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 -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 -dimensions into the nonlinear two-component system of TFPDEs in -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 -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.

中文翻译：

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[式略]维非线性二分量时间分数偏微分方程组：结合变量变换的不变子空间法


在本文中，我们开发了一种结合变量变换的不变子空间方法的系统方法，首次找到 - 维中时间分数偏微分方程（TFPDE）的非线性二分量系统的广义可分精确解。此外，我们还明确解释了如何使用不变子空间方法结合变量变换为给定系统构造各种有限维不变线性乘积空间。此外，我们还介绍了如何使用所获得的不变线性乘积空间来导出所讨论系统的广义可分精确解。我们还注意到，所讨论的方法将有助于将 - 维 TFPDE 的非线性二分量系统简化为 - 维 TFPDE 的非线性二分量系统，通过获得的不变量再次将其简化为时间分数 ODE 系统线性乘积空间。更具体地说，通过时间分数反应扩散方程（TFRDE）的广义非线性二元系统的初始和边值问题，研究了所讨论的方法的系统研究的意义和有效性，以找到广义可分离精确解，可以用指数函数、三角函数、多项式函数、Euler-Gamma 函数和 Mittag-Leffler 函数表示。此外，还针对不同的分数阶值给出了一些所获得的解决方案的 2D 和 3D 图形表示。