On the algorithmic linearizability of nonlinear ordinary differential equations

Abstract

Solving nonlinear ordinary differential equations is one of the fundamental and practically important research challenges in mathematics. However, the problem of their algorithmic linearizability so far remained unsolved. In this contribution, we propose a solution of this problem for a wide class of nonlinear ordinary differential equation of arbitrary order. We develop two algorithms to check if a nonlinear differential equation can be reduced to a linear one by a point transformation of the dependent and independent variables. In this regard, we have restricted ourselves to quasi-linear equations with a rational dependence on the occurring variables and to point transformations. While 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. The implementation of our algorithms is discussed and evaluated using several examples.

Probabilistic analysis of a class of compartmental models formulated by random differential equations

“…Despite their elementary form, ordinary and partial linear differential equations have remarkable efficacy in capturing the essence of real-world processes, encompassing applications within a great variety of scientific fields, such as chemical reactions [1], radioactive chains of decay [2], electrical networks [3], gene regulatory networks [4], or the absorption of medicine by various organs [5], just to mention a few. Linear differential equations also play a key role in the analytical and numerical analysis of models formulated via nonlinear differential equations [6,7].…”