ABSTRACT
This chapter provides solution to parabolic problems in the spectral methods using multivariate polynomials on the unit ball. In addition, assume that there is a unique solution to the parabolic problem. Some of these assumptions are stronger than needed, but they simplify the presentation. The chapter gives additional assumptions and discusses the problem of handling a nonhomogeneous boundary condition. A spectral method is applied, leading to a system of ordinary differential equations, for which there is much excellent software. The convergence analysis of the method depends on the landmark paper of Douglas and Dupont. The chapter also discusses reformulation and numerical approximation with numerical examples.
