Solution of Nonlinear Volterra-hammerstein Integral Equations via Single-term Walsh Series Method

Several numerical methods for approximating the solution of Hammerstein integral equations are known. For Fredholm-Hammerstein integral equations, the classical method of successive approximations was introduced in [16]. A variation of the Nystrom method was presented in [11]. A collocation-type method was developed in [9]. In [3], Brunner applied a collocation-type method to nonlinear Volterra-Hammerstein integral equations and integro-differential equations, and discussed its connection with the iterated collocation method. Han [6] introduced and discussed the asymptotic error expansion of a collocation-type method for Volterra-Hammerstein integral equations. The methods in [6, 9] transform a given integral equation into a system of nonlinear equations, which has to be solved with some kind of iterative method. In [9], the definite integrals involved in the solution may be evaluated analytically only in favorable cases, while in [6] the integrals involved in the solution have to be evaluated at each time step of the iteration. Orthogonal functions, often used to represent arbitrary time functions, have received considerable attention in dealing with various problems of dynamic systems. The main characteristic of this technique is that it reduces these problems to those of solving a system of algebraic equations, thus greatly simplifying the problem. The approach is based on converting the underlying differential equations into integral equations through integration, approximating various signals involved in the equations by truncated orthogonal series and using the operational matrix of integration P to eliminate the integral operations. The matrix P can be uniquely determined based on the particular orthogonal functions. Orthogonal functions have also been proposed to solve linear integral