Spectral parameter power series for Sturm–Liouville problems

摘要

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 the 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.