Lie Symmetry Analysis of Some Conformable Fractional Partial Differential Equations

TL;DR

This paper applies Lie symmetry analysis to nonlinear conformable fractional PDEs, deriving symmetries and exact solutions, including reductions to Painlevé equations, for equations like KdV and Burgers.

Contribution

It introduces a systematic Lie symmetry analysis for conformable fractional PDEs and finds explicit reductions and solutions, extending classical methods to fractional derivatives.

Findings

Lie symmetries for fractional PDEs are obtained.

Exact solutions are expressed via Painlevé equations.

Fractional KdV reduces to Painlevé equations.

Abstract

In this article, Lie symmetry analysis is used to investigate invariance properties of some nonlinear fractional partial differential equations with conformable fractional time and space derivatives. The analysis is applied to Korteweg-de Vries, modified Korteweg-de Vries, Burgers, and modified Burgers equations with conformable fractional time and space derivatives. For each equation, all of the vector fields and the Lie symmetries are obtained. Moreover, exact solutions are given to these equations in terms of solutions of ordinary differential equations. In particular, it is shown that the fractional Korteweg-de Vries can be reduced to the first Painlev\'{e} equation and to fractional second Painlev\'{e} equation. In addition a solution of the fractional modified Korteweg-de Vries is given in terms of solutions of fractional second Painlev\'{e} equation.