{k=0}^n}(\mu -{\mu }_k))w_j}+\frac{(\mu -{\mu }_0)(\mu -{\mu }_2)\cdots (\mu -{\mu }_n){\mu }^{\beta }{w}_1{\varphi }_1}{{\displaystyle \sum _{k\ne j=0}^n}({\displaystyle \prod _{k=0}^n}(\mu -{\mu }_k))w_j}\cdots \hfill \\ \hfill & +\frac{(\mu -{\mu }_1)(\mu -{\mu }_2)\cdots (\mu -{\mu }_{n-1}){\mu }^{\beta }{w}_n{\varphi }_n}{{\displaystyle \sum _{k\ne j=0}^n}({\displaystyle \prod _{k=0}^n}(\mu -{\mu }_k))w_j}=y(\mu )\hfill \\ \hfill & +{\int }_0^{\mu }{(\mu -t)}^{-\gamma }W(\mu ,t)[\frac{(t-{\mu }_1)(t-{\mu }_2)\cdots (t-{\mu }_n)w_0{\varphi }_0}{{\displaystyle \sum _{k\ne j=0}^n}({\displaystyle \prod _{k=0}^n}(t-{\mu }_k))w_j}\hfill \\ \hfill & +\frac{(t-{\mu }_0)(t-x_2)\cdots (t-{\mu }_n)w_1{\varphi }_1}{{\displaystyle \sum _{k\ne j=0}^n}({\displaystyle \prod _{k=0}^n}(t-{\mu }_k))w_j}+\cdots +\frac{(t-{\mu }_1)(t-{\mu }_2)\cdots (t-{\mu }_{n-1})w_n{\phi }_n}{{\displaystyle \sum _{k\ne j=0}^n}({\displaystyle \prod _{k=0}^n}(t-{\mu }_k))w_j}]dt,\hfill \\ \hfill & {\varphi }_0[\frac{(\mu -{\mu }_1)(\mu -{\mu }_2)\cdots (\mu -{\mu }_n){\mu }^{\beta }{w}_0}{{\displaystyle \sum _{k\ne j=0}^n}({\displaystyle \prod _{k=0}^n}(\mu -{\mu }_k))w_j}\hfill \\ \hfill & -{\int }_0^{\mu }{(\mu -t)}^{-\gamma }W(\mu ,t)\frac{(t-{\mu }_1)(t-{\mu }_2)\cdots (t-{\mu }_n)w_0}{{\displaystyle \sum _{k\ne j=0}^n}({\displaystyle \prod _{k=0}^n}(t-{\mu }_k))w_j}dt]\hfill \\ \hfill & +{\varphi }_1[\frac{(\mu -{\mu }_0)(\mu -{\mu }_2)\cdots (\mu -{\mu }_n){\mu }^{\beta }{w}_1}{{\displaystyle \sum _{k\ne j=0}^n}({\prod }_{k=0}^n(\mu -{\mu }_k))w_j}\hfill \\ \hfill & -{\int }_0^{\mu }{(\mu -t)}^{-\gamma }W(\mu ,t)\frac{(t-{\mu }_0)(t-{\mu }_2)\cdots (t-{\mu }_n)w_1}{{\displaystyle \sum _{k\ne j=0}^n}({\displaystyle \prod _{k=0}^n}(t-{\mu }_k))w_j}dt]\hfill \\ \hfill & +\cdots +{\varphi }_n[\frac{(\mu -{\mu }_0)(\mu -{\mu }_1)\cdots (\mu -{\mu }_{n-1}){\mu }^{\beta }{w}_n}{{\displaystyle \sum _{k\ne j=0}^n}({\displaystyle \prod _{k=0}^n}(\mu -{\mu }_k))w_j}\hfill \\ \hfill & -{\int }_0^{\mu }{(\mu -t)}^{-\gamma }W(\mu ,t)\frac{(t-{\mu }_0)(t-{\mu }_1)\cdots (t-{\mu }_{n-1})w_n}{{\displaystyle \sum _{k\ne j=0}^n}({\displaystyle \prod _{k=0}^n}(t-{\mu }_k))w_j}dt]=y(\mu ).\hfill \end{array}$$

with $\mu ={\mu }_i$, $i=0,1,2,\dots ,n$, we have the following system of $n+1$ equations

$$A_1\overrightarrow{v}=\overrightarrow{B_1},$$

(18)

where    $A_1$ = $\left[a_{ij}\right]$, $B_1$ = $b_i$,

$$b_i=y_i,i=0,1,2,3,\dots ,n,$$

(19)

and

$$\begin{array}{c}\hfill a_{ij}=\left\{\begin{array}{c}{\mu }_i^{\beta }-\left[{\int }_0^{{\mu }_i}{({\mu }_i-t)}^{-\gamma }W({\mu }_i,t)\times \frac{q}{q_1}dt\right],\mathrm{if}i=j\hfill \\ -1\times \left[{\int }_0^{{\mu }_i}{({\mu }_i-t)}^{-\gamma }W({\mu }_i,t)\times \frac{q}{q_1}dt\right],\mathrm{if}i\ne j\hfill \end{array}\right.\end{array}$$

(20)

where

$$q=\prod _{k=0,j\ne k}^n\frac{(t-{\mu }_k)}{({\mu }_j-{\mu }_k)},$$

(21)

and

$$q_1=\sum _{j\ne k}\left(\prod _{k=0,}^n(t-{\mu }_k)\right)*w_j,$$

(22)

for all $i,j=0,1,2,\cdots n$.

Methodology:   

Using the barycentric Lagrange polynomial, the numerical solution of third kind of VIEs is described in the following steps:

Step 1: 

Put $h=\frac{b-a}{n}$,   $n\in \mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers.

Step 2: 

Set ${\mu }_i=a+ih$, with ${\mu }_0=a$ and ${\mu }_n=b$,   $i=0,1,\cdots ,n.$

Step 3: 

Using Step 1, Step 2, and Equation  (20) to find $a_{ij}$ ( note that in Equation  (20) the exact value of integral is to be considered).

Step 4: 

Using Equation  (19) for determining $b_i^{\prime }s$.

Step 5: 

To solve the system Equation  (18) use Steps 3 and 4 and the Biconjugate Gradient Stabilized Method (BiCGSTAB) [45].

4. Convergence Furthermore, Error Analysis

Theorem 3. 

Let $\varphi (\mu )$ be a function in $C^{n+1}[0,T]$, and let $p_n(\mu )$ be a polynomial of degree $\le n$ interpolating the function $\varphi (\mu )$ at distinct $n+1$ points ${\mu }_0,{\mu }_1,\cdots {\mu }_n$$\in [0,T]$. Then to each μ ∈ $[0,T]$ there exists a point $\eta (\mu )$$\in (0,T)$ such that

$$\varphi (\mu )-p_n(\mu )=\frac{{\varphi }^{n+1}(\eta (\mu ))}{\mathsf{\Gamma }(n+2)}\prod _{i=0}^n(\mu -{\mu }_i),$$

where $p_n(\mu )$ is Lagrange interpolating polynomial.

Proof. 

For $\mu ={\mu }_i$, then the above equality is true since both sides vanish identically. Now suppose that $\mu \ne {\mu }_i$, $i=0,1,\cdots ,n$. Let

$$w(t)=\prod _{i=0}^n(t-{\mu }_i).$$

Define the function ζ for t in $[0,T]$ as follows:

$${\zeta }_{\mu }(t)=\varphi (t)-p_n(t)-\frac{\varphi (\mu )-p_n(\mu )}{w(\mu )}w(t).$$

Then ${\zeta }_{\mu }(t)$∈$C^{n+1}[0,T]$, and ${\zeta }_{\mu }(t)$ vanishes at distinct $n+2$ points, i.e., when $t={\mu }_0,{\mu }_1,\cdots ,{\mu }_n$ or μ. Now, by Rolle’s theorem, which asserts that if for a differentiable function $\varphi (\mu )$ with distinct n zeros, then its derivative must have at least $n-1$ zeros. As a result ${\zeta }_{\mu }^{\prime }(t)$ has at least $n+1$ distinct zeros, ${\zeta }_{\mu }^{″}(t)$ has at least n distinct zeros, and so on ${\zeta }_x^{n+1}(t)$ has at least one distinct zero in $[0,T]$; say $\eta (\mu )$. Now

$$\begin{array}{cc}\hfill {\zeta }_{\mu }^{n+1}(t)& =\frac{d^{n+1}}{d{t}^{n+1}}\left(\varphi (t)-p_n(t)-\frac{\varphi (\mu )-p_n(\mu )}{w(\mu )}w(t)\right)\hfill \\ \hfill & ={\zeta }^{n+1}(t)-p_n^{n+1}(t)-\left(\frac{\varphi (\mu )-p_n(\mu )}{w(\mu )}\right)\frac{d^{n+1}}{d{t}^{n+1}}(w(t))\hfill \\ \hfill & ={\varphi }^{n+1}(t)-p_n^{n+1}(t)-\left(\frac{\varphi (\mu )-p_n(\mu )}{w(\mu )}\right)\frac{d^{n+1}}{d{t}^{n+1}}(\prod _{i=0}^n(t-{\mu }_i))\hfill \\ \hfill & ={\varphi }^{n+1}(t)-p_n^{n+1}(t)-\left(\frac{\varphi (\mu )-p_n(\mu )}{w(\mu )}\right)\mathsf{\Gamma }(n+2).\hfill \end{array}$$

Hence,

$${\zeta }_x^{n+1}(\eta (\mu ))={\varphi }^{n+1}(\eta (\mu ))-p_n^{n+1}(\eta (\mu ))-\left(\frac{\varphi (\mu )-p_n(x)}{w(x)}\right)\mathsf{\Gamma }(n+2)=0.$$

Now, $p_n^{n+1}(\mu )=0$, because $p_n(\mu )$ is a polynomial of degree n. Hence, we have

$${\varphi }^{n+1}(\eta (\mu ))-\left(\frac{\varphi (\mu )-p_n(x)}{w(\mu )}\right)\mathsf{\Gamma }(n+2)=0,$$

or

$$\varphi (\mu )-p_n(\mu )=\frac{{\phi }^{n+1}(\eta (\mu ))}{(n+1)!}w(\mu )=\frac{{\phi }^{n+1}(\eta (\mu ))}{\mathsf{\Gamma }(n+2)}\prod _{i=0}^n(\mu -{\mu }_i),$$

or

$$\begin{array}{c}\hfill |\varphi (\mu )-p_n(\mu )|=\left|\frac{{\phi }^{n+1}(\eta (\mu ))}{\mathsf{\Gamma }(n+2)}\prod _{i=0}^n(\mu -{\mu }_i)\right|,\end{array}$$

$$|\varphi (\mu )-p_n(\mu )|\le \frac{W^{*}(\mu )}{\mathsf{\Gamma }(n+2)}M,$$

(23)

where $M=\underset{{\mu }_0\le \mu \le {\mu }_n}{max}|{\phi }^{n+1}(\eta (\mu )|$ and $W^{*}(\mu )=(\mu -{\mu }_0)(\mu -{\mu }_1)\cdots (\mu -{\mu }_n)$.

Hence the theorem is proved.    □

Theorem 4. 

Assume that $p_n(\mu )$ be the approximate solution and $\varphi (\mu )$ be the exact solution of Equation   (1), then $p_n(\mu )$ converges to $\varphi (\mu )$ as $n\to \infty$.

Proof. 

Suppose that $p_n(\mu )$ is the approximation of $\varphi (\mu )$ by Lagrange polynomials. Then $p_n(\mu )$ must satisfy the following equation:

$$\begin{array}{cc}\hfill & {\mu }^{\beta }{p}_n(\mu )=y(\mu )+{\int }_0^{\mu }{(\mu -t)}^{-\gamma }W(\mu ,t)p_n(t)dt.\hfill \end{array}$$

(24)

Subtracting Equation (28) from Equation  (1) and let $E_n(\mu )=\varphi (\mu )-p_n(\mu )$.

$$\begin{array}{cc}\hfill & {\mu }^{\beta }(\varphi (\mu )-p_n(\mu ))={\int }_0^{\mu }{(\mu -t)}^{-\gamma }W(\mu ,t)(\varphi (t)-p_n(t))dt,\hfill \\ \hfill & E_n(\mu )=\frac{1}{{\mu }^{\beta }}{\int }_0^{\mu }{(\mu -t)}^{-\gamma }W(\mu ,t)(\varphi (t)-p_n(t))dt,\hfill \end{array}$$

$$\begin{array}{cc}\hfill \left|E_n(\mu )\right|& =\left|\frac{1}{{\mu }^{\beta }}{\int }_0^{\mu }{(\mu -t)}^{-\gamma }W(\mu ,t)(\varphi (t)-p_n(t))dt\right|,\hfill \\ \hfill & \le \left|\frac{1}{{\mu }^{\beta }}\right|{\int }_0^{\mu }\left|{(\mu -t)}^{-\gamma }W(\mu ,t)(\varphi (t)-p_n(t))\right|dt,\hfill \\ \hfill & \le \left|\frac{1}{{\mu }^{\beta }}\right|{\int }_0^{\mu }{|(\mu -t)}^{-\gamma }W(\mu ,t)\left|\right|\varphi (t)-p_n(t)|dt,\hfill \\ \hfill & \le \left|\frac{1}{{\mu }^{\beta }}\right|M_1{\int }_0^{\mu }\left|{(\mu -t)}^{-\gamma }\right||\varphi (t)-p_n(t)|dt,\hfill \\ \hfill & \le \left|\frac{1}{{\mu }^{\beta }}\right|\frac{{(T)}^{1-\gamma }}{(1-\gamma )}{\int }_0^{\mu }|\varphi (t)-p_n(t)|dt.\hfill \end{array}$$

Using (23), and note that $\mu \in [0,T]$, it follows that

$$\begin{array}{c}\hfill \left|E_n(\mu )\right|\le \frac{W^{*}(t){(T)}^{1-\gamma }}{\mathsf{\Gamma }(n+2)(1-\gamma )\left|{\mu }^{\beta }\right|-1}M{M}_{1}T,\end{array}$$

where $M_1={max}_{\mu \in [0,t]}\left|W(\mu ,t)\right|$, $M=\underset{{\mu }_0\le \mu \le {\mu }_n}{max}\left|{\phi }^{n+1}(t)\right|$ and $W^{*}(t)=(t-{\mu }_0)(t-{\mu }_1)\cdots (t-{\mu }_n)$.

Thus, it suggests that, if $n\to \infty$, $\left|E_n(\mu )\right|\to 0$. Therefore, $p_n(\mu )$ converges to $\varphi (\mu )$.

This proves the theorem.    □

Theorem 5 

(Error bounds for Lagrange interpolation at equally spaced points [43]). Assume that $\varphi (\mu )$ is defined on the interval $[0,T]$, which contains equally spaced points ${\mu }_k={\mu }_0+hk$. Furthermore, assume that $\varphi (\mu )$ and its $(n+1)$ order derivatives are continuous and bounded on the intervals $[{\mu }_0,{\mu }_n]$, i.e.,

$$\left|{\varphi }^{n+1}(\mu )\right|\le Mfor{\mu }_0\le \mu \le {\mu }_n,$$

then the error bounds for $n=1,2,\cdots ,n$, we have

$$\left|E_n(\mu )\right|\le \frac{M}{4(n+1)}{\left(\frac{T}{n}\right)}^{n+1}=\frac{M}{4(n+1)}{h}^{n+1},for{\mu }_0\le \mu \le {\mu }_n,$$

(25)

where $\left|W^{*}(t)\right|\le \frac{h^{n+1}\mathsf{\Gamma }(n+1)}{4}$.

Theorem 6. 

Assume that $p_n(\mu )$ be the approximate solution and $\varphi (\mu )$ be the exact solution of Equation   (1). Then the error function is as follows:

$$\begin{array}{c}\hfill \left|E_n(\mu )\right|\le \frac{M{M}_1{(T)}^{-\gamma }}{\left|{\mu }^{\beta }\right|(1-\gamma )4(n+1)-1}{\left(\frac{T}{n}\right)}^{(n+1)}=\frac{M^{*}{(T)}^{-\gamma }}{\left|{\mu }^{\beta }\right|(1-\gamma )4(n+1)-1}{h}^{(n+1)}.\end{array}$$

(26)

Proof. 

Suppose that $p_n(\mu )$ is the approximation of $\varphi (\mu )$ by Lagrange polynomials. Then $p_n(\mu )$ must satisfy the following equation:

$$\begin{array}{cc}\hfill & {\mu }^{\beta }{p}_n(\mu )=y(\mu )+{\int }_0^{\mu }{(\mu -t)}^{-\gamma }W(\mu ,t)p_n(t)dt.\hfill \end{array}$$

(27)

Subtracting Equation (27) from Equation (1) and let $E_n(\mu )=\varphi (\mu )-p_n(\mu )$, we obtain

$$\begin{array}{cc}\hfill & {\mu }^{\beta }(\varphi (\mu )-p_n(\mu ))={\int }_0^{\mu }{(\mu -t)}^{-\gamma }W(\mu ,t)(\varphi (t)-p_n(t))dt,\hfill \\ \hfill & {\mu }^{\beta }{E}_n(\mu )={\int }_0^{\mu }{(\mu -t)}^{-\gamma }W(\mu ,t)(\varphi (t)-p_n(t))dt,\hfill \end{array}$$

$$\begin{array}{cc}\hfill \left|E_n(\mu )\right|& =\left|\frac{1}{{\mu }^{\beta }}{\int }_0^{\mu }{(\mu -t)}^{-\gamma }W(\mu ,t)(\varphi (t)-p_n(t))dt\right|,\hfill \\ \hfill & \le \left|\frac{1}{{\mu }^{\beta }}\right|{\int }_0^{\mu }\left|{(\mu -t)}^{-\gamma }W(\mu ,t)(\varphi (t)-p_n(t))\right|dt,\hfill \\ \hfill & \le \left|\frac{1}{{\mu }^{\beta }}\right|{\int }_0^{\mu }\left|{(\mu -t)}^{-\gamma }W(\mu ,t)\left|\right|\varphi (t)-p_n(t)\right|dt,\hfill \\ \hfill & \le \left|\frac{1}{{\mu }^{\beta }}\right|M_1{\int }_0^{\mu }\left|{(\mu -t)}^{-\gamma }\right||\varphi (t)-p_n(t)|dt,\hfill \\ \hfill & \le \left|\frac{1}{{\mu }^{\beta }}\right|\frac{{(T)}^{1-\gamma }}{(1-\gamma )}{\int }_0^{\mu }|\varphi (t)-p_n(t)|dt.\hfill \end{array}$$

Using (23), and note that $\mu \in [0,T]$, it follows that,

$$\begin{array}{c}\hfill \left|E_n(\mu )\right|\le \frac{W^{*}(t){(T)}^{-\gamma }}{\mathsf{\Gamma }(n+2)(1-\gamma )\left|{\mu }^{\beta }\right|-1}M{M}_{1}T,\end{array}$$

$$\begin{array}{c}\hfill \left|E_n(\mu )\right|\le \frac{W^{*}(t){(T)}^{-\gamma }}{\mathsf{\Gamma }(n+2)(1-\gamma )\left|{\mu }^{\beta }\right|-1}{M}^{*}T,\end{array}$$

where $M^{*}=M{M}_1$.

Using Equation (29), the inequalty follows that,

$$\begin{array}{c}\hfill \left|E_n(\mu )\right|\le \frac{M^{*}{(T)}^{-\gamma }}{\left|{\mu }^{\beta }\right|(1-\gamma )4(n+1)-1}{h}^{(n+1)}.\end{array}$$

where $\left|W^{*}(t)\right|\le \frac{h^{n+1}\mathsf{\Gamma }(n+1)}{4}$.

This proves the theorem. □

5. Applications

In this section, the proposed methods are applied to the four considered examples of third-kind VIEs. The kernels in the following examples are of both symmetric and non-symmetric types.

Example 1. 

Consider the VIE of third type as follows:

$${\mu }^{\frac{2}{3}}\varphi (\mu )=y(\mu )+{\int }_0^{\mu }\frac{1}{\sqrt{3}\pi }{t}^{\frac{1}{3}}{(\mu -t)}^{-\frac{2}{3}}\phi (t)dt,\mu \in [0,1],$$

(28)

where

$$y(\mu )={\mu }^{\frac{47}{12}}\left(1-\frac{\mathsf{\Gamma }(\frac{1}{3})\mathsf{\Gamma }(\frac{55}{12})}{\pi \sqrt{3}\mathsf{\Gamma }(\frac{59}{12})}\right).$$

The exact solution of above equation is $\varphi (\mu )={\mu }^{\frac{13}{4}}$ and the graphs of $\varphi (\mu )$ are shown in Figure 1. In

Table 1. Absolute errors of $\varphi (\mu )$ at $n=8$ for Example 1.

$\mathit{\mu }$	Exact Solution	Lagrange Approximation Error	Barycentric Lagrange Approximation Error	Modified Lagrange Approximation Error
0	0	0	0	0
$1.25\times {10}^{-1}$	$1.161335\times {10}^{-3}$	$3.775106\times {10}^{-7}$	$3.775101\times {10}^{-1}$	$3.775106\times {10}^{-7}$
$2.50\times {10}^{-1}$	$1.104854\times {10}^{-2}$	$1.255260\times {10}^{-1}$	$1.255240\times {10}^{-7}$	$1.255260\times {10}^{-7}$
$3.75\times {10}^{-1}$	$4.126687\times {10}^{-2}$	$2.254094\times {10}^{-8}$	$2.254073\times {10}^{-8}$	$2.254094\times {10}^{-8}$
$5.00\times {10}^{-1}$	$1.051120\times {10}^{-1}$	$6.798550\times {10}^{-10}$	$6.798540\times {10}^{-10}$	$6.798550\times {10}^{-10}$
$6.25\times {10}^{-1}$	$2.170751\times {10}^{-1}$	$1.976464\times {10}^{-8}$	$1.976453\times {10}^{-8}$	$1.976464\times {10}^{-8}$
$7.50\times {10}^{-1}$	$3.925989\times {10}^{-1}$	$1.430837\times {10}^{-8}$	$1.430828\times {10}^{-8}$	$1.430837\times {10}^{-8}$
$8.75\times {10}^{-1}$	$6.479271\times {10}^{-1}$	$8.293714\times {10}^{-9}$	$8.293704\times {10}^{-9}$	$8.293714\times {10}^{-9}$
$1.00\times {10}^0$	$1.000000\times {10}^0$	$1.041666\times {10}^{-7}$	$1.041655\times {10}^{-7}$	$1.041666\times {10}^{-7}$
$R.T.$	−	$1.676966\times {10}^2$	$1.567832\times {10}^3$	$2.693308\times {10}^{-1}$

, we present the exact solution, absolute errors, and the CPU time for the solution of the Example 1.

Example 2. 

Consider the VIE of third type as follows:

$${\mu }^{\frac{3}{2}}\varphi (\mu )=y(\mu )+{\int }_0^{\mu }\frac{1}{\sqrt{2}\pi }t{(\mu -t)}^{-\frac{1}{2}}\varphi (t)dt,\mu \in [0,1],$$

(29)

where

$$y(\mu )={\mu }^{\frac{33}{10}}\left(1-\frac{\mathsf{\Gamma }(\frac{19}{5})}{\sqrt{2\pi }\mathsf{\Gamma }(\frac{43}{10})}\right).$$

The exact solution of above equation is $\varphi (\mu )={\mu }^{\frac{9}{5}}$ and the graphs of $\varphi (\mu )$ are shown in Figure 2. In

Table 2. Absolute errors of $\varphi (\mu )$ at $n=8$ for Example 2.

$\mathit{\mu }$	Exact Solution	Lagrange Approximation Error	Barycentric Lagrange Approximation Error	Modified Lagrange Approximation Error
0	0	0	0	0
$1.25\times {10}^{-1}$	$2.368307\times {10}^{-2}$	$1.417307\times {10}^{-5}$	$1.417305\times {10}^{-5}$	$1.417307\times {10}^{-5}$
$2.50\times {10}^{-1}$	$8.246924\times {10}^{-2}$	$2.548815\times {10}^{-6}$	$2.548805\times {10}^{-6}$	$2.548815\times {10}^{-6}$
$3.75\times {10}^{-1}$	$1.711025\times {10}^{-1}$	$1.673861\times {10}^{-6}$	$1.673860\times {10}^{-6}$	$1.673861\times {10}^{-6}$
$5.00\times {10}^{-1}$	$2.871746\times {10}^{-1}$	$8.636553\times {10}^{-7}$	$8.636550\times {10}^{-7}$	$8.636553\times {10}^{-7}$
$6.25\times {10}^{-1}$	$4.291252\times {10}^{-1}$	$7.052558\times {10}^{-7}$	$7.052553\times {10}^{-7}$	$7.052558\times {10}^{-7}$
$7.50\times {10}^{-1}$	$5.958134\times {10}^{-1}$	$3.820696\times {10}^{-7}$	$3.820652\times {10}^{-7}$	$3.820696\times {10}^{-7}$
$8.75\times {10}^{-1}$	$7.863475\times {10}^{-1}$	$5.785149\times {10}^{-7}$	$5.785130\times {10}^{-7}$	$5.785149\times {10}^{-7}$
$1.00\times {10}^0$	$1.000000\times {10}^0$	$8.037568\times {10}^{-7}$	$8.037545\times {10}^{-7}$	$8.037568\times {10}^{-7}$
$R.T.$	−	$1.209714\times {10}^2$	$1.012352\times {10}^3$	$8.144942\times {10}^1$

, we present the exact solution, absolute errors, and the CPU time for the solution of the Example 2.

Example 3. 

Consider the VIE of third type as follows:

$$\mu \varphi (\mu )={\mu }^2(1-\frac{\mu }{3})+{\int }_0^{\mu }t\varphi (t)dt,\mu \in [0,1].$$

(30)

The exact solution of above equation is $\varphi (\mu )=\mu$ and the graphs of $\varphi (\mu )$ are shown in Figure 3. In

Table 3. Absolute errors of $\varphi (\mu )$ at $n=8$ for Example 3.

$\mathit{\mu }$	Exact Solution	Lagrange Approximation Error	Barycentric Lagrange Approximation Error	Modified Lagrange Approximation Error
0	0	0	0	0
$1.25\times {10}^{-1}$	$1.25\times {10}^{-1}$	0	0	0
$2.50\times {10}^{-1}$	$2.50\times {10}^{-1}$	0	0	0
$3.75\times {10}^{-1}$	$3.75\times {10}^{-1}$	0	0	0
$5.00\times {10}^{-1}$	$5.00\times {10}^{-1}$	0	0	0
$6.25\times {10}^{-1}$	$6.25\times {10}^{-1}$	0	0	0
$7.50\times {10}^{-1}$	$7.50\times {10}^{-1}$	0	0	0
$8.75\times {10}^{-1}$	$8.75\times {10}^{-1}$	0	0	0
$1.00\times {10}^0$	$1.00\times {10}^0$	0	0	0
$R.T.$	−	$6.187438\times {10}^0$	$1.853612\times {10}^2$	$1.771958\times {10}^0$

, we present the exact solution, absolute errors, and the CPU time for the solution of the Example 3.

Example 4. 

Consider the VIE of third type as follows:

$$\mu \varphi (\mu )=\frac{6}{7}{\mu }^3\sqrt{\mu }+{\int }_0^{\mu }\frac{1}{2}\varphi (t)dt,x\in [0,1].$$

(31)

The exact solution of above equation is $\varphi (\mu )={\mu }^{\frac{5}{2}}$ and the graphs of $\varphi (\mu )$ are shown in Figure 4. In

Table 4. Absolute errors of $\varphi (\mu )$ at $n=8$ for Example 4.

$\mathit{\mu }$	Exact Solution	Lagrange Approximation Error	Barycentric Lagrange Approximation Error	Modified Lagrange Approximation Error
0	0	0	0	0
$1.25\times {10}^{-1}$	$5.524271\times {10}^{-3}$	$1.866470\times {10}^{-5}$	$1.866450\times {10}^{-5}$	$1.866470\times {10}^{-5}$
$2.50\times {10}^{-1}$	$3.125000\times {10}^{-2}$	$1.263452\times {10}^{-5}$	$1.263402\times {10}^{-5}$	$1.263452\times {10}^{-5}$
$3.75\times {10}^{-1}$	$8.611487\times {10}^{-2}$	$1.040901\times {10}^{-5}$	$1.040900\times {10}^{-5}$	$1.040901\times {10}^{-5}$
$5.00\times {10}^{-1}$	$1.767766\times {10}^{-1}$	$8.981114\times {10}^{-6}$	$8.981102\times {10}^{-6}$	$8.981114\times {10}^{-6}$
$6.25\times {10}^{-1}$	$3.088161\times {10}^{-1}$	$8.054609\times {10}^{-6}$	$8.054601\times {10}^{-6}$	$8.054609\times {10}^{-6}$
$7.50\times {10}^{-1}$	$4.871392\times {10}^{-1}$	$7.328624\times {10}^{-6}$	$7.328601\times {10}^{-6}$	$7.328624\times {10}^{-6}$
$8.75\times {10}^{-1}$	$7.161766\times {10}^{-1}$	$6.833018\times {10}^{-6}$	$6.833005\times {10}^{-6}$	$6.833018\times {10}^{-6}$
$1.00\times {10}^0$	$1.000000\times {10}^0$	$6.192288\times {10}^{-6}$	$6.192205\times {10}^{-6}$	$6.192288\times {10}^{-6}$
$R.T.$	−	$6.179349\times {10}^0$	$5.6320462\times {10}^2$	$1.838735\times {10}^0$

, we present the exact solution, absolute errors, and the CPU time for the solution of the Example 4.

6. Conclusions

In this work, Lagrange polynomial, barycentric Lagrange polynomial, and modified Lagrange polynomial methods are applied to find the numerical solutions of third kind of VIEs. By approximating the unknown function, a system of algebraic equations has been obtained. The error estimation and convergence analysis were discussed for the proposed methods. The illustrative examples with kernels of both types, symmetric and non-symmetric, were discussed to examine the practicality and efficiency of the technique. Furthermore, we draw the following conclusions from the outcomes of the examples. The capability of proposed barycentric Lagrange polynomial and modified Lagrange polynomial approaches to reduce error and computational time, respectively, while approximating the solution, is equivalent to that of recently developed algorithms in the literature. In all techniques, the error term decreases with increasing n (the degree of polynomials). In this paper, we have used the MATLAB software for the computations and programming. The proposed methods can be applied to integral equations, especially with symmetric kernels. This will be addressed in future articles.