  In this article, we develop a systematic approach of the invariant subspace
method combined with variable transformation to find the generalized separable
exact solutions of the nonlinear two-component system of time-fractional PDEs
(TFPDEs) in (2+1)-dimensions for the first time. Also, we explicitly explain
how to construct various kinds of finite-dimensional invariant linear product
spaces for the given system using the invariant subspace method combined with
variable transformation. Additionally, we present how to use the obtained
invariant linear product spaces to derive the generalized separable exact
solutions of the discussed system. We also note that the discussed method will
help to reduce the nonlinear two-component system of TFPDEs in (2+1)-dimensions
into the nonlinear two-component system of TFPDEs in (1+1)-dimensions, which
again reduces to a system of time-fractional ODEs through the obtained
invariant linear product spaces. More specifically, the significance and
efficacy of the systematic investigation of the discussed method have been
investigated through the initial and boundary value problems of the generalized
nonlinear two-component system of time-fractional reaction-diffusion equations
(TFRDEs) in (2+1)-dimensions for finding the generalized separable exact
solutions, which can be expressed in terms of the exponential, trigonometric,
polynomial, Euler-Gamma and Mittag-Leffler functions. Also, 2D and 3D graphical
representations of some of the obtained solutions are presented for different
values of fractional orders.
