Closed form Solutions to Some Nonlinear equations by a Generalized Cole-Hopf Transformation

TL;DR

This paper introduces a generalized Cole-Hopf transformation that linearizes and solves various nonlinear equations, including Van der Pol, Lienard, Painleve III, and higher-order Burger's equations, providing explicit solutions.

Contribution

The paper presents a novel generalized Cole-Hopf transformation applicable to multiple complex nonlinear differential equations, extending existing linearization techniques.

Findings

Linearized Van der Pol and Lienard equations

Linearized Painleve III for specific parameters

Solved higher-order Burger's and convective equations

Abstract

In the first part of this paper we linearize and solve the Van der Pol and Lienard equations with some additional nonlinear terms by the application of a generalized form of Cole-Hopf transformation. We then show that the same transformation can be used to linearize Painleve III equation for certain combinations of its parameters. Finally we linearize new forms of Burger's and related convective equations with higher order nonlinearities.