Group classification and symmetry reduction of three-dimensional nonlinear anomalous diffusion equation

Abstract

The work is devoted to studying symmetry properties of a nonlinear anomalous diffusion equation involving a Riemann-Liouville fractional derivative with respect to the time. We resolve a problem on group classification with respect to the diffusion coefficient treated as a function of the unknown. We show that for an arbitrary function, the equation admits a seven-dimensional Lie algebra of infinitesimal operators corresponding to the groups of translations, rotations and dilations. In contrast to the symmetries of the equations with integer order derivatives, the translation in time is not admitted. Moreover, the coefficients of the group of dilations are different. If the coefficient is power, the admissible algebra is extended to a eight-dimensional one by an additional operator generating the group of dilatations. For two specific values of the exponent in the power, the algebra can be further extended to a nine-dimensional one or to a eleven-dimensional one and at that, additional admissible operators correspond to various projective transformations. For the obtained Lie algebras of symmetries with dimensions from seven to nine, we construct optimal systems of subalgebras and provide ansatzes for corresponding invariant solutions of various ranks. We also provide general forms of invariant solutions convenient for the symmetry reduction as the fractional Riemann-Liouville derivative is present. We make a symmetry reduction on subalgebras allowing one to find invariant solutions of rank one. We provide corresponding reduced ordinary fractional differential equations.