A Spectral Method for Parabolic Differential Equations

TL;DR

This paper introduces a spectral method for solving parabolic PDEs with zero boundary conditions, demonstrating spectral convergence for smooth solutions through theoretical analysis and numerical experiments in 2D and 3D.

Contribution

It develops a spectral approach tailored for parabolic equations with rigorous error analysis and validation via numerical examples in multiple dimensions.

Findings

Spectral convergence achieved for smooth solutions

Method validated in 2D and 3D numerical experiments

Error analysis confirms theoretical predictions

Abstract

We present a spectral method for parabolic partial differential equations with zero Dirichlet boundary conditions. The region {\Omega} for the problem is assumed to be simply-connected and bounded, and its boundary is assumed to be a smooth surface. An error analysis is given, showing that spectral convergence is obtained for sufficiently smooth solution functions. Numerical examples are given in both R^2 and R^3.