Nonlinear two-component system of time-fractional PDEs in (2+1)-dimensions: Invariant subspace method combined with variable transformation

TL;DR

This paper introduces a combined invariant subspace and variable transformation method to find exact solutions of nonlinear two-component time-fractional PDEs in (2+1) dimensions, simplifying the system to lower dimensions and providing explicit solutions.

Contribution

It presents a novel systematic approach for solving complex fractional PDE systems using invariant subspaces combined with variable transformation, leading to explicit solutions and dimensional reduction.

Findings

Derived generalized separable exact solutions involving special functions.

Reduced (2+1)-dimensional systems to (1+1)-dimensional systems and ODEs.

Provided graphical representations of solutions for different fractional orders.

Abstract

In this article, we develop a systematic approach of the invariant subspace method combined with variable transformation to find the generalized separable exact solutions of the nonlinear two-component system of time-fractional PDEs (TFPDEs) in (2+1)-dimensions for the first time. Also, we explicitly explain how to construct various kinds of finite-dimensional invariant linear product spaces for the given system using the invariant subspace method combined with variable transformation. Additionally, we present how to use the obtained invariant linear product spaces to derive the generalized separable exact solutions of the discussed system. We also note that the discussed method will help to reduce the nonlinear two-component system of TFPDEs in (2+1)-dimensions into the nonlinear two-component system of TFPDEs in (1+1)-dimensions, which again reduces to a system of time-fractional ODEs through the obtained invariant linear product spaces. More specifically, the significance and efficacy of the systematic investigation of the discussed method have been investigated through the initial and boundary value problems of the generalized nonlinear two-component system of time-fractional reaction-diffusion equations (TFRDEs) in (2+1)-dimensions for finding the generalized separable exact solutions, which can be expressed in terms of the exponential, trigonometric, polynomial, Euler-Gamma and Mittag-Leffler functions. Also, 2D and 3D graphical representations of some of the obtained solutions are presented for different values of fractional orders.