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Mathematical Problems in Engineering
Volume 2010, Article ID 537909, 8 pages
Research Article

Block-by-Block Method for Solving Nonlinear Volterra-Fredholm Integral Equation

Department of Mathematics, Faculty of Science, Alexandria University, Alexandria, Egypt

Received 21 April 2010; Accepted 27 December 2010

Academic Editor: Christos H. Skiadas

Copyright © 2010 Abdallah A. Badr. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider a nonlinear Volterra-Fredholm integral equation (NVFIE) of the second kind. The Volterra kernel is time dependent, and the Fredholm kernel is position dependent. Existence and uniqueness of the solution to this equation, under certain conditions, are discussed. The block-by-block method is introduced to solve such equations numerically. Some numerical examples are given to illustrate our results.

1. Introduction

Different methods are used to solve integral equations which are investigated from many physical applications such as the mixed problems in the theory of elasticity. Popov [1] applied the orthogonal polynomials method to solve the mixed problem in the mechanics of continuous media. Badr [2] applied Toeplitz matrix method to solve a NVFIE. Abdou et al. [3] discussed the solution of Harmmerstein-Volterra integral equation of the second kind. In [4], Haci obtained, numerically, the solution of a system of Harmmerstein integral equations in the space . The equivalence between Volterra integral equation with degenerate kernel and a linear system of differential equations is mentioned by Cochran [5]. Although there are some works on Hermite-type collocation method for the second-kind VIEs with smooth kernels, not too many studies have dealt with weakly singular kernel. For example, Papatheodorou and Jesanis [6] used the collocation method and obtained the solution of Volterra integrodifferential equation with weakly singular kernels. More information about different analytical and numerical solutions of Volterra equations can be found in Davis [7], Linz [8], Volterra [9], and Wolkenfelt [10].

In this paper, we consider the following NVFIE: The existence of a unique solution for the above equation, under certain conditions, is granted using fixed point theorem, where is the Fredholm kernel and is the Volterra kernel. is called the free term, and the unknown function, , is called the potential function in the applied mathematics, and it will be determined. Both two functions and are assumed in the same space. The parameter has many physical meanings. A numerical method is applied to this equation, and it is reduced it to a system of Volterra integral equations of the second kind. Finally, the block-by-block method is used to obtain the numerical solution of this system. Some examples are stated to illustrate the results.

2. Existence and Uniqueness of Solution

To guarantee the existence and uniqueness of solution to (1.1), we write (1.1) in the integral operator form where Also, we assume the following conditions: (i) and satisfies, in general, the condition ( is a constant). (ii) and satisfies , where is a constant. (iii)The given function with its partial derivatives with respect to position and time is continuous in the space , and its norm is defined as ( is a constant). (iv)The known function satisfies the following conditions:(a), (b).

Under these conditions, using the Cauchy-Schwarz and Minkowski inequalities in conjunction with Schauders fixed point theorem, we easily can prove the following theorem.

Theorem 2.1. If the conditions (i)–(iv) are satisfied, then (1.1) has a unique solution in the space .

3. The Numerical Solution

In this section, we present a numerical method to solve (1.1). This method consists of two phases.

In phase one, we rewrite this equation as a system of linear Volterra integral equations. To do so, we choose a sufficiently small step size , and we assume is a partition of the interval with . Then, at each point , (1.1) becomes where

Replacing the integral by a numerical integration rule of the form , where , then an approximate to (1.1) can be found as a system of Volterra integral equations: In phase two, we use the block-by-block method to solve such system of integral equations of Volterra type. The following subsection explains how this method works.

3.1. Block-by-Block Method, See [8]

Assume that we need to solve an equation of the form The idea behind the block-by-block method is to divide the interval into a mesh , and then we try to evaluate the value of the unknown function at these points except at , where we have that . Using any known rule, say Simpson's rule, we have To obtain a value for , we introduce the point ,  , and then we use Simpson's rule again to obtain

Replacing the value by a quadratic interpolation using the values , , and , we have so that we can compute by Equations (3.5) and (3.8) are a pair of simultaneous equations for and . For sufficiently small , and can be found uniquely using any procedure such as Netwon's method.

In general, for , the approximate solution of (3.4) is evaluated using the following two equations: where

At each subinterval , we solve these two equations simultaneously for the unknowns and , so that we obtain a block of unknowns at a time.

3.2. Application of the Block-by-Block Method on the NVFIE

Combining the previous results, we obtain a new scheme defined by the following two equations: where .

These two equations are two nonlinear equations in the two unknown and , and so they can be determined uniquely using any procedure such as Netwons method. By repetitions, we get the solution for each .

4. Examples

We use the block-by-block method to solve numerically the following examples when (i)case 1: ,(ii)case 2: .

Example 4.1. Solve the Volterra-Fredholm integral equation The exact solution of this integral equation is . Tables 1 and 2 list the error at different values of and for different value of (number of partitions of the interval and (number of partitions of the interval ).

Table 1
Table 2

Example 4.2. Solve the Volterra-Fredholm integral equation The exact solution of this integral equation is . Tables 3 and 4 list the error at different values of and for different value of and , .

Table 3
Table 4

5. Conclusion

Two rules of numerical integration methods are used to solve (1.1). One of these rules is of order , the rectangle method, which is applied on the integral with respect to the variable . The other method is of higher-order, , the block-by-block method, and is applied on the integral with respect to the variable t where we used the fact that . More information about the solution could allow us to use higher-order method with the block-by-block method.


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