The inverse source problem for time-fractional diffusion equation: stability analysis and regularization

Abstract

In the present paper, we consider an inverse source problem for a fractional diffusion equation. This problem is ill-posed, i.e. the solution (if it exists) does not depend continuously on the data. Based on an a priori assumption, we give the optimal error bound analysis and a conditional stability result. Moreover, we use the Fourier regularization method to deal with this problem. An a priori error estimate between the exact solution and its regularized approximation is obtained. Meanwhile, a new a posteriori parameter choice rule is also proposed. For the a priori and the a posteriori regularization parameters choice rules, we all obtain the convergence error estimates which are all order optimal. Numerical examples are presented to illustrate the validity and effectiveness of this method.

Fourier Truncation Regularization Method for a Time-Fractional Backward Diffusion Problem with a Nonlinear Source

“…The Fourier truncation regularization method is a very effective method for dealing with ill-posed problems. Many authors have used it to deal with different ill-posed problems, such as in [23][24][25][26][27][28][29][30][31][32]. In [33], the authors extended the Fourier method to the general filtering method and solved the semi-linear ill-posed problem in the general framework.…”

Fourier Truncation Regularization Method for a Three-Dimensional Cauchy Problem of the Modified Helmholtz Equation with Perturbed Wave Number

“…Recently, Fourier regularization method has been effectively applied to solve different inverse problem: The sideways heat equation [18,19], a more general sideways parabolic equation [20], numerical differentiation [21], a posteriori Fourier method for solving ill-posed problems [22], the unknown source in the Poisson equation [23], the time-dependent heat source for heat equation [24], the heat source problem for time fractional diffusion equation [25,26], the semi-linear backward parabolic problems [27], the unknown source for time-fractional diffusion equation in bounded domain [28], the Cauchy problem for the Helmholtz equation [29], the a posteriori truncation method for some Cauchy problems associated with Helmholtz-type equations [30], the Cauchy problem of the inhomogeneous Helmholtz equation [31].…”

Inverse source problem for a distributed-order time fractional diffusion equation

“…Jiang et al [13] reformulated the space-dependent source problem as an optimization problem with Tikhonov regularization method and proposed an iterative thresholding algorithm to solve it. Yang, Fu and Li [31,32] and Wei, Li and Li [28] applied the Fourier regularization method, a modification regularization method and the Tikhonov regularization method, respectively, to recover the time-dependent source from additional data at some fixed locations in the domain. Bazhlekova [1] proved the uniqueness and a conditional stability of an inverse source problem for a distributed-order time fractional diffusion equation.…”