  In this article, we systematically explain how to apply the analytical
technique called the invariant subspace method to find various types of
analytical solutions for a coupled nonlinear time-fractional system of partial
differential equations with time delays. Also, the present work explicitly
studies a systematic way to obtain various kinds of finite-dimensional
invariant vector spaces for the coupled nonlinear time-fractional
diffusion-reaction (DR) system with time delays under the two distinct
fractional derivatives, namely (a) the Riemann-Liouville fractional partial
time derivative and (b) the Caputo fractional partial time derivative.
Additionally, we provide details of deriving analytical solutions in the
generalized separable form for the initial and boundary value problems (IBVPs)
of the coupled nonlinear time-fractional DR system with multiple time delays
through the obtained invariant vector spaces under the considered two
time-fractional derivatives.
