Two regularization methods for identifying the initial value of Caputo–Hadamard time-fractional diffusion equation

In this paper, the inverse problem of identifying the unknown initial value for time fractional diffusion equation with Caputo-Hadamard derivative is considered. This problem is illposed and two regularization methods are used to solve it. Firstly, we prove that this problem is ill-posed. Secondly, the conditional stability result and the optimal error bound are given. Then, the error estimates of the Quasi-boundary regularization method and the fractional Landweber iterative regularization meth