Time-fractional inhomogeneous nonlinear diffusion equation: Symmetries, conservation laws, invariant subspaces, and exact solutions

Abstract

In this paper, a class of time-fractional inhomogeneous nonlinear diffusion equation (tFINDE) with Riemann–Liouville fractional derivative is studied. All point symmetries admitted by this equation are derived. The optimal system of one-dimensional subalgebras is classified to perform the symmetry reductions. It is shown that the tFINDE can be reduced to fractional ordinary differential equations (FODEs), including Erdélyi–Kober fractional derivatives. As the results, some explicit group-invariant solutions are obtained. Through nonlinear self-adjointness, all conservation laws admitted by tFINDE arising from these point symmetry groups are listed. The method of invariant subspace is also applied to reduce the tFINDE to a two-dimensional dynamical system (DS). The admitted point symmetries of DS are used to derive the exact solutions of DS, which determine the exact solutions of the original tFINDE.

Lie Symmetry Analysis and Conservation Laws of Fractional Benjamin–Ono Equation

“…Zhang [24] provided the symmetric determining equation and nonlinear method for solving fractional nonlinear partial differential equations. Feng [25][26][27] investigated the symmetry and conservation laws of various classes of time-fractional nonhomogeneous nonlinear diffusion equations. Chen [28] expanded the coefficients of fractional-order equation from constant coefficients to variable coefficients while conducting Lie symmetry analysis.…”

Applications of the invariant subspace method on searching explicit solutions to certain special-type non-linear evolution equations

“…Very recently, Refs. [15][16][17][18][19][20][21][22][23] generalized this method to resolve fractional non-linear partial differential equations (fNPDEs). It is verified that by applying ISM, a fNPDE can be reduced to a system of fractional nonlinear ordinary differential equations (fNODEs), which can be solved by known analytical approaches.…”

Lie Symmetry Group, Invariant Subspace, and Conservation Law for the Time-Fractional Derivative Nonlinear Schrödinger Equation

“…Recently, Gazizov et al established the modification of the invariant subspace method for FDEs [19], where they also found that the jointness of invariant subspace and the Lie symmetry group is helpful for finding exact solutions of FDEs. Based on these established schemas, numerous works have focused on FDEs, including scalar-time FDEs [20,21], multidimensional-time FDEs [22,23], coupled-time FDEs [24][25][26], and space-time FDEs [27][28][29].…”