Abstract: Analytical solutions for the one-dimensional discrete ordinates equation (S_N equations) in planar, cylindrical and spherical geometry are computed by the integral transform technique, namely the Laplace and Hankel transform techniques. To this end, and for sake of completeness, the solution of the S_N equations in a slab by the Laplace transform technique (LTS_N approach) is presented [1, 2, 3, 4]. The main idea relies on the application of the Laplace transform to the set of ordinary differential equations (S_N equations). The analytical solution of the resulting linear algebraic equation is obtained by employing the Heaviside expansion technique to calculate the resolvent of the algebraic transformed equation. To this point, it is important to recall that the S_N equation is an approximation of the one-dimensional linear Boltzmann transport equation, and the convergence of the LTS_N solution is also discussed. The solution of the S_N equations in spherical geometry is displayed by transforming it to an equivalent problem in planar geometry, in which the LTS_N solution can be applied in a straightforward manner. On the other hand, for S_N problem in cylindrical geometry and isotropic scattering, an analytical solution, dubbed as HTS_N solution, is reported. In latter geometry the one-dimensional S_N equation is solved like in the LTS_N approach, with a main difference that now the Hankel transform technique is applied. An outline of the convergence of the HTS_N approach is sketched.

[1] G.A. Gonçalves, C.F. Segatto, M.T. Vilhena (2000): The LTS_N particular solution in a slab for an arbitrary source and large order of quadrature, J. Quant. Spec. and Radiative Transfer, 66(3), 271-276.

[2] R.P. Pazos, M.T. Vilhena (1999): Convergence of the LTS_N method: approach of C₀ semi-groups, Progress in Nuclear Energy, 34(1), 77-86.

[3] M.T. Vilhena, L.B. Barichello, J.R. Zabadal, C.F. Segatto, A.V. Cardona (1998): General solution of one-dimensional approximations to the transport equation, Progress in Nuclear Energy, 33(1/2), 99-115.

[4] C.F. Segatto, M.T. Vilhena (1999): "The state-of-the-art of the LTS_N method - mathematics and computation, Reactor Physics and Environmental Analysis in Nuclear Applications - International Conference, vol. 2, 1618-1631, Madrid-Spain.