Numerical Solution for IVP in Volterra Type Linear Integrodifferential Equations System

and Applied Analysis 3 Table 1: Computed errors for Example 1. t Absolute error for y 1 (t) Absolute error for y 2 (t) 0.0 0.000000 0.0000000000 0.2 1.4801 × 10−10 2.2475 × 10−11 0.4 0.162735585 × 10−5 3.12780502 × 10−7 0.6 0.251133963 × 10−5 0.1536077787 × 10−5 0.8 1.8337 × 10−10 0.8864238659 × 10−5 1.0 4.5905 × 10−10 7.897 × 10−12 Table 2: Computed errors for Example 2. t Absolute error for y 1 (t) Absolute error for y 2 (t) 0.0 0.000000 0.0000000000 0.2 3.840 × 10−11 1.5360 × 10−11 0.4 0.5791064832 × 10−3 0.156041748480 × 10−3 0.6 0.17373195072 × 10−2 0.156041748480 × 10−3 0.8 0.69492781056 × 10−2 1.5360 × 10−11 1.0 0.000 0.000 with the initial conditionsy 1 (0) = y 󸀠 1 (0) = 1, y 2 (0) = y 󸀠󸀠 2 (0) = 0, and y 2 (0) = 1, where g 1 (t) = (2 + t 2 ) e t − t − cos (t) + sin (t) , g 2 (t) = 7 sin (t) − (1 + 2t) cos (t) + et (1 + 4t2 + 4t3) + t − 1. (10) The exact solution of this system is y 1 (t) = e , y 2 (t) = sin(t). With N = 5, the approximated solutions for y 1 (t) and y 2 (t) 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]): − y 󸀠 1 − 1 2 ty 1 + 3 2 y 2 = 5 2 − t − 27 2 t 2 + t 4 + 3 2 (−1 + 2t 2 ) − 1 2 (−3t + 4t 3 )


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 [0, 1) (see [6]).Yusufoglu [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.
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.

Function Approximation
Our strategy is to use Bezier curves to approximate the solutions   (), for 1 ≤  ≤ , which are given below.Define the Bezier polynomials of degree  that approximate, respectively, the actions of   () over the interval [ 0 ,   ] as follows: where ℎ =   −  0 and   is the control point of Bezier curve, and is the Bernstein polynomial of degree  over the interval [ 0 ,   ] (see [17]).By substituting (3) in (2),   () can be defined for  ∈ [ 0 ,   ] 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 [ 0 ,   ] as follows: where ‖ ⋅ ‖ is the Euclidean norm.Our aim is to solve the following problem over the interval [ 0 ,   ]: min  () ( 0 ) =  1 , . . ., The mathematical programming problem ( 7) can be solved by many subroutine algorithms, and we used Maple 12 to solve this optimization problem.

Applications and Numerical Results
Consider the following examples which can be solved by using the presented method.
With  = 5, the approximated solutions for  1 () and  2 () 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.
With  = 5, the computed errors are shown in Table 2 which show the high accuracy of the proposed method.

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.