In this paper, the inverse problem of initial value identification for homogeneous anomalous diffusion equation with Riemann-Liouville fractional derivative in time is studied. We prove that this kind of problem is ill-posed. We analyze the optimal error bound of the problem under the source condition and apply the quasi-boundary regularization method, fractional Landweber iterative regularization method, and Landweber iterative regularization method to solve this inverse problem. Based on the results of conditional stability, the error estimates between the exact solution and the regular solution are given under the priori and posteriori regularization parameter selection rules. Finally, three examples are given to illustrate the effectiveness and feasibility of these methods.

中文翻译：

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齐次异常二次扩散方程初值的三种正则化方法

本文研究了具有Riemann-Liouville时间分数阶导数的齐次反常扩散方程初值辨识的反问题。我们证明这种问题是不适定的。我们分析了问题在源条件下的最优误差界，并应用拟边界正则化方法、分数Landweber迭代正则化方法和Landweber迭代正则化方法来解决这个逆问题。根据条件稳定性的结果，在先验和后验正则化参数选择规则下，给出了精确解与正则解之间的误差估计。最后，通过三个例子来说明这些方法的有效性和可行性。