(2+1)-dimensional Burgers equation with a Caputo fractional derivative: Lie symmetry analysis, optimal system, exact solutions and conservation laws

Abstract

In this paper, the Lie group method is applied to analyze the (2+1)-dimensional Burgers equation with Caputo fractional derivative. The infinitesimal generators of the equation are investigated. Then, using the Lie point symmetry of the equation, a one-dimensional optimal system is constructed. Subsequently, within the framework of the optimal system, the equation is symmetrically simplified and the exact solutions of the original equation are determined using the invariant subspace method and the homogeneous balance method. In the process of simplifying the equation to a (1+1)-dimensional fractional differential equation, it is deliberately avoided to use complex Erdélyi-Kober fractional derivative operator. Instead, a simple transformation is utilized to obtain a low-dimensional fractional differential equation. Finally, based on the concept of nonlinear self-adjointness, the conservation laws of the fractional differential equation are obtained.