We explain how the invariant subspace method can be extended to a scalar and coupled system of time-space fractional partial differential equations. The effectiveness and applicability of the method have been illustrated using time-space (i) fractional diffusion-convection equation, (ii) fractional reaction-diffusion equation, (iii) fractional diffusion equation with source term, (iv) two-coupled system of fractional diffusion equation, (v) two-coupled system of fractional stationary transonic plane-parallel gas flow equation and (vi) three-coupled system of fractional Hirota–Satsuma KdV equation. Also, we explicitly showed how to derive more than one exact solution of the equations as mentioned above using the invariant subspace method.

中文翻译：

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某些类型的标量和耦合时空分数扩散方程的不变子空间和精确解

我们解释了如何将不变子空间方法扩展到时空分数偏微分方程的标量和耦合系统。该方法的有效性和适用性已通过时空 (i) 分数阶扩散-对流方程，(ii) 分数阶反应-扩散方程，(iii) 带源项的分数阶扩散方程，(iv) 两耦合系统说明分数扩散方程，（v）分数平稳跨音速平面平行气流方程的两耦合系统和（vi）分数Hirota-Satsuma KdV方程的三耦合系统。此外，我们明确展示了如何使用不变子空间方法推导出上述方程的多个精确解。