Spectral parameter power series method for discontinuous coefficients

TL;DR

This paper introduces a spectral parameter power series method to solve differential equations with discontinuous coefficients, providing a convergent series representation and applications to eigenvalue problems.

Contribution

It develops a novel power series approach for differential equations with discontinuous coefficients, enabling explicit solution construction and eigenvalue problem solutions.

Findings

Series converge uniformly on [a,b]

Recursive procedure for coefficients

Numerical tests validate method

Abstract

Let (a,b) be a finite interval and 1/p, q, r be functions from L1(a,b). We show that a general solution (in the weak sense) of the equation (pu')'+qu = zru on (a,b) can be constructed in terms of power series of the spectral parameter z. The series converge uniformly on [a,b] and the corresponding coefficients are constructed by means of a simple recursive procedure. We use this representation to solve different types of eigenvalue problems. Several numerical tests are discussed.