We consider a time-space fractional diffusion equation with a variable coefficient and investigate the inverse problem of reconstructing the source term, after regularizing the problem with the quasiboundary value method to mitigate the ill-posedness. The equation involves a Caputo fractional derivative in the space variable and a tempered fractional derivative in the time variable, both of order in (0, 1). A finite difference approximation leads to a two-by-two block linear system of large dimensions. We conduct a spectral analysis of the associated matrix sequences, employing tools from Generalized Locally Toeplitz (GLT) theory, and construct the preconditioner guided by the GLT analysis. Numerical experiments are reported and commented, followed by concluding remarks.

翻译：本文研究含变系数的时空分数阶扩散方程，并探讨源项重构的反问题。为缓解不适定性，我们采用拟边界值法对问题进行正则化处理。该方程包含空间变量的Caputo分数阶导数与时间变量的缓变分数阶导数，二者阶数均位于(0,1)区间。通过有限差分近似，我们得到具有大规模二维块结构的线性系统。运用广义局部Toeplitz(GLT)理论工具对相关矩阵序列进行谱分析，并基于GLT分析构建预处理器。文中报告并讨论了数值实验结果，最后给出结论性评述。