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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.

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 [

Let

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 [

In this paper, we consider combined Walsh function with the Sumudu transform in order to extend the transport problem for the three-dimensional case by following the similar method that was proposed in [

Note that, in the case of one-speed neutron transport equation; taking the angular variable in a disc, this problem will corresponds to a three dimensional case with all functions being constant in the azimuthal direction of the

Given the functions

For the boundary terms in

Consider the integrodifferential equation (

Expanding the angular flux

If we deal with different type of boundary conditions, then we consider the first components

Now, we solve the first-order linear differential equation system with isotropic scattering, that is,

Then, we have the following theorem that is subject to the boundary conditions (

Consider the integrodifferential equation (

For this problem we expand the angular flux in terms of the Walsh function in the angular variable with its domain extended into the interval

The operational approach to the Tau method proposed by [

Let

Now we solve (

In order to convert (

Let

For any linear differential operator

Let us assume that

Consider now the discrete ordinates

In general, obtaining solutions of some integrodifferential equations are usually difficult. In our recent works we have used Walsh functions, Chebyshev polynomials and Lengendre polynomials in order to reduces these kind of equations. However our present work suggests that the Tau method can be a good approximation to the exact solutions. The application of the Tau method by using the orthogonal polynomials will be considered as a future work.

The authors thank the referee(s) for the helpful and significant comments that bring the attention of authors to the references [

_{2}-error estimates for the discrete ordinates method for three-dimensional neutron transport

_{p}and eigenvalue error estimates for the discrete ordinates method for two-dimensional neutron transport

_{N}method for the neutron transport equation