In this paper, we develop an efficient finite difference/spectral method to solve a coupled system of nonlinear multi-term time-space fractional diffusion equations. In general, the solutions of such equations typically exhibit a weak singularity at the initial time. Based on the L1 formula on nonuniform meshes for time stepping and the Legendre–Galerkin spectral method for space discretization, a fully discrete numerical scheme is constructed. Taking into account the initial weak singularity of the solution, the convergence of the method is proved. The optimal error estimate is obtained by providing a generalized discrete form of the fractional Grönwall inequality which enables us to overcome the difficulties caused by the sum of Caputo time-fractional derivatives and and the positivity of the reaction term over the nonuniform time mesh. The error estimate reveals how to select an appropriate mesh parameter to obtain the temporal optimal convergence. Furthermore, numerical experiments are presented to confirm the theoretical claims.

中文翻译：

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非线性多项时空分数阶扩散方程耦合系统的分级网格离散化

在本文中，我们开发了一种有效的有限差分/谱方法来求解非线性多项时空分数扩散方程的耦合系统。通常，此类方程的解在初始时通常表现出弱奇异性。基于用于时间步长的非均匀网格的 L1 公式和用于空间离散化的 Legendre-Galerkin 谱方法，构建了一个完全离散的数值方案。考虑到解的初始弱奇异性，证明了该方法的收敛性。通过提供分数 Grönwall 不等式的广义离散形式获得最佳误差估计，这使我们能够克服由 Caputo 时间分数导数的总和和非均匀时间网格上的反应项的正性引起的困难。误差估计揭示了如何选择合适的网格参数以获得时间最优收敛。此外，还提出了数值实验来证实理论主张。