Approximation Solution to Solving Linear Volterra-Fredholm Integro-Differential Equations of the Second Kind by Using Bernstein Polynomials Method

Mathematical modelling of real-life problems usually results in functional equations, such an ordinary or partial differential equations, integral and integro-differential equations and stochastic equations. Many mathematical formulations of physical phenomena contain integro-differential an equation, these equations arises in many fields like fluid dynamics, biological models and chemical kinetics. In fact, integro-differential equations are usually difficult to solve analytically so it is required to obtain an efficient approximate or numerical solution [1,2].


Introduction
Mathematical modelling of real-life problems usually results in functional equations, such an ordinary or partial differential equations, integral and integro-differential equations and stochastic equations. Many mathematical formulations of physical phenomena contain integro-differential an equation, these equations arises in many fields like fluid dynamics, biological models and chemical kinetics. In fact, integro-differential equations are usually difficult to solve analytically so it is required to obtain an efficient approximate or numerical solution [1,2].
In this study Bernstein polynomial method (BPM) is used to solve the linear Volterra-Fredholm integro-differential equation of the second kind: with the initial condition y (i) (a)=y i , i=1,…., m where a,b,λ 1 ,λ 2 ,y i are constants values, f(x),k 1 (x,t),k 2 (x,t) and µ i ,i=1,...,m with µ i (x)≠0 are known functions that have derivatives on an interval a ≤ x ≤ t ≤ b and y(x) is the unknown function which must be determined.

Bernstein Polynomials Method
Polynomials are incredibly useful mathematical tools as they are simple to define, can be calculated quickly on computer systems and represent a tremendous variety of functions. The Bernstein polynomials of degree-n are defined by [9]: where ( ) , n is the degree of polynomials, i is the index of polynomials and t is the variable.
The exponents on the t term increase by one as i increase, and the exponents on the (1-t) term decrease by one as i increases.
The Bernstein polynomials of degree-n can be defined by blending together two Bernstein polynomials of degree-(n-1) that is, the k th n thdegree Bernstein polynomial can be written as [9]: Bernstein polynomials of degree-n can be written in terms of the power basis. This can be directly calculated using equation (2) and the Binomial theorem as follows [9]: where the Binomial theorem is used to expand (1-t) n-k .
The derivatives of the n th -degree Bernstein polynomials are polynomials of degree-(n-1).

A Matrix Representation for Bernstein Polynomials
In many applications, a matrix formulation for the Bernstein polynomials is useful. These are straight forward to develop if only looking at a linear combination in terms of dot products. Given a polynomial written as a linear combination of the Bernstein basis functions [10]: The dot product of two vectors *Corresponding author: Shahooth MK, Department of Mathematics, Faculty Science and Technology, National University of Malaysia, Malaysia, Tel: +60389215555; E-mail: moha861122@yahoo.com which can be converted to the following form: where b nn are the coefficients of the power basis that are used to determine the respective Bernstein polynomials, we note that the matrix in this case is lower triangular. The matrix of derivatives of Bernstein polynomials

Solution for Volterra-Fredholm Integro-Differential Equations of the Second Kind
In this section, Bernstein polynomials method is proposed to find the approximate solution for Volterra-Fredholm integro-differential equations of the second kind.
Consider the Volterra-Fredholm integro-differential equations of the second kind in equation (1): Substituting (9) into equation (8), we get: Applying equation (7) into equation (10), we have: Now to find all integration in equation (11). Then in order to determine C 0 ,C 1 ,…,C n we need n equations. Now chose x i ,i=1,2,3,…,n in the interval [a,b], which gives n equations. Solve the n equations by Gauss elimination to find the values of C 0 ,C 1 ,…,C n . The following algorithm summarizes the steps for finding the approximate solution for the second kind of linear Volterra-Fredholm integro-differential equations.

Numerical Examples
In this section, two numerical examples are exhibited to illustrate the Bernstein polynomials method. The computations associated with the examples were performed using Matlab ver.2013a.

Example 1:
Consider the linear Volterra-Fredholm integrodifferential equation of the second kind [11].
x u x f x x yu y dy x x y u y dy Tables 1 and 2 show that numerical results and the error respectively with the exact solution for Example 1 for n=1,4 and 7 by using BPM (Figure 1).

Example 2:
Consider the linear Volterra-Fredholm integrodifferential equation of the second kind [12].

Comparison with other Methods
In this part, the BPM was compared its performance with Repeated Trapezoidal method and Repeated Simpson's 1/3 Method. A parameters here such as the degree of BPM and the error are considered as comparison. Throughout this manuscript, the convergence test with the proposed method is considered the last square error. We can notice that all the methods on the finite interval [a,b]. In BPM we proposed that we have a solution and we can develop it by increasing the degree of Bernstein polynomials method. Accordingly, the solution is convergence by increasing the number of the limits of Bernstein polynomials resulted from increasing the degree of Bernstein polynomials n, and the error decreases as results of that. As for Repeated Trapezoidal method and Repeated Simpson's 1/3 Method, the solution is a result of Trapezoidal and Simpson's 1/3 laws. Consequently, the error in this methods decrease in speed depending the h value which in its turn depends on the number of n points as mentioned earlier. Also, the accuracy of solution increases with the increase of n points number and the result will be a decrease at h value. Finally, the following table shows the error between these methods for example 1 (Table 3).

Conclusion
In the present study we have successfully used the proposed method to find an approximation solution for solving a second kind Volterra-Fredholm integro-differential equation. We noted from our results the approximation solution is close to the exact solution when we only used the degree of BPM is n=4 in example 1 and the error is small but still impossible to get satisfactory results with using this degree. When n=7 the result becomes so accuracy, so efficiency and the curve of an approximation solution is exactly over the curve of the exact solution. The figuring comes about additionally demonstrate that this strategy is so productive and it can be successfully use in the numerical arrangement of such sort mathematical statements. The integro differential equations are usually difficult to solve analytically so they are required to obtain an efficient approximated method. For this reason, the presented method have been proposed for approximated solution to the linear Volterra-Fredholm integro-differential equations of the second kind. From numerical examples it can be seen that the proposed numerical method is efficient and accurate to estimate the solution of these equations. Also we noted that when the degree of Bernstein polynomials is increasing the errors decrease to smaller values.