Non-point symmetry reduction method of partial differential equations

TL;DR

This paper introduces a novel non-point symmetry reduction method for nonlinear partial differential equations, enabling the derivation of solutions that are inaccessible via classical Lie symmetry techniques.

Contribution

It develops new ansätze based on non-point classical and conditional symmetries, expanding the toolkit for reducing and solving complex PDEs.

Findings

Derived solutions to nonlinear heat equations beyond classical Lie methods

Constructed solutions for hyperbolic PDEs depending on arbitrary functions

Demonstrated applicability to non-evolutionary PDEs

Abstract

We study the symmetry reduction of nonlinear partial differential equations with two independent variables. We propose new ans\"atze reducing nonlinear evolution equations to system of ordinary differential equations. The ans\"atze are constructed by using operators of non-point classical and conditional symmetry. Then we find solution to nonlinear heat equation which can not be obtained in the framework of the classical Lie approach. By using operators of Lie--B\"acklund symmetries we construct the solutions of nonlinear hyperbolic equations depending on arbitrary smooth function of one variable too. We show that the method can be applied to nonevolutionary partial differential equations.