We study a class of nonlinear evolution systems of time fractional partial differential equations using Lie symmetry analysis. We obtain not only infinitesimal symmetries but also a complete group classification and a classification of group invariant solutions of this class of systems. We find that the class of systems of differential equations studied is naturally divided into two cases on the basis of the type of a function that they contain. In each case, the dimension of the Lie algebra generated by the infinitesimal symmetries is greater than 2, and for this reason we present the structures and one-dimensional optimal systems of these Lie algebras. The reduced systems corresponding to the optimal systems are also obtained. Explicit group invariant solutions are found for particular cases.

It is well-known that one-dimensional time fractional diffusion-wave equations with variable coefficients can be reduced to ordinary fractional differential equations and systems of linear fractional differential equations via scaling transformations. We then derive exact solutions to classes of linear fractional differential equations and systems thereof expressed in terms of Mittag-Leffler functions, generalized Wright functions and Fox H-functions. These solutions are invariant solutions of diffusionwave equations obtained through certain transformations, which are briefly discussed. We show that the solutions given in this work contain previously known results as particular cases.

We explicitly give new group invariant solutions to a class of Riemann-Liouville time fractional evolution systems with variable coefficients. These solutions are derived from every element in an optimal system of Lie algebras generated by infinitesimal symmetries of evolution systems in the class. We express the solutions in terms of Mittag-Leffler functions, generalized Wright functions, and Fox H-functions and show that these solutions solve diffusion-wave equations with variable coefficients. These solutions contain previously known solutions as particular cases. Some plots of solutions subject to the order of the fractional derivative are illustrated.