ABSTRACT
This chapter seeks to develop methods of using operator eigenbases in which the original partial differential equations reduce to ordinary differential equations that are generally easier to solve. It looks at how series expansion using the eigenfunctions of Sturm–Liouville operators may be applied in order to solve more general physics problems, as the earlier examples were mainly restricted to cases where the differential equations were homogeneous. The chapter explores solutions in terms of series containing an infinite number of eigenmodes to the Laplace operator or other combinations of Sturm–Liouville operators. It deals with problems defined in a bounded spatial region, leading to the Sturm–Liouville operators of interest having a countable set of eigenfunctions with a corresponding discrete set of eigenvalues. An important observation is that the resulting function bases span the same vector space and are all eigenfunctions of the same operator.
