A Computational Method Based on Bernstein Polynomials for Solving FredholmIntegro-Differential Equations under Mixed Conditions

Corresponding Author: Miloud Moussai Laboratoire de Mathématiques Pures et Appliquées, University of M’sila, M’sila, 28000 Algeria Email: mi.moussai@gmail.com Abstract: In this study, a computational method for solving linear FredholmIntegro-Differential Equation (FIDE) of the first order under the mixed conditions using the Bernstein polynomials. First, we present some properties of these polynomials and the method is explained. These properties are then used to convert the integro-differential equation to a system of linear algebraic equations with unknown Bernstein coefficients. Using Galerkin method, we give an approximate solution. This method seems very attractive and simple to use. Illustrative examples show the efficiency and validity of the method we discuss the results using error analysis, the results are discussed.


Introduction
Integro-differential equations, which are composed by integral and differential equations, are a well-known mathematical tool and an important branch of modern mathematics. Among these equations, FredholmIntegro-Differential Equations (FIDE) are encountered in several areas such as biology, economics, engineering and many others, so as usual, there is no specific analytic method to solve this equations, several numerical methods are presented to approximate the solution of FIDEs. Recently, different methods based on basic functions have been proposed to approximate the solution of FIDE, such as orthogonal basis and wavelets. At the same time, various numerical methods take an important place in solving FIDE numerically, such as Legendre polynomials, which have been used for high-order linear FIDE (Yalçinbaş et al., 2009), rationalized Haar functions and Walsh series, differential transform method (Golubov et al., 1991) and many other known methods in the literature. Among these methods, the polynomials of Bernstein that have been widely used to solve both linear and non linear integro-differential equations. Very few paper using Bernstein polynomials to solve FIDE with mixed conditions have been considered in the literature.
The aim of this paper is to find the solution of the following integro-differential equation: where, Q,R,f and K are continuous functions on the interval [0,1]. Equation 1.1 can be found in (IdreesBhattia and Bracken, 2007;Swarup, 2007). This paper is organized as follows. In section 2, we introduce the properties of the Bernstein polynomials. In section 3, we present the numerical method to solve the problem (1.1-1.2). Illustrative examples are presented in section 4 and finally, in section 5 we give a brief conclusion.

Properties of Bernstein Polynomials
The Bernstein polynomials of degree n on [a,b] are defined by: In what follows, we give some basic properties of the Bernstein polynomials. Formore details, we refer to (Isik et al., 2012;Boyer and Thiel, 2002): In particular if [a,b] = [0, 1], we obtain: The linear combination of the basic Bernstein polynomials is given by the following formula: where, c i are the Bernstein coefficients. We have also the following proprieties: The derivatives of different degrees of Bernstein polynomials have an important role in the numerical solution of differential and integro-differential equations.

Lemma 2.1
The derivative of order k≥ 1 of the Bernstein polynomials is given by: (2.10)

Proof
We prove the relation by induction. We check the property for k = 1: Thus the relation holds for k = 1. It is assumed that the relation is true for k∈N * and we should prove it for (k + 1).
Indeed, we have: According to the above relations, we have: Substituting (2.16) in the relation (2.15), we obtain: It is easy to check that: By substituting (2.18) and (2.19) in the relation (2.17), we obtain: which proves that the relation is true for all k.

Approximation Function
A function f square integrable in the interval [0, 1] can be written as:

Numerical Method
Consider the linear FIDE of the first order (1.1) with the mixed conditions (1.2). The approximation of y(x) by the polynomials of Bernstein is given by: which can be written, after simplification, as: Multiplying both sides in (3.4) by b j,n (x) and integrating with respect to x from 0 to 1 we obtain: which can be written by the following system:

Remark 3.1
The square error E is defined by the formula: where, y(x) and ỹ(x) are the exact and approximate solutions respectively.

Illustrative Examples
The following examples illustrate the efficiency of the used method. Note that in these examples, the series (3.1) are truncated at levels 4 and 5, i.e., (n = 4) and (n = 5).

Example 4.1
Consider the following linear FIDE equation of the first order with mixed conditions:  3.0 1.0000 0.0000 0.0000 0.0000 1.0000 The graph of the exact and approximate solutions for Equation 4.1 is represented in Fig. 1. The results, together with the square error, are reported in Table 1 for different values of x.

Example 4.2
Consider the following linear FIDE of the first order with mixed conditions: The exact solution (4.2) is: We do the same as in example 1 by a comparison of the Equation 1.1 and 1.2 with (4.5), we find: We solve (4.5) by using the method described in Section (4), for (n = 5), we find: The approximate solution is: The graph of the exact and approximate solutions for Equation (4.5) is represented in Fig. 2 and the results are reported in Table 2 for different values of x.

Conclusion
We have applied the Galerkin's method by using the Bernstein polynomials to find the approximate solution of the linear FIDE of the first order. Using this procedure, the integro-differential form is reduced to solve a system of algebraic equations. Examples have been introduced to demonstrate the validity of the present technique. These examples also exhibit the accuracy and efficiency of the present method.