Solution of Fuzzy Volterra Integral Equations In A Bernstein Polynomial Basis

In this paper, we have used the parametric form of fuzzy number and convert a fuzzy Volterra integral equation to a system of integral equations in crisp case. We present a numerical method for solving fuzzy Volterra integral equations of the second kind. The proposed method is based on approximating unknown function with Bernstein’s approximation. This method using simple computation with quite acceptable approximate solution. However, accuracy and efficiency are dependent on the size of the set of Bernstein polynomials. Furthermore we get an estimation of error bound for this method.


I. INTRODUCTION
The solutions of integral equations have a major role in the field of science and engineering.A physical even can be modelled by the differential equation, an integral equation.Since few of these equations cannot be solved explicitly, it is often necessary to resort to numerical techniques which are appropriate combinations of numerical integration and interpolation [1,2].There are several numerical methods for solving linear Volterra integral equation [3].Kauthen in [4] used a collocation method to solve the Volterra-Fredholm integral equation numerically.Maleknejad and et al. in [5] obtained a numerical solution of Volterra integral equations by using Bernstein Polynomials.
The concept of fuzzy numbers and fuzzy arithmetic operations were first introduced by Zadeh [6], Dubois and Prade [7].We refer the reader to [8] for more information on fuzzy numbers and fuzzy arithmetic.The topics of fuzzy integral equations (FIE) which growing interest for some time, in particular in relation to fuzzy control, have been rapidly developed in recent years.The fuzzy mapping function was introduced by Chang and Zadeh [9].Later, Dubois and Prade [10] presented an elementary fuzzy calculus based on the extension principle also the concept of integration of fuzzy functions was first introduced by Dubois and Prade [10].Babolian et al., Abbasbandy et al. in [11,12] obtained a numerical solution of linear Fredholm fuzzy integral equations of the second kind.Also, the fuzzy integral equations have been studied by several authors [13,14,15].
In this paper, we present a novel and very simple numerical method based upon Bernstein's approximation for solving Volterra fuzzy integral equations.

II. PRELIMINARIES
In this section the basic notations used in fuzzy calculus and Bernstein polynomials are introduced.We start by defining the fuzzy number.Definition 1. [16] A fuzzy number is a fuzzy set ) (x u The set of all the fuzzy numbers is denoted by we define addition and multiplication by k as and it is shown that ) , ( 1 D E is a complete metric space [20].
The Bernstein's approximation, We suppose .be the max norm on [0,1] , then the given in [22], shows that the rate of convergence is at (3) due to Voronovskaya [23] shows that for (0,1) the rate of convergence is precisely . 1 n

III. FUZZY VOLTERRA INTEGRAL EQUATION
The Fuzzy Volterra integral equations of the second kind (FVIE-2) is [24] 1. 0 is a fuzzy function these equation may only possess fuzzy solution.Sufficient conditions for the existence of a unique solution to the fuzzy Volterra integral equation are given in [24].Now, we introduce parametric form of a FVIE-2 with respect to Definition 2. Let  5) and ( 6), we have By referring to Remark 2 we have ( 7) It is clear that we must solve two crisp Volterra integral equation of the second kind provided that each of Eqs. ( 7) and ( 8) have solution.
We consider the Volterra integral equations of the second kind given by, , ) To determine an approximate the unknown function of Eq. ( 4), we approximate with Bernstein's approximation , then we have: By referring to Remark 2, we have the following equations x s being chosen as suitable distinct points in ] (0, , and 0 x is taken near 0 such that 1. In general we cannot be able to carry out analytically the integrations, involved.We compute the integral that exist  7) and (8).
We give error bound for this solution in the following theorem.
Theorem 2. Consider the crisp Volterra integral equations of the second kind (7) and (8)

A is nonnegative if and only if 2
A is a A is a generalized permutation matrix, then Eq. ( 9) has a fuzzy Bernstein approximation.
Proof.By we have 0 2 ≥ A . By Theorem 3 and our hypotheses, proof is completed.

A. Comparison with Other Methods
In this subsection, the shortcomings of the existing methods [5,12,26] for solving fuzzy integral equations are pointed out.
Abbasbandy and et al. in [26] used the homotopy analysis method (HAM) to obtain solution of fuzzy integro-differential equation But, in this paper we used Bernstein Polynomials to obtain solution of equation (4).

IV. NUMERICAL EXAMPLES
To illustrate the technique proposed in this paper, consider the following examples.
Example 4.1.We consider the fuzzy Volterra integral equation of the second kind given by, The exact solution in this case is given by 1 According to Eqs. (7) and (8) we have the following two crisp Volterra integral equations Now we approximate the unknown functions where we calculate the error of the exact solution and obtained solution of fuzzy Volterra integral equation with Bernstein approximation.Table 1 show the convergence behavior for      Here a very simple and straight method, based on approximation of the fuzzy unknown function of an fuzzy Volterra integral equation on the Bernstein polynomial basis is used.Our achieve results in this paper, show that Bernstein's approximation method for solving fuzzy Volterra integral equations of second kind, is very effective and the answers are trusty and their accuracy are high and we can execute this method in a computer simply.
are real numbers b and c , ,

.
The set of all the fuzzy numbers (as given in definition 1monotonically increasing, left continuous function on (0,1] and right continuous at 0 ; ii.) (r uis a bounded monotonically decreasing, left continuous function on (0,1] and right continuous at 0 For this example, we use

2 .
We consider the fuzzy Volterra integral equation of the second kind given by, According to Eqs. (7) and (8) we have the following two crisp Volterra integral equations 1we calculate the error of the exact solution and obtained solution of fuzzy Volterra integral equation with Bernstein approximation. Table

1
and obtained solution of fuzzy Volterra integral equation in this example at 0

Figure 1 .
Figure 1.Compares the exact solution and obtained solutions

Figure 2 .
Figure 2. Compares the exact solution and obtained solutions