Symmetry Group Classification and Conservation Laws of the Nonlinear Fractional Diffusion Equation with the Riesz Potential

Nikita S Belevtsov,Stanislav Yu Lukashchuk

doi:10.3390/sym12010178

Abstract

Symmetry properties of a nonlinear two-dimensional space-fractional diffusion equation with the Riesz potential of the order α ∈ ( 0 , 1 ) are studied. Lie point symmetry group classification of this equation is performed with respect to diffusivity function. To construct conservation laws for the considered equation, the concept of nonlinear self-adjointness is adopted to a certain class of space-fractional differential equations with the Riesz potential. It is proved that the equation in question is nonlinearly self-adjoint. An extension of Ibragimov’s constructive algorithm for finding conservation laws is proposed, and the corresponding Noether operators for fractional differential equations with the Riesz potential are presented in an explicit form. To illustrate the proposed approach, conservation laws for the considered nonlinear space-fractional diffusion equation are constructed by using its Lie point symmetries.

Highlights

Fractional differential equations (FDEs) with multi-dimensional spatial fractional differential operators have attracted considerable attention during the last decade due to the possibility to describe power-law long-range interactions in complex systems [1,2,3]
We present the results of Lie point symmetry group classification for a nonlinear space-fractional diffusion equation containing the Riesz potential with respect to diffusivity function
In [27,28,29], it is shown that the concept of nonlinear self-adjointness can be enhanced to FDEs with the Riemann–Liouville and Caputo fractional derivatives. We extend this approach to FDEs with the Riesz potential

Summary

Introduction

Fractional differential equations (FDEs) with multi-dimensional spatial fractional differential operators have attracted considerable attention during the last decade due to the possibility to describe power-law long-range interactions in complex systems [1,2,3] Such equations can be efficiently used for modelling a fluid flow in naturally fractured porous media, which is a very important problem for the oil industry. Finding exact solutions to nonlinear space-fractional FDEs is a sufficiently complex problem This problem can be significantly simplified if symmetry properties of the considered equation are known. We present the results of Lie point symmetry group classification for a nonlinear space-fractional diffusion equation containing the Riesz potential with respect to diffusivity function.

Preliminaries

Group Classification of the Nonlinear Space-Fractional Porous Medium Equation

Nonlinear Self-Adjointness

Conservation Laws