Generalized separable solutions for (2+1) and (3+1)-dimensional m-component coupled nonlinear systems of PDEs under three different time-fractional derivatives

In this article, we explain the invariant subspace approach for $(2+1)$ and $(3+1)$-dimensional $m$-component nonlinear coupled systems of PDEs with and without time delays under three different time-fractional derivatives. Also, we explain how this method can be used to derive different types of generalized separable solutions for the nonlinear systems mentioned above through the obtained invariant subspaces. More precisely, we show the applicability of this method using the general class of coupled 2-component nonlinear $(2+1)$-dimensional reaction-diffusion system under three time-fractional derivatives. Moreover, we provide a detailed description for obtaining the various types of different dimensional invariant linear 2-component subspaces and their solutions for the underlying coupled 2-component nonlinear $(2+1)$-dimensional reaction-diffusion system with appropriate initial-boundary conditions under the three time-fractional derivatives known as (a) Riemann–Liouville (RL) fractional derivative, (b) Caputo fractional derivative, and (c) Hilfer fractional derivative, as examples. Furthermore, we observe that the derived separable solutions under three fractional-order derivatives consist of trigonometric, polynomial, exponential, and Mittag–Leffler functions. Additionally, we present a comparative study of the obtained solutions and results of the discussed nonlinear systems under the three considered fractional derivatives through the corresponding two and three-dimensional plots for various values of fractional orders as well as with the existing literature.