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,