  This article develops how to generalize the invariant subspace method for
deriving the analytical solutions of the multi-component (N+1)-dimensional
coupled nonlinear time-fractional PDEs (NTFPDEs) in the sense of Caputo
fractional-order derivative for the first time. Specifically, we describe how
to systematically find different invariant product linear spaces with various
dimensions for the considered system. Also, we observe that the obtained
invariant product linear spaces help to reduce the multi-component
(N+1)-dimensional coupled NTFPDEs into a system of fractional-order ODEs, which
can then be solved using the well-known analytical methods. More precisely, we
illustrate the effectiveness and importance of this developed method for
obtaining a long list of invariant product linear spaces for the
multi-component (2+1)-dimensional coupled nonlinear time-fractional
diffusion-convection-wave equations. In addition, we have shown how to find
different kinds of generalized separable analytical solutions for a
multi-component (2 + 1)-dimensional coupled nonlinear time-fractional
diffusion-convection-wave equations along with the initial and boundary
conditions using invariant product linear spaces obtained. Finally, we provide
appropriate graphical representations of some of the derived generalized
separable analytical solutions with various fractional-order values.
