NUMERICAL SOLUTION OF NONLINEAR VOLTERRA-FREDHOLM INTEGRO DIFFERENTIAL EQUATION BY USING WALSH FUNCTIONS

In this article, we present a numerical method to solve the nonlinear Volterra-Fredholm integro-differential equations by new basis functions, in particular, Walsh functions. Then the operational matrix and direct method for solving the equations are investigated. Finally error estimates and numerical express are also reported. AMS Subject Classification: 45G10, 45D05


Introduction
Several numerical methods for solving linear and nonlinear integro-differential equations have been presented.In most methods, a set of basis functions and an appropriate projection method such as Galerkin, collocation, eac.or a direct method have been applied, see [4].
In this work, we consider the Volterra-Fredholm integro-differential equations of the form n i=0 r i (t)x (i) (t) + λ 1 s 0 k 1 (s, t)F (x(t))dt + 1 0 k 2 (x, t)G(x(t))dt = y(t), (1) where the functions F (x(t)) and G(x(t)) are polynomials of x(t) with constant coefficients.For convenience, we put F (x(t)) = [x(t)] p and G(x(t)) = [x(t)] q where p and q are positive integers.
The presented method in this article can be extended and applied to solve Ordinary differential equations, nonlinear integral, and integro differential equation (1).
Many authors applied block pulse functions for solving differential problem, see [1,6].For solving these equations, this paper uses the orthogonal Walsh functions and operational matrix for differential and integral equations.

Block Pulse Functions (BPF)
An m-set of BPF is defined as the functions are disjoint and orthogonal.That is, The set {φ i (t)} may be normalized to { φ i (t)} by letting φ i (t) = √ mφ i (t) for all i.Thus { φ i (t)} is a disjoint orthogonal system.

Rademacher Functions (RF)
RF's {r i (t)} on unit time are defined in the interval [0, 1).In general r i (t) is a train of unit pulses with 2 m−1 cycles in [0, 1) taking alternately 1 and -1.An exception is r 0 (t) which is the unit pulse over [0, 1).This system of square waves may be generated in many ways physically.The system {r i (t)} is orthonormal but not complete.

Walsh Functions (WF)
WFs may be generated from RFs using the relation where w n (t) is the (n + 1)-th member of w i (t) ordered in a particular way, and q = |[log n 2 ]| + 1 in which for natural number m by [m] least integer larger then over equal m.That is n = d q 2 q−1 + d q−1 2 q−2 + . . .and d k , k = 0, 1 are binary digits of n.The system of WF's are orthonormal and complete.

Relationships Among the Various Systems
Consider the first m = 2 k (k an integer) terms in each of the series of PCBF defined above and write them concisely as m-vectors All the PCBF may be expressed as linear combinations of BPF.As a result of such a possibility it can shown that T BW is constant invertible matrixe.It is well known that the systems {w i (t)} is complete.With respect to the normalised system if we expand a function f (t), the i-th Fourier coefficient is given by by virtue of the mean value theorem, we have The Fourier series with respect to {φ i (t)} of f (t) truncated to retain the first m terms, may be compactly written as where the vector F W represents the related spectra.
which may be concisely written as , an m-term approximation, the integral may be written as where E is defined as above constant invertible matrix.

Differentiation
In view of the fact that t 0 f ′ (s)ds = f (t) − f (0) by virtue of the approximation discussed, we get f d = E −1 (F − f (0)U ) where f d and U represent the PCBF spectral vectors of f (t) and unit step function [1, 0, 0, ..., 0] respectively.and by applying for hight-order we have

Error Estimated
i) Error in PCBF approximation: ii) Error representation of the integral of PCBF: iii)Error representation of the Differentiation of PCBF:

Vector Forms
Consider the m terms of BPFs and write them concisely as m-vector: φ(t) = [φ 0 (t), φ 1 (t), ..., φ m−1 (t)] T .Above representation and disjointness property, follows: where, V is an m-vector and V = diag(V ).Moreover, it can be clearly concluded that for every m × m matrix B: where, B is an m=vector with elements equal to the diagonal entries of matrix B.
So for Walsh functions we have:

Method of Solution
We consider special cases of Volterra-Fredholm integro-differential equations of the following form: In this section, we solve Volterra-Fredholm integro-differential equations by using Walsh functions.We now approximate functions x(t), x ′ (t), [x(t)] p ,[x(t)] q , k 1 (s, t) and k 2 (s, t) with respect to walsh functions.
where w(T ) is defined above, the vectors X, Y , R are walsh functions coefficients of x(t), y(t), r(t), and , are column vectors whose elements are p-th and q-th powers of the vector X and matrixes K 1 and K 2 are m × m walsh coefficients of k 1 (s, t) and k 2 (s, t) respectively.
To approximate the integrals, we get by substituting above equations in Volterra-Fredholm integro-differential equations give , where given equation is a nonlinear system of m algebraic equations for the m unknowns x 0 , x 1 , ..., x m−1 components of X can be obtained by an iterative method.Hence, an approximate x(t) ≃ X T Φ(t) can be computed for Volterra-Fredholm integro-differential equation without any projection method.

Error Estimated for Method
Now we consider the following erro estimates: x q (t) ≃ x q m (t).By applying this approximations in the equation (1), we get so by applying errors in (2), ( 3) and (4) we have with the initial condition x(0) = 1, and the exact solution x(t) = e t .Table 1 shows the numerical results.

Conclusion
The approach presented transforms a nonlinear Volterra-Fredholm integro differential equation into a system of nonlinear algebric equations.The numerical results show that the accuracy of the solutions obtained is good.The approximate solutions by MATLAB software show the validity and efficiency of the proposed method.The above example shows that error will decrease as m increases.This method can be easily extended and applied to system of nonlinear Volterra-Fredholm integro differential equation.
3.1.Integration with Respect to t