Spectral parameter power series for arbitrary order linear differential equations

TL;DR

This paper generalizes the spectral parameter power series (SPPS) method to arbitrary order linear differential equations, providing a recursive integral approach for solutions and numerical applications.

Contribution

It introduces a new recursive integral representation for solutions of high-order linear differential equations, extending the SPPS method beyond second order.

Findings

Provides a recursive integral method for solutions

Enables numerical solutions of high-order spectral problems

Generalizes SPPS to arbitrary order equations

Abstract

Let $L$ be the $n$-th order linear differential operator $Ly = \phi_0y^{(n)} + \phi_1y^{(n-1)} + \cdots + \phi_ny$ with variable coefficients. A representation is given for $n$ linearly independent solutions of $Ly=\lambda r y$ as power series in $\lambda$, generalizing the SPPS (spectral parameter power series) solution which has been previously developed for $n=2$. The coefficient functions in these series are obtained by recursively iterating a simple integration process, begining with a solution system for $\lambda=0$. It is shown how to obtain such an initializing system working upwards from equations of lower order. The values of the successive derivatives of the power series solutions at the basepoint of integration are given, which provides a technique for numerical solution of $n$-th order initial value problems and spectral problems.