Group classification and symmetry reductions of a nonlinear Fokker-Planck equation based on the Sharma-Taneja-Mittal entropy

TL;DR

This paper conducts a Lie symmetry analysis of a nonlinear Fokker-Planck equation derived from Sharma-Taneja-Mittal entropy, classifying its symmetries and reducing it to simpler ODEs for invariant solutions.

Contribution

It provides a complete group classification of the equation based on entropy parameters and identifies all cases with additional symmetries, enabling systematic solution reduction.

Findings

The equation admits a two-dimensional symmetry algebra for general parameters.

Additional symmetries are identified for specific parameter values.

Symmetry reductions lead to second-order ODEs describing invariant solutions.

Abstract

A nonlinear Fokker-Planck equation that arises in the framework of statistical mechanics based on the Sharma-Taneja-Mittal entropy is studied via Lie symmetry analysis. The equation describes kinetic processes in anomalous mediums where both super-diffusive and subdiffusive mechanisms arise contemporarily and competitively. We perform complete group classification of the equation based on two parameters that characterise the underlying Sharma-Taneja-Mittal entropy. For arbitrary values of the parameters the equation is found to admit a two-dimensional symmetry Lie algebra. We identify and catalogue all the cases in which the equation admits additional symmetries. We also perform symmetry reductions of the equation and obtain second-order ODEs that describe all essentially different invariant solutions of the Fokker-Planck equation.