ABSTRACT
This chapter presents the approximation by polynomials and orthogonal functions for determining the solution of integral equations. Nonlinear integral equations appear in many problems of physical phenomena and engineering [208]. In recent years, many different polynomials and basic functions have been used to estimate the solution of integral equations. The approximate solutions for a system of nonlinear Fredholm integral equations of second kind are available in open literature. The learned researchers Biazer et al. have solved the system of nonlinear Fredholm integral equations of second kind by the Adomian decomposition method [22] and Legendre wavelets method [38]. In [19], the nonlinear Volterra integral equations system has been solved by Adomian decomposition method. Nonlinear Fredholm-Voltera integral equations system has been solved by homotopy perturbation method [223]. Hammerstein
The integral equations has been solved by B-spline wavelet method [180] in a previous chapter. In this chapter, we have implemented Bernstein polynomials and hybrid Legendre Block-Pulse functions to approximate the solution of the nonlinear Fredholm integral equations system. Since, the polynomials are differentiable and integrable, the Bernstein polynomials are defined on an interval to form a complete basis over the finite interval. On the other hand, each basis function of hybrid Legendre-BlockPulse functions are piecewise continuous functions and also, these functions are orthonormal. It approximates any function defined on the interval [0, 1] very accurately. The numerical technique based on hybrid Legendre-BlockPulse function has been developed to approximate the solution of the system of nonlinear Fredholm-Hammerstein integral equations. These functions are formed by the hybridization of Legendre polynomials and Block-Pulse functions. These functions are orthonormal and have compact support on [0, 1]. The numerical results obtained by the present method have been compared with other methods. These proposed methods reduce the system of integral equations to a system of algebraic equations that can be solved easily by any of the usual numerical methods. Numerical examples are presented to illustrate the accuracy of the method.
