Lie Symmetry Analysis of Some Conformable Fractional Partial Differential 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.