Algorithmic Verification of Linearizability for Ordinary Differential Equations

TL;DR

This paper presents two algorithms, implemented in Maple, for algorithmically verifying whether a nonlinear ordinary differential equation can be transformed into a linear form through point transformations, based on symmetry analysis and differential Thomas decomposition.

Contribution

It introduces novel algorithms for linearizability verification of ODEs using Lie symmetries and differential Thomas decomposition, with practical implementation and examples.

Findings

Algorithms successfully verify linearizability of various ODEs.

Implementation in Maple demonstrates practical applicability.

Algorithms generate explicit transformation systems for linearization.

Abstract

For a nonlinear ordinary differential equation solved with respect to the highest order derivative and rational in the other derivatives and in the independent variable, we devise two algorithms to check if the equation can be reduced to a linear one by a point transformation of the dependent and independent variables. The first algorithm is based on a construction of the Lie point symmetry algebra and on the computation of its derived algebra. The second algorithm exploits the differential Thomas decomposition and allows not only to test the linearizability, but also to generate a system of nonlinear partial differential equations that determines the point transformation and the coefficients of the linearized equation. Both algorithms have been implemented in Maple and their application is illustrated using several examples.