Spectral parameter power series for Sturm-Liouville problems

TL;DR

This paper introduces the spectral parameter power series (SPPS) method for solving Sturm-Liouville problems, providing an efficient numerical approach for initial, boundary, and spectral problems, including singular cases.

Contribution

It presents a novel SPPS representation for Sturm-Liouville solutions, enabling effective numerical solutions and root-finding for spectral problems.

Findings

SPPS method efficiently solves Sturm-Liouville problems.

Applicable to singular and spectral parameter-dependent boundary conditions.

Numerical examples demonstrate the method's effectiveness.

Abstract

We consider a recently discovered representation for the general solution of the Sturm-Liouville equation as a spectral parameter power series (SPPS). The coefficients of the power series are given in terms of a particular solution of the Sturm-Liouville equation with the zero spectral parameter. We show that, among other possible applications, this provides a new and efficient numerical method for solving initial value and boundary value problems. Moreover, due to its convenient form the representation lends itself to numerical solution of spectral Sturm-Liouville problems, effectively by calculation of the roots of a polynomial. We discuss examples of the numerical implementation of the SPPS method and show it to be equally applicable to a wide class of singular Sturm-Liouville problems as well as to problems with spectral parameter dependent boundary conditions.