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

and Applied Analysis 3 where we assume that the spatial variable x : x, y, z varies in the cubic domain Ω : { x, y, z : −1 ≤ x, y, z ≤ 1}, andΨ x, μ, θ : Ψ x, y, z, μ, θ is the angular flux in the direction defined by μ ∈ −1, 1 and θ ∈ 0, 2π . σt and σs denote the total and the differential cross section, respectively, σs μ′, φ′ → μ, φ describes the scattering from an assumed pre-collision angular coordinates μ′, θ′ to a postcollision coordinates μ, θ and S is the source term. See 12 for further details. Note that, in the case of one-speed neutron transport equation; taking the angular variable in a disc, this problemwill corresponds to a three dimensional case with all functions being constant in the azimuthal direction of the z variable. In this way the actual spatial domain may be assumed to be a cylinder with the cross-section Ω and the axial symmetry in z. Then D will correspond to the projection of the points on the unit sphere the “speed” onto the unit disc which coincides with D . See 13 for the details. Given the functions f1 y, z, μ, φ , f2 x, z, μ, φ , and f3 x, y, μ, φ describing the incident flux, we seek for a solution of 2.1 subject to the following boundary conditions. For the boundary terms in x, for 0 ≤ θ ≤ 2π , let Ψ ( x ±1, y, z, μ, θ ⎧ ⎨ ⎩ f1 ( y, z, μ, θ ) , x −1, 0 < μ ≤ 1, 0, x 1, −1 ≤ μ < 0. 2.2 For the boundary terms in y and for −1 ≤ μ < 1, Ψ ( x, y ±1, z, μ, θ ⎧ ⎨ ⎩ f2 ( x, z, μ, θ ) , y −1, 0 < cos θ ≤ 1, 0, y 1, −1 ≤ cos θ < 0. 2.3 Finally, for the boundary terms in z, for −1 ≤ μ < 1, Ψ ( x, y, z ±1, μ, θ ⎧ ⎨ ⎩ f3 ( x, y, μ, θ ) , z −1, 0 ≤ θ < π, 0, z 1, π < θ ≤ 2π. 2.4 Theorem 2.1. Consider the integrodifferential equation 2.1 under the boundary conditions 2.2 , 2.3 and 2.4 , then the function Ψ x, y, z, μ, θ satisfy the following first-order linear differential equation system for the spatial component Ψi,j x, μ, θ μ ∂Ψi,j ∂x ( x, μ, θ ) σtΨi,j ( x, μ, θ ) G i,j ( x;μ, θ )∫∫1 −1 σs ( μ′, θ′ −→ μ, θΨi,j ( x, μ′, θ′ ) dθ′dμ′, 2.5 with the boundary conditions Ψi,j −1, μ, η f 1 ( μ, θ ) , 2.6 4 Abstract and Applied Analysis where f i,j 1 ( μ, θ ) 4 π2 ∫∫1 −1 Ti ( y ) Rj z √( 1 − y2 1 − z2 f1 ( y, z, μ, θ ) dzdy,


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 f ∈ L 2 0, 1 can be expanded as a series of Walsh functions where f j is the average value of the function f x in the jth interval of width 1/m in the interval 0, 1 , and W ij is the value of the ith Walsh function in the jth subinterval.The order m Walsh matrix, W m , has elements W ij .
Let f x have a Walsh series with coefficients c i and its integral from 0 to x have a Walsh series with coefficients b i : 2 n terms and use the obvious vector notation, then integration is performed by matrix multiplication b P T m c, where and I m is the unit matrix, O m is the zero matrix of order m , see 6 .

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 method in an infinite cylindrical domain, see 9 , similarly by using Chebyshev spectral-S N method, see 10 , and the discrete ordinates in the infinite cylindrical domain, see 11 .
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 7 .This method is based on expansion of the angular flux in a truncated series of Walsh function in the angular variable.By replacing this development in the transport equation, this will result a first-order linear differential system.First of all we consider the three-dimensional linear, steady state, transport equation which is given by where we assume that the spatial variable x : x, y, z varies in the cubic domain Ω : { x, y, z : −1 ≤ x, y, z ≤ 1}, and Ψ x, μ, θ : Ψ x, y, z, μ, θ is the angular flux in the direction defined by μ ∈ −1, 1 and θ ∈ 0, 2π .σ t and σ s denote the total and the differential cross section, respectively, σ s μ , φ → μ, φ describes the scattering from an assumed pre-collision angular coordinates μ , θ to a postcollision coordinates μ, θ and S is the source term.See 12 for further details.
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 z variable.In this way the actual spatial domain may be assumed to be a cylinder with the cross-section Ω and the axial symmetry in z.Then D will correspond to the projection of the points on the unit sphere the "speed" onto the unit disc which coincides with D .See 13 for the details.
Given the functions f 1 y, z, μ, φ , f 2 x, z, μ, φ , and f 3 x, y, μ, φ describing the incident flux, we seek for a solution of 2.1 subject to the following boundary conditions.
For the boundary terms in x, for 0 ≤ θ ≤ 2π, let

2.2
For the boundary terms in y and for −1 ≤ μ < 1,

2.4
Theorem 2.1.Consider the integrodifferential equation 2.1 under the boundary conditions 2.2 , 2.3 and 2.4 , then the function Ψ x, y, z, μ, θ satisfy the following first-order linear differential equation system for the spatial component with the boundary conditions where

2.8
Proof.Expanding the angular flux Ψ x, y, z, μ, φ in a truncated series of Chebyshev polynomials T i y and R j z leads to We insert Ψ x, y, z, μ, θ given by 2.9 into the boundary condition in 2.3 , for y ±1.Multiplying the resulting expressions by R j z / √ 1 − z 2 and integrating over z, we get the components Ψ 0,j x, μ, θ for j 0, . . ., J,

2.10
Similarly, we substitute Ψ x, y, z, μ, θ from 2.9 into the boundary conditions for z ±1, multiply the resulting expression by T i y / 1 − y 2 , i 0, . . ., I and integrating over y, to define the components where dy.

2.17
Now, starting from the solution of the problem given by 2.13 -2.17 for Ψ I,J x, μ, θ , we then solve the problems for the other components, in the decreasing order in i and j.Recall that Hence, solving I × J one-dimensional problems, the angular flux Ψ x, μ, θ is now completely determined through 2.9 .
Remark 2.2.If we deal with different type of boundary conditions, then we consider the first components Ψ i,0 x, μ, θ and Ψ 0,j x, μ, θ for i 1, . . ., I and j 1, . . ., J will satisfy onedimensional transport problems subject to the same of boundary conditions of the original problem in the variable x.

Analysis
Now, we solve the first-order linear differential equation system with isotropic scattering, that is, σ s μ , φ → μ, φ ≡ σ s constant.Assuming isotropic scattering, the equation 2.13 is written as , and θ ∈ 0, 2π .Then, we have the following theorem that is subject to the boundary conditions 2.14 .
Theorem 3.1.Consider the integrodifferential equation 3.1 under the boundary conditions 2.14 , then the function Ψ i,j x, μ, θ satisfy the following linear system of algebraic equations: D n,m α n,i,j 0, θ .

3.2
Proof.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 −1, 1 .To end this, the Walsh function W n μ are extended in an even and odd fashion as follows, see 14 : for n 0, 1, . . ., N. The important feature of this procedure relies on the fact that a function f μ is defined in the interval −1, 1 it can be expanded in terms of these extended functions in the manner: where the coefficients a n and b n are determined as

3.5
So, in order to use the Walsh function for the solution of the problem 3.1 , the angular flux is approximated by the truncated expansion: α n,i,j x, θ W e n μ β n,i,j x, θ W o n μ .

3.6
Inserting this expansion into the linear transport 3.1 , it turns out

3.9
The integrals appearing in 3.8 and 3.9 are known and are given 14 as pα n,i,j p, θ σ t α n,i,j p, θ D n,m α n,i,j 0, θ . 3.12

Operational Tau Method and Converting Transport Equation
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: We recall the following properties 17 .

4.6
we have where

Matrix Representation for the Integral Form
Let us assume that 4.9 then we can write

Conclusion
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.
n m mod 2 denotes the mod 2 sum of the binary digits n and m 15 N n 0 D n,m pβ n,i,j p, θ − σ s N n 0 , θ Ψ i,j x, μ , θ dθ dμ 4.3 for x ∈ Ω : { x, y : 0 ≤ x ≤ 1, −1 ≤ y ≤ 1}, μ ∈ −1, 1 , and θ ∈ 0, 2π subject to the following boundary conditions 2.14 .In order to convert 5.1 to a system of linear algebraic equations we define the linear differential operator D of order ζ with polynomial coefficients defined by D : In this part, we evaluate an error estimator for the approximate solution of 2.13 , we suppose that equations 2.13 and 5.1 have the same boundary conditions.Let us call m x, μ, θ Ψ i,j x, μ m , θ − Ψ i,j,m x, μ, θ 5.2 this error function of the approximate solution Ψ i,j,m to Ψ i,j where Ψ i,j is the exact solution of 2.13 .Hence, Ψ i,j x, μ m , θ satisfies the following problem: 1,j x, θ , . . ., Ψ i,0 x, θ , Ψ i,1 x, θ , . . .M n 1