Analytical solutions of higher-dimensional coupled system of nonlinear time-fractional diffusion-convection-wave equations

TL;DR

This paper extends the invariant subspace method to derive analytical solutions for multi-component higher-dimensional nonlinear time-fractional PDEs, reducing them to fractional ODEs and finding generalized solutions with graphical illustrations.

Contribution

It introduces a systematic approach to find invariant product linear spaces for multi-component fractional PDEs, enabling analytical solutions and their graphical representations.

Findings

Invariant product linear spaces help reduce PDEs to fractional ODEs.

Method successfully derives generalized separable solutions.

Graphical plots illustrate solutions for various fractional orders.

Abstract

This article develops how to generalize the invariant subspace method for deriving the analytical solutions of the multi-component (N+1)-dimensional coupled nonlinear time-fractional PDEs (NTFPDEs) in the sense of Caputo fractional-order derivative for the first time. Specifically, we describe how to systematically find different invariant product linear spaces with various dimensions for the considered system. Also, we observe that the obtained invariant product linear spaces help to reduce the multi-component (N+1)-dimensional coupled NTFPDEs into a system of fractional-order ODEs, which can then be solved using the well-known analytical methods. More precisely, we illustrate the effectiveness and importance of this developed method for obtaining a long list of invariant product linear spaces for the multi-component (2+1)-dimensional coupled nonlinear time-fractional diffusion-convection-wave equations. In addition, we have shown how to find different kinds of generalized separable analytical solutions for a multi-component (2 + 1)-dimensional coupled nonlinear time-fractional diffusion-convection-wave equations along with the initial and boundary conditions using invariant product linear spaces obtained. Finally, we provide appropriate graphical representations of some of the derived generalized separable analytical solutions with various fractional-order values.