Proposed Development of Scattering Problem Solution based on Walsh Function

Walsh function are new widely used in the analysis of communication systems, since they have most of the properties of Fourier Series but are more suited to nonlinear studies. The using of Walsh series in the simulation of the wave form is considered, when it is truncated at the end of any group of terms of a given order, and the partial sum will be a stair step approximation to the waveform. The height of each step will be the average value of the waveform over the same interval. The incident wave and scattering wave which satisfied the Maxwell's equation in the incident and scattered field, respectively, have a periodic form and can expressed as a series of Walsh function. The scattered wave is often taken into account, and its problem which is termed " Scattering problem " is classified either differential or integral equations. Both kind can be solved by a Walsh series. The differential equation is solved for the highest derivatives first and the result is then integrated numbers of times to get a required solution. Where as, the integral equation can be solved by direct substitution of Walsh functions in the equation with some iterative fashion. Accordingly, the above characteristics may have considered with some efficient way to solve the scattering problems. This paper provides, studying the Rademacher functions in order to construct the set of Walsh functions, studying the set of Walsh functions and its properties, generalizing the definition of Walsh functions to form a complete set of Walsh functions for the space of square-integrable functions defined over a region [0,1]x[0,1]. Solution of one dimensional linear integral equation of second kind (Fredholm type) using Walsh functions and finally, the results for the solution of one dimensional linear integral equation and the validity of this solution with the exact solution.