Solving of Linear Volterra-Fredholm Integral Equations via Modification of Block Pulse Functions

A computational method based on modification of block pulse functions is proposed for solving numerically the linear Volterra-Fredholm integral equations. We obtain integration operational matrix of modification of block pulse functions on interval [0, T ). A modification of block pulse functions and their integration operational matrix can be reduced to a linear upper triangular system. Then, the problem under study is transformed to a system of linear algebraic equations which can be used to obtain an approximate solution of linear Volterra-Fredholm integral equations. Furthermore, the rate of convergence is O(h) and error analysis of the proposed method are investigated. The results show that the approximate solutions have a good of efficiency and accuracy.


Introduction
An integral equation is defined as an equation in which the unknown function X(t) to be determined appear under one or more integral signs. The subject of integral equations is one of the most useful mathematical tools in pure and applied mathematics. It arise naturally in physics, chemistry, biology, and engineering applications modelled by initial value problems for a finite interval [a, b]. It also arise as representation formulas for the solutions of differential equations. Indeed, a differential equation can be replaced by an integral equation that incorporates its boundary conditions [4]. It has enormous applications in many physical problems. Many initial and boundary value problems associated with ordinary differential equation (ODE) and partial differential equation (PDE) can be transformed into problems of solving some approximate integral equations [8].
The Volterra-Fredholm integral equation, which is a combination of disjoint Volterra and Fredholm integrals, appears in one integral equation. The Volterra-Fredholm integral equations arise from parabolic boundary value problems, mathematical modelling of the spatiotemporal development of an epidemic, various physical, biological, and chemical applications [10,11]. There are several techniques for approximating the solution such as moving least square method and Chebyshev polynomials [1], collocation and Galerkin methods [2], parameterized pseudospectral integration matrices [9], triangular functions [5], Taylor polynomial [12], and Legendre collocation method [7]. In this paper, we apply a numerical method based on the modification of block pulse functions and integration operational matrix to consider the following linear Volterra-Fredholm integral equation: where X(t) is the unknown function, f (t) is analytic function, while K 1 (s, t) andK 2 (s, t) are the kernels of L 2 functions. In order to obtain an approximate solution for Eq. (1) based on modification of block pulse functions, we derive a new integration operational matrix and reduce our problem to solving a system of linear algebraic equations. Moreover, a new technique for computation of the linear terms in such equations is presented. Furthermore, convergence analysis of modification of block pulse functions is investigated. We also demonstrate the efficiency and accuracy of the proposed method.

Materials and Methods
and ∆(l i ) is length of interval I i 2.3. Function Approximation. Rewriting Eq. (2) in the vector form we have ) can be expanded with respect to εMBPFs such as where Ψ(s) and Ψ(t) are m 1 and m 2 dimensional εMBPFs vectors respectively, and K = (k ij ), i = 0, 1, . . . , m 1 , j = 0, 1, . . . , m 2 is the m 1 × m 2 ε modified block pulse coefficient matrix with where D F usually denotes a diagonal matrix whose diagonal entries are related to a constant vector where the integration operational matrix Q of εMBPFs is given by .
So, the integral of every function f (t) can be approximated as follows

Solving Volterra-Fredholm Integral Equations by Modification of Block Pulse
Functions. We consider following linear Volterra-Fredholm integral equation We approximate functions X(t), f (t), k 1 (s, t), and k 2 (s, t) by εMBPFs as follows In the above approximation, W and F are modified block pulse coefficients vector, K 1 and K 2 are modified block pulse coefficients matrix.
Substituting above approximation in Eq. (3), we get Let K i j be the ith row of the constant matrices K j , j = 1, 2, 3.R i be the ith row of the integration operational matrix Q, D K i j be diagonal matrices with K i j as its diagonal entries. By the relation β α Φ(s)Φ T (s)ds = hI (m1+1)×(m2+1) and assuming m 1 = m 2 = m, we have where where , with substituting (5) and (6) in (4), we get Then, Which is a linear system of equations with upper triangular coefficients matrix that gives the approximate modified block pulse coefficient of the unknown X(t).

Error Analysis.
In the following theorems, for simplicity we assume T = 1 and h = 1 m .

Proof. Proof is like similar theorem in [3] but intervals of integration have to redefine as
Then

Proof. Trapezoidal rule for integral is
where E is error of integration. Suppose t i = i m = ih and The representation error when f (t) is represented by a series of BPFs over every subinterval t i , t i + h k , i = 0, . . . , m − 1 is (7), It is obvious that if f (t) = C (constant), then e i (t) = 0. So, this error is computed for For this function E = 0, so then this error with BPFs is h 2 M . Similarly, the error when f (t) is represented in a series of εMBPFs over every subinterval So, the error with εMBPFs is h 2k M .
So, the error is O h 4 also for I n Now, We define the representation error between f (s, t) and its 2D-εMBPFs expansion f ij over every subregion D ij , is defined as Based on Taylors expansion and similarity to the above discussion, Theorem 3.3. Assume that i P (ω ∈ Ω : u(ω, t) < C) = 1 ii k i < C, i = 1, 2. Then Proof. For a complete proof see [6].

Examples of linear Volterra-Fredholm integral equations.
Example 3.4. consider the linear Volterra-Fredholm integral equation with the exact solution is f (t) = cos t.

Conclusions
The εMBPFs and their integration operational matrix are used to obtain the solution of linear Volterra-Fredholm integral equations. The present method reduces a linear Volterra-Fredholm integral equations into a system of algebraic equations. The convergence and error analysis of the proposed method are investigated. Some numerical examples are given, we plot approximate and exact solution to demonstrate the efficiency and accuracy of the proposed method. The results show that the approximate solutions of the proposed method have a good of efficiency and accuracy.