Spectral parameter power series for arbitrary order linear differential equations
This work provides a theoretical extension of the SPPS method to higher-order differential equations, which is incremental for mathematicians working on spectral problems.
The authors generalize the spectral parameter power series (SPPS) method from second-order to arbitrary n-th order linear differential equations, providing a representation of n linearly independent solutions as power series in λ. The method uses recursive integration and yields explicit derivative values at the basepoint, enabling numerical solutions for initial value and spectral problems.
Let $L$ be the $n$-th order linear differential operator $Ly = ϕ_0y^{(n)} + ϕ_1y^{(n-1)} + \cdots + ϕ_ny$ with variable coefficients. A representation is given for $n$ linearly independent solutions of $Ly=λr y$ as power series in $λ$, 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 $λ=0$. 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.