Note on the Solution of Transport Equation by Tau Method and Walsh Functions

We consider the combined Walsh function for the three-dimensional case. A method for the solution of the neutron transport equation in three-dimensional case by using the Walsh function, Chebyshev polynomials, and the Legendre polynomials are considered. We also present Tau method, and it was proved that it is a good approximate to exact solutions. This method is based on expansion of the angular flux in a truncated series of Walsh function in the angular variable. The main characteristic of this technique is that it reduces the problems to those of solving a system of algebraic equations; thus, it is greatly simplifying the problem. 1. Introduction The Walsh functions have many properties similar to those of the trigonometric functions. For example, they form a complete, total collection of functions with respect to the space of square Lebesgue integrable functions. However, they are simpler in structure to the trigonometric functions because they take only the values 1 and ？1. They may be expressed as linear combinations of the Haar functions [1], so many proofs about the Haar functions carry over to the Walsh system easily. Moreover, the Walsh functions are Haar wavelet packets. For a good account of the properties of the Haar wavelets and other wavelets, see [2]. We use the ordering of the Walsh functions due to Paley [3, 4]. Any function can be expanded as a series of Walsh functions In [5], Fine discovered an important property of the Walsh Fourier series: the th partial sum of the Walsh series of a function is piecewise constant, equal to the mean of , on each subinterval . For this reason, Walsh series in applications are always truncated to terms. In this case, the coefficients of the Walsh (-Fourier) series are given by where is the average value of the function in the th interval of width in the interval , and is the value of the th Walsh function in the th subinterval. The order Walsh matrix, , has elements . Let have a Walsh series with coefficients and its integral from 0 to have a Walsh series with coefficients : . If we truncate to terms and use the obvious vector notation, then integration is performed by matrix multiplication , where and is the unit matrix, is the zero matrix (of order ), see [6]. 2. The Three-Dimensional Spectral Solution In the literature there several works on driving a suitable model for the transport equation in 2 and 3-dimensional case as well as in cylindrical domain, for example, see [7], and by using the eigenvalue error estimates for two-dimensional neutron transport, see [8], by applying the finite element