Spectral parameter power series for arbitrary order linear differential equations

Abstract

Let L be the n‐th order linear differential operator Ly=ϕ0y(n)+ϕ1y(n−1)+⋯+ϕny with variable coefficients. A representation is given for n linearly independent solutions of Ly=λry as power series in λ, generalizing the SPPS (spectral parameter power series) solution that has been previously developed for n=2. The coefficient functions in these series are obtained by recursively iterating a simple integration process, beginning with a solution system for λ=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.

Spectral parameter power series representation for solutions of linear system of two first order differential equations

“…Later the SPPS representation was extended to solutions of singular second order differential equations [10], of equations with polynomial dependence on the spectral parameter [21] and recently of linear differential equations of arbitrary order [18]. However the SPPS representation for the solutions of linear systems of differential equations was constructed only for Zakharov-Shabat system and a particular case of one dimensional Dirac system [21], in both cases by transforming the system into a certain Sturm-Liouville equation.…”