Extensions of a New Algorithm for the Numerical Solution of Linear Differential Systems on an Infinite Interval

TL;DR

This work extends a Levinson-based algorithm for solving linear differential systems on an infinite interval by incorporating a large, growing scalar factor $\lambda(x)$ and allowing $\rho(x)$ to include the case $\alpha=-1$. It develops a symbolic–numeric framework that generates transformation terms and computes solutions with explicit error bounds, applying a sequence of refinements that reduce the perturbation $R(x)$. The generalized hypergeometric equation serves as a rigorous testbed, with a concrete $\rho(x)=x^{-1}$ example yielding asymptotic forms $y_{\infty1}(x)\sim x^{-3}e^{x^3/3}$ and $y_{\infty2}(x)\sim x^{-1}$, and analytic PW86 relations providing exact cross-checks between asymptotic and small-$x$ behavior. The results demonstrate the method’s accuracy, including back-propagation from large-$x$ asymptotics to $x=0$ with sub-10^{-7} precision and interval-enclosed validation of the computed values.

Abstract

This paper is part of a series of papers in which the asymptotic theory and appropriate symbolic computer code are developed to compute the asymptotic expansion of the solution of an n-th order ordinary differential equation. The paper examines the situation when the matrix that appears in the Levinson expansion has a double eigenvalue. Application is made to a fourth-order ODE with known special function solution.