Initial value problem for the two-dimensional time-fractional generalized convection-reaction-diffusion-wave equation: Invariant subspaces and exact solutions

TL;DR

This paper extends the invariant subspace method to solve two-dimensional time-fractional nonlinear PDEs, constructing invariant subspaces and exact solutions, including equations with time delay, using Mittag-Leffler functions and other special functions.

Contribution

It introduces a systematic approach for constructing invariant subspaces and exact solutions for complex two-dimensional time-fractional PDEs, including those with time delay.

Findings

Constructed invariant subspaces for various types of fractional PDEs.

Derived exact solutions involving Mittag-Leffler, polynomial, exponential, and trigonometric functions.

Demonstrated the method's effectiveness on a nonlinear PDE with time delay.

Abstract

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