Symmetry classification of time-fractional diffusion equation

• Fractional differential equations arise in fractals, acoustics, control theory, signal processing and many other important areas.
• The fractional differential equations can be expressed in terms of different differential operators e.g. Riemann-Liouville, Caputo, Weyl and many others.
• Numerous methods have been developed and used to find approximate and analytical solutions of these equations e.g. Laplace transform method, Green’s function method [1], variational iteration method, Adomian decomposition method, Finite Sine transform method.
• A new way of computing symmetries for fractional differential equations is given which is easier and more efficient whereas in the direct symmetry methods, one has to deal with the determining equations which are fractional itself are of very difficult to handle.
• These symmetries can be utilized to compute the exact solutions of fractional differential equations.
• In order to understand the phenomena of anomalous diffusion we discussed important symmetries.

In this article, a new approach is proposed to construct the symmetry groups for a class of fractional differential equations which are expressed in the modified Riemann-Liouville fractional derivative. We perform a complete group classification of a nonlinear fractional diffusion equation which arises in fractals, acoustics, control theory, signal processing and many other applications. Introducing the suitable transformations, the fractional derivatives are converted to integer order derivatives and in consequence the nonlinear fractional diffusion equation transforms to a partial differential equation (PDE). Then the Lie symmetries are computed for resulting PDE and using inverse transformations, we derive the symmetries for fractional diffusion equation. All cases are discussed in detail and results for symmetry properties are compared for different values of α. This study provides a new way of computing symmetries for a class of fractional differential equations.