Numerical Solution of Nonlinear Integro-Differential Equations with Initial Conditions by Bernstein Operational Matrix of Derivative

In this paper, we present a practical matrix method for solving nonlinear Volterra-Fredholm integro-differential equations under initial conditions in terms of Bernstein polynomials on the interval [0,1]. The nonlinear part is approximated in the form of matrices’ equations by operational matrices of Bernstein polynomials, and the differential part is approximated in the form of matrices’ equations by derivative operational matrix of Bernstein polynomials. Finally, the main equation is transformed into a nonlinear equations system, and the unknown of the main equation is then approximated. We also give some numerical examples to show the applicability of the operational matrices for solving nonlinear Volterra-Fredholm integro-differential equations (NVFIDEs).


Introduction
Some of the phenomena in physics, electronics, biology, and other applied sciences lead to nonlinear Volterra-Fredholm integro-differential equations.Of course, these equations can also appear when transforming a differential equation into an integral equation [1][2][3][4][5][6][7].In these kinds of equations, the unknown function   u x appears on both sides; it appears under the integral sign on one side, while it appears as an ordinary derivative on the other side.Investigating the results of nonlinear integrodifferential equations, we observe that no obvious algebraic method has been recognized to solve these equations, and thus, approximation methods are used to solve such equations [8][9][10].Bernstein polynomials (B-plynomials) have many useful properties [11,12].In this paper, we use Bernstein polynomials and their resulting operational matrices to solve the following nonlinear Volterra-Fredholm integro-differential equations with the initial conditions indicated as: x m u x q x u x r x u x k x s u s s k x s u s s f x where the functions         1 2 , , , , , q x r x k x s k x s and   f x are known, whereas is unknown.We approximate phrases including derivative of unknown function by operational matrix of derivative ; and approximate phrases including powers of unknown function by Bernstein polynomial matrix . This yields an equation based on the unknown which is the approximation   u x .To examine the accuracy of the formula, we present some examples.We will show the difference between the accurate and approximation solutions in these examples upon knowing.

Definition of Bernstein Polynomials Basis
The Bernstein basis polynomials of degree are defined by: By using binomial expansion of   With the aid of Equation (3), the Bernstein vector , , , can be written in the form where

Function Approximation
, square integrable in (0,1), may be expressed in terms of Bernstein basis [13].In practice, only the first  terms Bernstein polynomials are considered.Hence, if we write where then where is said dual matrix of and is given in [13].We can also approximate the function   by double Fourier expansion as follows where 00 01 0 for , 0,1, , i j n   .Due of (9), we obtain

Operational Matrix of Integration
In performing arithmetic and other operations on Bernstein basis, we frequently encounter the integration of the vector where is the operational matrix for integration and is given in [13,14].

Product Operational Matrix
It is always necessary to evaluate the product of , which is called the product matrix of Bernstein polynomials basis.Let   x  Then, multiplying the matrix with the vector which is defined in Equation ( 8), we obtain where is an Û matrix and is called the coefficient matrix.The details of obtaining this matrix are given in [13].

Operational Matrix of Derivative
The differentiation of vector in Equation ( 4) can be expressed as: where is the operational matrix of derivatives for Bernstein polynomials.From (5), we have   Defining matrix V and vector Equation ( 17) may then be restated as We now expand vector n . We get where Therefore, we have the operational matrix of derivative as .
If we approximate , then for ( n is the order of derivatives), we get

Operational Matrices for Numerical Solution of NVFIDE
To obtain the approximate solution of the (1) under the initial conditions, the following matrix method is used.Function is approximated by using a finite number of terms in (7) as The approximate functions and by Bernstein polynomials can be framed by: , , , , where 1 K and 2 K are defined with (12).For numerical implementation of the method explained in this section, we need to evaluate  and where and  are positive integers, as follows From ( 14) and ( 16), we have where the vector Therefore, with this method we can approximate We obtain it for as follows We have a similar relation for .So, the components of m and can be computed in terms of components of unknown vector U .In this case, for the Volterra and Fredholm part of (1), we have and Also, to approximate the differential part of Equation (1), making use of Equation ( 23) we have After substituting the approximate Equations ( 24)-( 32) in (1), we get And also, for the initial conditions, we have The initial conditions Equation (34) give two linear equations.Since the number of unknowns for a vector in (33) is , then we collocate Equation (33) in Newton-Cotes points as For collocating Equation (33), we have used the Newton-Cotes points because of their simplicity and their good utility in our implementation as regards the speed and accuracy of answers.However, we can use other points like the Gauss points, Clenshaw-Curtis points, Lobatto points, etc.Here we present the final system: After solving nonlinear system (36), we get ; then, U we have the approximate solution of Equation (1).

Numerical Examples
To illustrate the effectiveness of the proposed method in the present paper, several examples are presented in this section.
By applying the method in Section 4, we have for 2 n  , .
We obtain the approximate solutions of the problem   where the exact solution is . Table 3 with 4 choices of n illustrates the numerical results for this example.By applying the method in Section 4, we have for .

  u x x 
where   The exact solution is   2 u x x 

5
. The results of proposed method on this example are exhibited in Table 4 with 4 choices of .For this complicated NVFIDE, we get acceptable results with a small number of Bernstein basis functions.We obtain the approximate solutions of the problem for 0.9997098280 , 1.000309299 0.0002921170000 , 1.000029079 0.00013307 0.000218757 0.000117802 , 1.000000001 0.000000001 0.00000002 0.00000003 0.000000014 .
at the selected points of the given interval.
We have zoomed out Figures 1 and 2 to a great extent because the exact and approximate solutions are very close.
where,   The exact solution is   . The results of the proposed method for this example are exhibited in Table 5 with 4 choices of .For this complicated NVFIDE, we get acceptable results with a small number of Bernstein basis functions.We obtain the approximate solutions of the problem for and respectively,    We have zoomed out Figures 3 and 4 to a great extent because the exact and approximate solutions are very close.While in [17], the adopted method yields no solution for , takes time to give the approximate answer for , and is proper only for , our method is significant not only because it yeilds solutions for any n, but also because the approximate solutions it gives are very close to exact ones.

Conclusion
In this paper, we have proposed a numerical solution to solve nonlinear Volterra-Fredholm integro-differential equations with initial condition by Bernstein polynomials operational matrices and derived operational matrix.We use formula for numerical examples and it is obvious that the numerical solution coincides with exact solution even with a few Bernstein polynomials used in the approximation.Finally, errors show that the approximation becomes more accurate when n is increased.Therefore, for better results, it is recommended to use a larger n.

Example 3 .
Consider the nonlinear Volterra integrodifferential equation, as follows:

Figure 1 .
Figure 1.Comparison of the exact solution and the approximate solutions for Example 5.

Figure 2 .
Figure 2. Comparison of the absolute error functions for Example 5.

Figure 3 .
Figure 3.Comparison of the exact solution and the approximate solutions for Example 5.

Figure 4 .
Figure 4. Comparison of the absolute error functions for Example 5.

Table 1
with 4 choices of .For this complicate NFIDE with a small number of Bernstein basis functions, we get acceptable results.

Table 4 . Approximate and exact solutions for Example 4.
In this regard, we have reported in figures and a table, the values of the exact solution , the approximate solution , and the absolute error function