Approximate Solution of Volterra-Stieltjes Linear Integral Equations of the Second Kind with the Generalized Trapezoid Rule

Various issues concerning Volterra and Volterra-Stieltjes integral equations were studied in [1–13]. Some practical and theoretical investigations were made in paper [1] for nonclassical Volterra integral equations of the first kind. Also, the approximate solution for the integral equation considered is obtained. In paper [2], various inverse problems including Volterra operator equations were studied. Some properties for Volterra-Stieltjes integral operators were given in [3]. In the studies [6, 7], existence and uniqueness of the solutions were given for Volterra integral and Volterra operator equations of the first and the second kinds. In papers [4, 6], quadratic integral equations of Urysohn-Stieltjes type and their applications were investigated. Various numerical solutionmethods for integral equations were presented in the studies [8–13].Thenotion of derivative of a function bymeans of a strictly increasing function was given by Asanov in [14]. In the study [15], the generalized trapezoid rule was proposed to evaluate the Stieltjes integral approximately by employing the notion of derivative of a function by means of a strictly increasing function. In this study, we investigate the numerical solution of linear Volterra-Stieltjes integral equations of the second kind by using the generalized trapezoid rule. Therefore, we need the concept of the derivative defined in the works [14, 15] and theorems connected with it.


Introduction
Various issues concerning Volterra and Volterra-Stieltjes integral equations were studied in [1][2][3][4][5][6][7][8][9][10][11][12][13].Some practical and theoretical investigations were made in paper [1] for nonclassical Volterra integral equations of the first kind.Also, the approximate solution for the integral equation considered is obtained.In paper [2], various inverse problems including Volterra operator equations were studied.Some properties for Volterra-Stieltjes integral operators were given in [3].In the studies [6,7], existence and uniqueness of the solutions were given for Volterra integral and Volterra operator equations of the first and the second kinds.In papers [4,6], quadratic integral equations of Urysohn-Stieltjes type and their applications were investigated.Various numerical solution methods for integral equations were presented in the studies [8][9][10][11][12][13].The notion of derivative of a function by means of a strictly increasing function was given by Asanov in [14].In the study [15], the generalized trapezoid rule was proposed to evaluate the Stieltjes integral approximately by employing the notion of derivative of a function by means of a strictly increasing function.
In this study, we investigate the numerical solution of linear Volterra-Stieltjes integral equations of the second kind by using the generalized trapezoid rule.Therefore, we need the concept of the derivative defined in the works [14,15] and theorems connected with it.

Approximating Volterra-Stieltjes Integral Equations
Consider the linear integral equation of the second kind where where () is a given strictly increasing continuous function in (, ).
If the limit in (2) exists, we say that () has a derivative (is differentiable) with respect to ().The first derivative    () may also be a differentiable function with respect to () at every point  ∈ (, ).Then, its derivative is called the second derivative of () with respect to ().
Consequently, the th derivative of () with respect to () is defined by We need the following theorem which is given in [15].
Corollary 3. Let () be a strictly increasing continuous function on [, ], () = 0 for all  ∈ [, ] and where where Then, we will need the following theorem which is given in [16]. where can be evaluated numerically by employing the generalized trapezoid rule.
Let us assume that where   (ℎ) denotes the modulus of continuity of the function ; that is, Under condition (23), the system of (22) has a unique solution which is given by the formulas for  = 2, 3, . . ., .We give a concrete example below.
Using the proposed method of this study, we get the following results.Here, 20 nodes are selected; that is,  = 20.
In Table 1, we give the values of the approximate solution obtained by the proposed method of this study and the error in absolute values at the given nodes.

Estimation of the Error
In this section, we investigate the problem of convergence of the approximate solution   to the solution of integral (1) at the nodes as  → ∞.Theorem 9. Let () be a strictly increasing continuous function on [, ] and for all ,  ∈ [, ] the following inequality holds: where  > 0 and  is independent of the variables  and .

Table 1 :
The values of approximate solution, analytical solution, and the error at the nodes.