Analytical solutions of time-space fractional, advection-dispersion and Whitham-Broer-Kaup equations

In this article, we study time-space fractional advection-dispersion (FADE) equation and time-space fractional Whitham-Broer-Kaup (FWBK) equation that have significant roles in hydrology. We introduce suitable transformations to convert fractional-order derivatives to integerorder derivatives and as a result these equations transform into partial differential equations (PDEs). Then the Lie symmetries and the corresponding optimal systems of the resulting PDEs are derived. The symmetry reductions and exact independent solutions based on optimal system are investigated which constitute the exact solutions of original fractional differential equations.