Construction of Special Solutions for the Maxwell Equations

Abstract

In this paper, we consider homogeneous polynomial solutions for the classical Maxwell equations in a time-space domain with time variable $$t<t_0$$, where $$t_0$$ is an arbitrary constant. The technique is motivated by the study of the so-called generalized Maxwell operators, which are constructed as conformally invariant differential operators in the framework of Clifford analysis. We firstly show that solutions of the classical Maxwell equations are equivalent to null solutions of the generalized Maxwell operator, then a direct decomposition and a partially orthogonal polynomial basis for the space of homogeneous polynomial null solutions of the generalized Maxwell operator are provided. At the end, a basis of homogeneous polynomial solutions for the source-free Maxwell equations on a bounded time-space domain is given.