Nonclassical Symmetries of a Nonlinear Diffusion–Convection/Wave Equation and Equivalents Systems

Daniel J Arrigo ,Seth J Bloomberg ,Brandon P Ashley ,Thomas W Deatherage

doi:10.3390/sym8120140

Daniel J Arrigo , Seth J Bloomberg + Show 2 more

Open Access

https://doi.org/10.3390/sym8120140

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Abstract

It is generally known that classical point and potential Lie symmetries of differential equations (the latter calculated as point symmetries of an equivalent system) can be different. We question whether this is true when the symmetries are extended to nonclassical symmetries. In this paper, we consider two classes of nonlinear partial differential equations; the first one is a diffusion–convection equation, the second one a wave, where we will show that the majority of the nonclassical point symmetries are included in the nonclassical potential symmetries. We highlight a special case were the opposite is true.

Highlights

Symmetry analysis plays a fundamental role in the construction of exact solutions to nonlinear partial differential equations
We have considered the symmetries of a nonlinear diffusion–convection and wave equation and equivalent systems
It is well known that classical Lie symmetries of differential equations and equivalent systems can be different

Summary

Introduction

Symmetry analysis plays a fundamental role in the construction of exact solutions to nonlinear partial differential equations. Where F satisfies Ft = Fvv. Clearly, the powers m = −4/3 and m = −2 show themselves as special, and—as this example demonstrates—the symmetries of equations and equivalent systems can be different. Of particular interest here is the paper by Bluman and Yan [24] They consider two algorithms that extend the nonclassical method to potential systems and potential equations. They consider the nonlinear diffusion Equation (1), an equivalent potential system (Algorithm 1). We will not address this question in general here, we will use Algorithm 1 to consider a large class of nonlinear diffusion–convection and wave equations to show that—in the majority of cases—the nonclassical potential system symmetries contain the nonclassical symmetries of the original equation. We highlight a special case where the opposite is true

Nonclassical Symmetries

Nonlinear Diffusion–Convection Equation

Nonlinear Wave Equation

Nonlinear Diffusion Equation

Conclusions