An Efficient Method for Deriving Normalization Constants for Eigenfunctions of Sturm-Liouville Problems and Its Application to the Graetz Problem for Diffusive and Convection Heat/Mass Transfer

摘要

Sturm-Liouville (SL) problems arise in the solution methods to an important class of partial differential equations (PDEs) relevant in many industrial and scientific applications [1,2,3]. The importance of SL theory is rooted in the fact that the solutions of certain PDEs can be expanded as an infinite series of normalized eigenfunctions of the SL operator multiplied by spectral coefficients [4]. The spectral coefficients can be computed by matching the infinite series at the initial/boundary conditions. Although methods for determining the eigenfunctions (and corresponding eigenvalues) of the SL operator are well-known, few procedures are available for normalizing this set of eigenfunctions (i.e., ensuring the inner product of an eigenfunction with itself is unity). Most normalization procedures require a large amount of time and effort and may not yield an analytic expression in a simple and readily computable form.