  This work investigates how we can extend the invariant subspace method to
two-dimensional time-fractional non-linear PDEs. More precisely, the systematic
study has been provided for constructing the various dimensions of the
invariant subspaces for the two-dimensional time-fractional generalized
convection-reaction-diffusion-wave equation along with the initial conditions
for the first time. Additionally, the special types of the above-mentioned
equation are discussed through this method separately such as
reaction-diffusion-wave equation, convection-diffusion-wave equation and
diffusion-wave equation. Moreover, we explain how to derive variety of exact
solutions for the underlying equation along with initial conditions using the
obtained invariant subspaces. Finally, we extend this method to two-dimensional
time-fractional non-linear PDEs with time delay. Also, the effectiveness and
applicability of the method have been illustrated through the two-dimensional
time-fractional cubic non-linear convection-reaction-diffusion-wave equation
with time delay. In addition, we observe that the obtained exact solutions can
be viewed as the combinations of Mittag-Leffler function and polynomial,
exponential and trigonometric type functions.
