In the present paper, we devote our effort to a nonlinear inverse problem for simultaneously recovering the potential function and the fractional orders in a multi-term time-fractional diffusion equation from the noisy boundary Cauchy data in the one-dimensional case. The uniqueness for the inverse problem is derived based on the analytic continuation, the Laplace transformation and the Gel’fand–Levitan theory. Finally, the Levenberg–Marquardt regularization method with a regularization parameter chosen by a sigmoid-type function is applied for finding a stable approximate solution. Three numerical examples are provided to show the effectiveness of the proposed method.

在本文中，我们致力于从一维情况下的噪声边界柯西数据中同时恢复多项时间分数扩散方程中的势函数和分数阶的非线性逆问题。逆问题的唯一性是基于解析延拓、拉普拉斯变换和 Gel'fand-Levitan 理论推导出来的。最后，使用由 sigmoid 型函数选择的正则化参数的 Levenberg-Marquardt 正则化方法用于寻找稳定的近似解。提供了三个数值例子来证明所提出方法的有效性。