Abstract and Applied Analysis

Volume 2013 (2013), Article ID 490689, 4 pages

http://dx.doi.org/10.1155/2013/490689

## Numerical Solution for IVP in Volterra Type Linear Integrodifferential Equations System

^{1}Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran^{2}Department of Mathematics, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia

Received 23 May 2013; Accepted 9 July 2013

Academic Editor: Santanu Saha Ray

Copyright © 2013 F. Ghomanjani 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

A method is proposed to determine the numerical solution of system of linear Volterra integrodifferential equations (IDEs) by using Bezier curves. The Bezier curves are chosen as piecewise polynomials of degree *n*, and Bezier curves are determined on by control points. The efficiency and applicability of the presented method are illustrated by some numerical examples.

#### 1. Introduction

Integrodifferential equations (IDEs) have been found to describe various kinds of phenomena, such as glass forming process, dropwise condensation, nanohydrodynamics, and wind ripple in the desert (see [1, 2]).

There are several numerical and analytical methods for solving IDEs. Some different methods are presented to solve integral and IDEs in [3, 4]. Maleknejad et al. [5] used rationalized Haar functions method to solve the linear IDEs system. Linear IDEs system has been solved by using Galerkin methods with the hybrid Legendre and block-Pulse functions on interval (see [6]). Yusufoğlu [7] presented an application of He's homotopy perturbation (HPM) method to solve the IDEs system. He's variational iteration method has been used for solving two systems of Volterra integrodifferential equations (see [8]). Arikoglu and Ozkol [9] presented differential transform method (DTM) for integrodifferential and integral equation systems. He's homotopy perturbation (HPM) method was proposed for system of integrodifferential equations (see [10]). A numerical method based on interpolation of unknown functions at distinct interpolation points has been introduced for solving linear IDEs system with initial values (see [11]). Recently, Biazar introduced a new modification of homotopy perturbation method (NHPM) to obtain the solution of linear IDEs system (see [12]). Taylor expansion method has been used for solving IDEs (see [13, 14]). Rashidinia and Tahmasebi developed and modified Taylor series method (TSM) introduced in [15] to solve the system of linear Volterra IDEs.

In the present work, we suggest a technique similar to the one which was used in [16] for solving the system of linear Volterra IDEs in the following form: with the initial conditions where stands for th-order derivative of . , , and are known functions (), and , , and () are appropriate constants.

The current paper is organized as follows. In Section 2, function approximation will be introduced. Numerical examples will be stated in Section 3. Finally, Section 4 will give a conclusion briefly.

#### 2. Function Approximation

Our strategy is to use Bezier curves to approximate the solutions , for , which are given below. Define the Bezier polynomials of degree that approximate, respectively, the actions of over the interval as follows: where and is the control point of Bezier curve, and is the Bernstein polynomial of degree over the interval (see [17]). By substituting (3) in (2), can be defined for as where (2) is satisfied. The convergence was proved in the approximation with Bezier curves when the degree of the approximate solution, , tends to infinity (see [18]).

Now, the residual function is defined over the interval as follows: where is the Euclidean norm. Our aim is to solve the following problem over the interval :

The mathematical programming problem (7) can be solved by many subroutine algorithms, and we used Maple 12 to solve this optimization problem.

*Remark 1. *In Chapter 1 of [19], it was proved that satisfies
where , and because of this reason that is uniformly continuous on , we have that and , for more details see [19].

#### 3. Applications and Numerical Results

Consider the following examples which can be solved by using the presented method.

*Example 1. *Consider a system of third-order linear Volterra IDEs on the interval (see [4]):
with the initial conditions , , and , where

The exact solution of this system is , .

With , the approximated solutions for and are shown, respectively, in Figures 1 and 2, and the computed errors are shown in Table 1 which show the high accuracy of the proposed method.

*Example 2. *Consider the following system of linear Volterra IDEs equations as follows (see [4]):
under the conditions and , with the exact solution , .

With , the computed errors are shown in Table 2 which show the high accuracy of the proposed method.

#### 4. Conclusions

In this paper, Bernstein's approximation is used to approximate the solution of linear Volterra IDEs. In this method, we approximate our unknown function with Bernstein's approximation. The present results show that Bernstein's approximation method for solving linear Volterra IDEs is very effective and simple, and the answers are trusty, and their accuracy is high, and we can execute this method in a computer simply. The numerical examples support this claim.

#### Acknowledgment

The authors are very grateful to the referees for their valuable suggestions and comments that improved the paper.

#### References

- T. L. Bo, L. Xie, and X. J. Zheng, “Numerical approach to wind ripple in desert,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 8, no. 2, pp. 223–228, 2007. View at Google Scholar - L. Xu, J. H. He, and Y. Liu, “Electro spun nano-porous spheres with Chinese drug,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 8, no. 2, pp. 199–202, 2007. View at Google Scholar - R. Agarwal and D. O’Regan,
*Integral and integro-differential equations theory, methods and applications*, vol. 2, The Gordon and breach science publishers, Singapore, 2000. - J. Rashidinia and A. Tahmasebi, “Taylor series method for the system of linear Volterra integro-differential equations,”
*The Journal of Mathematics and Computer Science*, vol. 4, no. 3, pp. 331–343, 2012. View at Google Scholar - K. Maleknejad, F. Mirzaee, and S. Abbasbandy, “Solving linear integro-differential equations system by using rationalized Haar functions method,”
*Applied Mathematics and Computation*, vol. 155, no. 2, pp. 317–328, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Maleknejad and M. Tavassoli Kajani, “Solving linear integro-differential equation system by Galerkin methods with hydrid functions,”
*Applied Mathematics and Computation*, vol. 159, no. 3, pp. 603–612, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - E. Yusufoğlu, “An efficient algorithm for solving integro-differential equations system,”
*Applied Mathematics and Computation*, vol. 192, no. 1, pp. 51–55, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Saberi-Nadjafi and M. Tamamgar, “The variational iteration method: a highly promising method for solving the system of integro-differential equations,”
*Computers & Mathematics with Applications*, vol. 56, no. 2, pp. 346–351, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - A. Arikoglu and I. Ozkol, “Solutions of integral and integro-differential equation systems by using differential transform method,”
*Computers & Mathematics with Applications*, vol. 56, no. 9, pp. 2411–2417, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Biazar, H. Ghazvini, and M. Eslami, “He's homotopy perturbation method for systems of integro-differential equations,”
*Chaos, Solitons & Fractals*, vol. 39, no. 3, pp. 1253–1258, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Yusufoğlu, “Numerical solving initial value problem for Fredholm type linear integro-differential equation system,”
*Journal of the Franklin Institute*, vol. 346, no. 6, pp. 636–649, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Biazar and M. Eslami, “Modified HPM for solving systems of Volterra integral equations of the second kind,”
*Journal of King Saud University*, vol. 23, no. 1, pp. 35–39, 2011. View at Publisher · View at Google Scholar - Y. Huang and X.-F. Li, “Approximate solution of a class of linear integro-differential equations by Taylor expansion method,”
*International Journal of Computer Mathematics*, vol. 87, no. 6, pp. 1277–1288, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Karamete and M. Sezer, “A Taylor collocation method for the solution of linear integro-differential equations,”
*International Journal of Computer Mathematics*, vol. 79, no. 9, pp. 987–1000, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - A. Tahmasbi and O. S. Fard, “Numerical solution of linear Volterra integral equations system of the second kind,”
*Applied Mathematics and Computation*, vol. 201, no. 1-2, pp. 547–552, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Ghomanjani and M. H. Farahi, “The Bezier control points method for solving delay differential equation,”
*Intelligent Control and Automation*, vol. 3, no. 2, pp. 188–196, 2012. View at Google Scholar - J. Zheng, T. W. Sederberg, and R. W. Johnson, “Least squares methods for solving differential equations using Bézier control points,”
*Applied Numerical Mathematics*, vol. 48, no. 2, pp. 237–252, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Ghomanjani, M. H. Farahi, and M. Gachpazan, “Bézier control points method to solve constrained quadratic optimal control of time varying linear systems,”
*Computational & Applied Mathematics*, vol. 31, no. 3, pp. 433–456, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. I. Berenguer, A. I. Garralda-Guillem, and M. Ruiz Galn, “An approximation method for solving systems of Volterra integro-differential equations,”
*Applied Numerical Mathematics*, vol. 67, pp. 126–135, 2013. View at Google Scholar