Abstract
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.
Highlights
The Walsh functions have many properties similar to those of the trigonometric functions
The operational approach to the Tau method proposed by 16 describes converting of a given integral, integrodifferential equation or system of these equations to a system of linear algebraic equations based on three simple matrices:
In our recent works we have used Walsh functions, Chebyshev polynomials and Lengendre polynomials in order to reduces these kind of equations
Summary
The Walsh functions have many properties similar to those of the trigonometric functions They form a complete, total collection of functions with respect to the space of square Lebesgue integrable functions. In 5 , Fine discovered an important property of the Walsh Fourier series: the m 2nth partial sum of the Walsh series of a function f is piecewise constant, equal to the L1 mean of f, on each subinterval i − 1 /m, i/m. For this reason, Walsh series in applications are always truncated to m 2n terms. 1.3 and Im is the unit matrix, Om is the zero matrix of order m , see 6
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