A ghost perturbation scheme to solve ordinary differential equations

TL;DR

This paper introduces a novel algebraic perturbation method for solving second-order ODEs with boundary conditions, achieving exponential convergence by optimizing parameters at each expansion order.

Contribution

The authors develop a ghost perturbation scheme that constructs a hierarchy of linear ODEs, enabling rapid convergence and solution identification for boundary value problems.

Findings

Exponential convergence of the perturbative solution in the number of terms.

Ability to determine the number of solutions by analyzing minima of a distance function.

Method successfully applied to various example problems.

Abstract

We propose an algebraic method that finds a sequence of functions that exponentially approach the solution of any second-order ordinary differential equation (ODE) with any boundary conditions. We define an extended ODE (eODE) composed of a linear generic differential operator that depends on free parameters, $p$, plus an $\epsilon$ perturbation formed by the original ODE minus the same linear term. After the eODE's formal $\epsilon$ expansion of the solution, we can solve order by order a hierarchy of linear ODEs, and we get a sequence of functions $y_n(x;\epsilon,p)$ where $n$ indicates the number of terms that we keep in the $\epsilon$-expansion. We fix the parameters to the optimal values $p^*(n)$ by minimizing a distance function of $y_n$ to the ODE's solution, $y$, over a given $x$-interval. We see that the eODE's perturbative solution converges exponentially fast in $n$ to the ODE solution when $\epsilon=1$: $\vert y_n(x;\epsilon=1,p^*(n))-y(x)\vert<C\delta^{n+1}$ with $\delta<1$. The method permits knowing the number of solutions for Boundary Value Problems just by looking at the number of minima of the distance function at each order in $n$, $p^{*,\alpha}(n)$, where each $\alpha$ defines a sequence of functions $y_n$ that converges to one of the ODE's solutions. We present the method by its application to several cases where we discuss its properties, benefits and shortcomings, and some practical algorithmic improvements.