Numerical Solution of Nonlinear Fredholm-volterra Integtral Equations via Piecewise Constant Function by Collocation Method

In this work, we present a computational method for solving nonlinear Fredholm-Volterra integral equations of the second kind which is based on replacement of the unknown function by truncated series of well known Block-Pulse functions (BPfs) expansion. Error analysis is worked out that shows efficiency of the method. Finally, we also give some numerical examples.


Introduction
The integral equation method is widely used for solving many problems in mathematical physics and engineering.This article proposes a computational method for solving nonlinear Fredholm-Volterra integral equations.Several numerical methods for approximating the solution of linear and nonlinear integral equations and specially Fredholm-Volterra integral equations are known [1][2][3][4][5][6][7][8][9][10].Also, Block-Pulse functions are studied by many authors and applied for solving different problems.In presented paper, by using vector forms of BPfs, the main problem can be easily reduced to a nonlinear system of algebraic equations which can be solved by Newton's iterative method.

Review of Some Related Papers
Some computational methods for approximating the solution of linear and nonlinear integral equations are known.The classical method of successive approximation for Fredholm-Hammerstein integral equations was introduced in [3].Brunner in [4] applied a collocation type method and Ordokhani in [8] applied rationalized Haar function to nonlinear Volterra-Fredholm-Hammerstein integral equations.A variation of the Nystrom method was presented in [5].A collocation type method was developed in [6].The asymptotic error expansion of a collocation type method for volterra-Hammerstein in-tegral equations has been considered in [7].Yousefi in [9] applied Legendre wavelets to a special type of nonlinear Volterra-Fredholm integral equations of the form.
where ( ) f t , and 1 ( , ) K t x and 2 ( , ) K t x are assumed to be in on the interval 2 Yalcinbas in [10] used Taylor polynomials for solving Equation (1) with ( ) Orthogonal functions and polynomials receive attention in dealing with various problems that one of those in integral equation.The main characteristic of using orthogonal basis is that it reduces these problems to solving a system of nonlinear algebraic equations.The aim of this work is to present a numerical method for approximating the solution of nonlinear Fredholm-Volterra integral equation of the form: , for all 1, 2, , ( ) 0, elsewhere The functions are disjoint and orthogonal.That is, ( ) A function defined over the interval [0, 1) may be expanded as: In practice, only k-term of ( 6) are considered, where k is a power of 2, that is, with matrix from: where, and In a similar manner, [ (  can be approximated in term of BPfs that we need to calculate vector whose elements are nonlinear combination of the elements of the vector u For this purpose, we can write now using (4) leads to where, is the identity matrix of order k.By incorporating these results we have Hence, So using (11) leads to t Now for evaluating the integral at the collocation points 0 ( ) ( )d we may proceed as follows where, in fact, the diagonal matrix j D , is defined as follows : Also, may be approximated as: [ ]

Solution of the Nonlinear Fredholm-Volterra Integral Equations
In order to use BPfs for solving nonlinear Fredholm-Voterra integral equations given in Equation ( 2), we first approximate the , ( ) where k-vectors , , 1 , 2 , and matrices and are BPfs coefficients of , K t x respectively.For solving Equation ( 2), we substitute (15-20) into (2), therefore We now collocate Equation (21) at k points , by using (10) and (13) and the fact that ( ) t  B e where, j e is the j-th column of the identity matrix of order k, Equation (22) may then be restated as Equation ( 23) gives nonlinear equations which can k k

Illustrative Examples
Consider the following nonlinear volterra-Fredholm integral equations.
We applied the method presented in this paper for solving Equation ( 2

Conclusions
The aim of present work is to apply a method for solving the nonlinear Volterra-Fredholm integral equations.The properties of the Block Pulse functions together with the collocation method are used to reduce the problem to the solution of nonlinear algebraic equations.Example 1 is solved in [2] using Chebyshev expansion method (Cem), comparing the results shows Cem is more accurate than BPfs method But, it seems the number of calculations of BPfs method is lower.Also, the benefits of this method are low cost of setting up the equations due to properties of BPfs mentioned in Section 2. In addition, the nonlinear system of algebraic equations is sparse.Finally, this method can be easily extended and applied to nonlinear Volterra-Fredholm integral equations of the form Equation (1).Illustrative examples are included to demonstrate the validity and applicability of the technique.

2 
are constants.For this purpose we define a k-set of BPfs as 1 1,