A modified kernel method for a time-fractional inverse diffusion problem

Abstract

In this paper, we consider a time-fractional inverse diffusion problem, where the data is given at x = 1 and the solution is sought in the interval 0 ≤ x < 1. Such a problem is obtained from the classical diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α ∈ (0, 1). We show that a time-fractional inverse diffusion problem is severely ill-posed and we further apply a modified kernel method to solve it based on the solution in the frequency domain. The corresponding convergence estimates are provided. Finally, an example is constructed to show the feasibility and efficiency of the proposed method. MSC: 35R25; 35R30; 47A52

An Inverse Source Problem of Space-Fractional Diffusion Equation

“…coefficient information, source term information might not be given and then we need to recover them by extra measured information which is able to yield some fractional diffusion inverse problems [12][13][14][15][16][17][18][19][20]. In recent years, inverse problems for fractional diffusion equation have become very active in various fields of sciences and engineering, such as biology [21,22], physics [23,24], chemistry [25], and hydrology [26].…”

Optimal error bound and modified kernel method for a space-fractional backward diffusion problem

“…Before this, the idea has also appeared in [31], in which the problem of high-order numerical differentiation was successfully solved. Then, the modified 'kernel' idea has been used for solving various types of ill-posed problems [32][33][34]. In order to examine whether a regularization method is optimal, we will give the optimal error bound for the problem under a source condition.…”