Initial value problem for the $$(2+1)$$-dimensional time-fractional generalized convection–reaction–diffusion wave equation: invariant subspaces and exact solutions

Abstract

This work investigates how we can extend the invariant subspace method to $$(2+1)$$ -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 $$(2+1)$$ -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 $$(2+1)$$ -dimensional time-fractional non-linear PDEs with time delay. Also, the effectiveness and applicability of the method have been illustrated through the $$(2+1)$$ -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 the Mittag-Leffler function and polynomial, exponential and trigonometric type functions.