In this paper, we consider the inverse problem for identifying the source term and initial value for time-fractional diffusion equation on spherically symmetric domain with Caputo–Hadamard fractional derivative. By solving the direct problem, the exact solutions of the problem can be calculated, and based on the expressions of the exact solutions, it can be analyzed that this problem is ill-posed. To address this, we employ the fractional Landweber iterative regularization method to restore the stability of the solutions. Furthermore, the error estimates under the priori regularization parameter choice rules and the posteriori regularization parameter choice rules are given, respectively. Finally, different numerical examples are presented to demonstrate the validity and effectiveness of our method.

在本文中，我们考虑用 Caputo-Hadamard 分数阶导数识别球对称域上时间分数扩散方程的源项和初始值的反问题。通过求解直接问题，可以计算出问题的精确解，并根据精确解的表达式，可以分析该问题是病态的。为了解决这个问题，我们采用分数 Landweber 迭代正则化方法来恢复解的稳定性。此外，还分别给出了先验正则化参数选择规则和后验正则化参数选择规则下的误差估计。最后，给出了不同的数值例子来证明我们方法的有效性和有效性。