Abstract
The symmetry group method is applied to study a class of time-fractional generalized porous media equations with Riemann–Liouville fractional derivatives. All point symmetry groups and the corresponding optimal subgroups are determined. Then, the similarity reduction is performed to the given equation and some explicit solutions are derived. The asymptotic behaviours for the solutions are also discussed. Through the concept of nonlinear self-adjointness, the conservation laws arising from the admitted point symmetries are listed.
Highlights
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Kac concluded that “success is characterized by the fidelity with which such models fit the observed phenomena, and by the sharpness of the questions they pose about the underlying physics.” [4]
Questions about how the Lie symmetry group works for fractional differential equation (FDE) and whether the effectiveness of the Lie symmetry group for FDEs is the same as it for usual differential equations (DEs) arise
Summary
The theory of differentiations and integrations of arbitrary order (real or complex) is named as fractional calculus, and was born on 30 September 1695 in a letter from Leibniz to L‘Hopital discussing the derivative of order 1/2. The study of Lie group theory for FDEs was initiated by Buckwar and Luchko in [14], where they mainly concentrated on the scaling invariance of a linear fractional diffusion equation and obtained several explicit solutions in terms of generalized Wright functions. As a matter of fact, up to now, the nonlinear self-adjointness method has been widely used to construct the conservation laws for FDEs. The present paper is devoted to studying the symmetry groups and conservation laws to a time-fractional generalized porous medium equation for u = u(t, x), given by. The generalized porous medium equation with an integer order time derivative provides models of many interesting physical phenomena, such as the flow of liquids in porous media and transport of thermal energy in plasma, which has been investigated from several points of view [33,34,35,36,37].
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