An iterative method for backward time‐fractional diffusion problem

Abstract

The aim of this work is to solve the backward problem for a time‐fractional diffusion equation with variable coefficients in a general bounded domain. The problem is ill‐posed in L 2 norm sense. An iteration scheme is proposed to obtain a regularized solution. Two kinds of convergence rates are obtained using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Numerical examples in one‐dimensional and two‐dimensional cases are provided to show the effectiveness of the proposed methods. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 2029–2041, 2014

Simultaneous inversion of two initial values for a time‐fractional diffusion‐wave equation

“…For the time-fractional diffusion equations ( ∈ (0, 1)), the inverse problems are studied in various methods, refer to refs. [15,[21][22][23][24][25][26]. The inverse initial value problem of diffusion process aims at detecting the previous status of physical field from additional measurements and is of great importance in engineering, where the physical field may be temperature or concentration.…”

“…Wang et al [30] used a Tikhonov method to solve the backward problem in a general bounded domain. The modified quasi-boundary value method [31], the iterative regularization method [23] have also been proposed to solve the backward problem for a time-fractional diffusion equation. Some authors use other additional measurement to recover the initial value.…”

Variational method for a backward problem for a time-fractional diffusion equation

“…Wei and Wang [28] used a modified quasi-boundary value regularization method to solve this ill-posed problem. In [24], Wang and Wei used an iterative regularization method to solve the backward problem. In this paper, we use the Tikhonov regularization method, but finding the minimizer by a conjugate gradient method without using any spectral information of the operator −L, therefore we can consider numerical examples in a general domain.…”

An Iterative Regularization Method for Identifying the Source Term in a Second Order Differential Equation

“…In the present work, we focus on an iterative method proposed by Kozlov and Maz'ya [13,14] for solving the problem; it is based on solving a sequence of well-posed boundary value problems such that the sequence of solutions converges to the solution for the original problem. It has been successfully used for solving various classes of ill-posed elliptic, parabolic, and hyperbolic problems [5,[15][16][17][18][19][20][21].…”