On the solutions of coupled nonlinear time-fractional diffusion-reaction system with time delays

TL;DR

This paper applies the invariant subspace method to find analytical solutions for a coupled nonlinear time-fractional diffusion-reaction system with time delays, considering two types of fractional derivatives, and derives solutions for initial-boundary value problems.

Contribution

It systematically constructs invariant vector spaces for the system under Riemann-Liouville and Caputo derivatives, enabling explicit analytical solutions for complex fractional PDEs with delays.

Findings

Derived invariant vector spaces for the system

Obtained explicit analytical solutions in generalized separable form

Addressed solutions for systems with multiple time delays

Abstract

In this article, we systematically explain how to apply the analytical technique called the invariant subspace method to find various types of analytical solutions for a coupled nonlinear time-fractional system of partial differential equations with time delays. Also, the present work explicitly studies a systematic way to obtain various kinds of finite-dimensional invariant vector spaces for the coupled nonlinear time-fractional diffusion-reaction (DR) system with time delays under the two distinct fractional derivatives, namely (a) the Riemann-Liouville fractional partial time derivative and (b) the Caputo fractional partial time derivative. Additionally, we provide details of deriving analytical solutions in the generalized separable form for the initial and boundary value problems (IBVPs) of the coupled nonlinear time-fractional DR system with multiple time delays through the obtained invariant vector spaces under the considered two time-fractional derivatives.