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
References
[1]
A. Haar, “Zur Theorie der orthogonalen Funktionensysteme,” Mathematische Annalen, vol. 69, no. 3, pp. 331–371, 1910.
[2]
P. Wojtaszczyk, A Mathematical Introduction to Wavelets, vol. 37 of London Mathematical Society Student Texts, Cambridge University Press, Cambridge, Mass, USA, 1997.
[3]
R. E. A. C. Paley, “A remarkable series of orthogonal functions I,” London Mathematical Society, vol. 34, pp. 241–264, 1932.
[4]
R. E. A. C. Paley, “A remarkable series of orthogonal functions II,” London Mathematical Society, vol. 34, pp. 265–279, 1932.
[5]
N. J. Fine, “On the Walsh functions,” Transactions of the American Mathematical Society, vol. 65, pp. 372–414, 1949.
[6]
C. F. Chen and C. H. Hsiao, “A Walsh series direct method for solving variational problems,” Journal of the Franklin Institute, vol. 300, no. 4, pp. 265–280, 1975.
[7]
M. Asadzadeh, “L2-error estimates for the discrete ordinates method for three-dimensional neutron transport,” Transport Theory and Statistical Physics, vol. 17, no. 1, pp. 1–24, 1988.
[8]
M. Asadzadeh, “Lp and eigenvalue error estimates for the discrete ordinates method for two-dimensional neutron transport,” SIAM Journal on Numerical Analysis, vol. 26, no. 1, pp. 66–87, 1989.
[9]
M. Asadzadeh, “A finite element method for the neutron transport equation in an infinite cylindrical domain,” SIAM Journal on Numerical Analysis, vol. 35, no. 4, pp. 1299–1314, 1998.
[10]
M. Asadzadeh and A. Kadem, “Chebyshev spectral-SN method for the neutron transport equation,” Computers & Mathematics with Applications, vol. 52, no. 3-4, pp. 509–524, 2006.
[11]
M. Asadzadeh, P. Kumlin, and S. Larsson, “The discrete ordinates method for the neutron transport equation in an infinite cylindrical domain,” Mathematical Models & Methods in Applied Sciences, vol. 2, no. 3, pp. 317–338, 1992.
[12]
E. E. Lewis and W. F. Miller Jr., Computational Methods of Neutron Transport, John Wiley & Sons, New York, NY, USA, 1984.
[13]
M. Asadzadeh, “Analysis of a fully discrete scheme for neutron transport in two-dimensional geometry,” SIAM Journal on Numerical Analysis, vol. 23, no. 3, pp. 543–561, 1986.
[14]
A. V. Cardona and M. T. Vilhena, “A solution of linear transport equation using Walsh function and laplace transform,” Annals of Nuclear Energy, vol. 21, pp. 495–505, 1994.
[15]
M. S. Corrington, “Solution of differential and integral equations with walsh functions,” IEEE Transactions on Circuit Theory, vol. 20, no. 5, pp. 470–476, 1973.
[16]
E. L. Ortiz, “The tau method,” SIAM Journal on Numerical Analysis, vol. 6, pp. 480–492, 1969.
[17]
E. L. Ortiz and H. Samara, “An operational approach to the tau method for the numerical solution of nonlinear differential equations,” Computing, vol. 27, no. 1, pp. 15–25, 1981.