In this paper, the problem of unknown source identification for the space-time fractional diffusion equation is studied. In this equation, the time fractional derivative used is a new fractional derivative, namely, Caputo-Fabrizio fractional derivative. We have illustrated that this problem is an ill-posed problem. Under the assumption of a priori bound, we obtain the optimal error bound analysis of the problem under the source condition. Moreover, we use a modified quasi-boundary regularization method and Landweber iterative regularization method to solve this ill-posed problem. Based on a priori and a posteriori regularization parameter selection rules, the corresponding convergence error estimates of the two regularization methods are obtained, respectively. Compared with the modified quasi-boundary regularization method, the convergence error estimate of Landweber iterative regularization method is order-optimal. Finally, the advantages, stability and effectiveness of the two regularization methods are illustrated by examples with different properties.

本文研究了时空分数阶扩散方程的未知源识别问题。在这个方程中，使用的时间分数阶导数是一个新的分数阶导数，即 Caputo-Fabrizio 分数阶导数。我们已经说明这个问题是一个不适定的问题。在先验界限的假设下，我们得到了源条件下问题的最优误差界限分析。此外，我们使用改进的准边界正则化方法和 Landweber 迭代正则化方法来解决这个不适定问题。基于先验和后验正则化参数选择规则，分别得到两种正则化方法对应的收敛误差估计。与改进的准边界正则化方法相比，Landweber迭代正则化方法的收敛误差估计是阶次最优的。最后，通过不同性质的例子说明了两种正则化方法的优点、稳定性和有效性。