Abstract
The numerical solution of linear Volterra-Stieltjes integral equations of the second kind by using the generalized trapezoid rule is established and investigated. Also, the conditions on estimation of the error are determined and proved. A selected example is solved employing the proposed method.
1. Introduction
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 solution methods for integral equations were presented in the studies [8–13]. 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.
2. Approximating Volterra-Stieltjes Integral Equations
Consider the linear integral equation of the second kindwhere is a given continuous function on are given continuous functions on is a given strictly increasing continuous function on , and is the sought function on .
Definition 1. The derivative of a function with respect to is the function , whose value at is the numberwhere 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 derivativeis 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].
Theorem 2. Let and be two strictly increasing continuous functions on and . Then,whereand ( denotes the set of natural numbers).
Corollary 3. Let be a strictly increasing continuous function on , for all and . Then,where
Theorem 4. Let be a strictly increasing continuous function on , and . Then, the integral equation (1) has a unique solution andwhere and .
Then, we will need the following theorem which is given in [16].
Theorem 5. Let be strictly increasing continuous functions on , and . Then,where
Corollary 6. Let be a solution of the integral equation (1), , and . Then, andwhere .
Corollary 7. Let be a solution of the integral equation (1), , and . Then, andwhere .
In this paper, we assume that , and . Then, using Theorems 4 and 5 (and Corollaries 6 and 7), we show that the number defined ascan be determined in terms of quantities , , , and .
Under these circumstances, using Theorem 2, the integralcan be evaluated numerically by employing the generalized trapezoid rule.
3. Numerical Solution
In order to obtain the approximate solution of (1), we employ the generalized trapezoid rule given in [15] to the integral in (1). Let ,where . Let us substitute in the integral equation (1) and examine the following system of equations:To evaluate the integral term in (17), we employ the generalized trapezoid rule given in [15] at the nodes . So, we getwhereSubstituting (18) in (17), we getwhere .
Omitting the terms appearing in each equation of system (21) and , we obtainwhere .
Let us assume thatwhere 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 formulasfor .
We give a concrete example below.
Example 8. Let us take the integral equation (1) for and withand usingIt is easily seen that , is the unique solution of the integral equation (1) and the conditions , and hold, where .
Using the proposed method of this study, we get the following results. Here, 20 nodes are selected; that is, . 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.
4. 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 and is independent of the variables and . Then, the inequalityholds in which , and the number is determined by (20).
Proof. Let the error be denoted by for . Taking into account (21) and (22), we have the following system of equations:where .
Rearranging the above system of equations, we getwhere .
Along with the inequality , using conditions (19) and (23), we get the following inequality for from (31):where .
Let the term for be determined byand as an initial condition.
It is easily seen that for . This can be verified by mathematical induction as follows: for , it is trivial. Let for . Then, using inequality (32), we getLet us show thatare the solution of the system of (33). Taking (35) into account, we getHere, we use the equalitywhere . Consequently, we get the following estimate for the error for all values :Using the fact that is increasing and approaches the number as , we get the following chain of inequalities:for . Hence, the proof is obtained.
Remark 10. The functionis a strictly increasing continuous function on . But, for all , the following inequality holds:
Theorem 11. Let be a strictly increasing continuous function on andThen, the inequalityholds in which .
Proof. Let the error be denoted by and set up the system of equationsfor . From this system of equations, we getfor . Using condition (42), we get inequality (43). Therefore, Theorem 11 is proved.
Competing Interests
The authors declare that they have no competing interests.