Advances in Mathematical Physics
Volume 2016 (2016), Article ID 1798050, 6 pages
http://dx.doi.org/10.1155/2016/1798050

Research Article

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

Avyt Asanov,¹ Elman Hazar,2,3 Mustafa Eroz,² Kalyskan Matanova,¹ and Elmira Abdyldaeva¹

¹Department of Mathematics, Kyrgyz-Turkish Manas University, Bishkek, Kyrgyzstan
²Department of Mathematics, Sakarya University, Sakarya, Turkey
³Department of Applied Mathematics and Informatics, Kyrgyz-Turkish Manas University, Bishkek, Kyrgyzstan

Received 21 June 2016; Revised 15 August 2016; Accepted 24 August 2016

Academic Editor: Soheil Salahshour

Copyright © 2016 Avyt Asanov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 kind

where 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 number

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].

Theorem 2. Let and be two strictly increasing continuous functions on and . Then,

where

and ( 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 and

where 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, and

where .

Corollary 7. Let be a solution of the integral equation (1), , and . Then, and

where .

In this paper, we assume that , and . Then, using Theorems 4 and 5 (and Corollaries 6 and 7), we show that the number defined as

can be determined in terms of quantities , , , and .

Under these circumstances, using Theorem 2, the integral

can 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 get

where

Substituting (18) in (17), we get

where .

Omitting the terms appearing in each equation of system (21) and , we obtain

where .

Let us assume that

where 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 .

We give a concrete example below.

Example 8. Let us take the integral equation (1) for and with

and using

It 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.

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

The nodes

Real value at  

Approx. value at  

The error at  

0
1
1
0

0.1
1
1.00271872
0.00271872

0.2
1
1.00331826
0.00331826

0.3
1
1.00376994
0.00376994

0.4
1
1.00417187
0.00417187

0.5
1
1.00455757
0.00455757

0.6
1
1.00494416
0.00494416

0.7
1
1.00534264
0.00534264

0.8
1
1.00576140
0.00576140

0.9
1
1.00620761
0.00620761

1.0
1
1.00668805
0.00668805

1.1
1
1.00720950
0.00720950

1.2
1
1.00777907
0.00777907

1.3
1
1.00840442
0.00840442

1.4
1
1.00909399
0.00909399

1.5
1
1.00985722
0.00985722

1.6
1
1.01070480
0.01070480

1.7
1
1.01164880
0.01164880

1.8
1
1.01270300
0.01270300

1.9
1
1.01388350
0.01388350

2.0
1
1.01520860
0.01520860

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 inequality

holds 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 get

where .
Along with the inequality , using conditions (19) and (23), we get the following inequality for from (31):

where .
Let the term for be determined by

and 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 get

Let us show that

are the solution of the system of (33). Taking (35) into account, we get

Here, we use the equality

where . 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 function

is a strictly increasing continuous function on . But, for all , the following inequality holds:

Theorem 11. Let be a strictly increasing continuous function on and

Then, the inequality

holds in which .

Proof. Let the error be denoted by and set up the system of equations

for . From this system of equations, we get

for . Using condition (42), we get inequality (43). Therefore, Theorem 11 is proved.

Competing Interests

The authors declare that they have no competing interests.

References

A. S. Apartsyn, Nonclassical Linear Volterra Equations of the First Kind, VSP, TB Utrecht, The Netherlands, 2003.

A. L. Bukhgeim, Volterra Equations and Inverse Problems, VSP, Utrecht, The Netherlands, 1999.

J. Banaś and D. O'Regan, “Volterra-Stieltjes integral operators,” Mathematical and Computer Modelling, vol. 41, no. 2-3, pp. 335–344, 2005.

J. Banaś, J. R. Rodriguez, and K. Sadarangani, “On a class of Urysohn-Stieltjes quadratic integral equations and their applications,” Journal of Computational and Applied Mathematics, vol. 113, no. 1-2, pp. 35–50, 2000.

L. M. Delves and J. Walsh, Numerical Solution of Integral Equations, Oxford University Press, New York, NY, USA, 1974.

M. Federson and R. Bianconi, “Linear Volterrra-Stieltjes integral equations in the sense of the Kurzweil-Henstock integral,” Archivum Mathematicum, vol. 37, no. 4, pp. 307–328, 2001.

M. Federson, R. Bianconi, and L. Barbanti, “Linear Volterra integral equations,” Acta Mathematicae Applicatae Sinica, vol. 18, no. 4, pp. 553–560, 2002.

A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2nd edition, 2008.

K. Maleknejad and N. Aghazadeh, “Numerical solution of Volterra integral equations of the second kind with convolution kernel by using Taylor-series expansion method,” Applied Mathematics and Computation, vol. 161, no. 3, pp. 915–922, 2005.

M. A. Darwish and J. Henderson, “Nondecreasing solutions of a quadratic integral equation of Urysohn-Stieltjes type,” Rocky Mountain Journal of Mathematics, vol. 42, no. 2, pp. 545–566, 2012.

S. A. Isaacson and R. M. Kirby, “Numerical solution of linear Volterra integral equations of the second kind with sharp gradients,” Journal of Computational and Applied Mathematics, vol. 235, no. 14, pp. 4283–4301, 2011.

T. Diogo, N. J. Ford, P. Lima, and S. Valtchev, “Numerical methods for a Volterra integral equation with non-smooth solutions,” Journal of Computational and Applied Mathematics, vol. 189, no. 1-2, pp. 412–423, 2006.

S. Zhang, Y. Lin, and M. Rao, “Numerical solutions for second-kind Volterra integral equations by Galerkin methods,” Applications of Mathematics, vol. 45, no. 1, pp. 19–39, 2000.

A. Asanov, “The derivative of a function by means of an increasing function,” Manas Journal of Engineering, no. 1, pp. 18–64, 2001 (Russian).

A. Asanov, M. H. Chelik, and A. Chalish, “Approximating the Stieltjes integral by using the generalized trapezoid rule,” Le Matematiche, vol. 66, no. 2, pp. 13–21, 2011.

A. Asanov, “Volterra-stielties integral equations of the second kind and the first kind,” Manas Journal of Engineering, no. 2, pp. 79–95, 2002 (Russian).