A Backward Identification Problem for an Axis-Symmetric Fractional Diffusion Equation

Abstract

We consider a backward ill-posed problem for an axis-symmetric fractional diffusion equation which is described in polar coordinates. A closed form solution of the inverse problem is obtained. However, this solution blows up. For numerical stability, a general regularization principle is presented for constructing regularization methods. Several numerical examples are conducted for showing the validity and effectiveness of the proposed methods.

The backward problem for radially symmetric time‐fractional diffusion‐wave equation under Robin boundary condition

“…8 In this paper, we are interested in studying the backward problem of the radially symmetric time-fractional diffusion-wave equation. There are only a few papers [9][10][11] on the inverse problem of the radially symmetric time-fractional diffusion or diffusion-wave equation, but these papers are limited to the Dirichlet boundary condition (1.1) or the Neumann boundary condition (1.2).…”

“…In this paper, we are interested in studying the backward problem of the radially symmetric time‐fractional diffusion‐wave equation. There are only a few papers 9–11 on the inverse problem of the radially symmetric time‐fractional diffusion or diffusion‐wave equation, but these papers are limited to the Dirichlet boundary condition () or the Neumann boundary condition (). $$$$ u\left(r,t\right)=\sigma (t),\kern2em \left(r,t\right)\in \partial D\times \left[0,T\right].$$…”

The backward problem for radially symmetric time-fractional diffusion-wave equation under Robin boundary condition

“…In this paper, we are interested in studying the backward problems of the radially symmetric time-fractional diffusion-wave equation. There are only a few papers [9,10,11] on the inverse problem of the radially symmetric time-fractional diffusion or diffusion-wave equation, but these papers are limited to the Dirichlet boundary condition (1.1) or the Neumann boundary condition (1.2). u(r, t) = σ(t), (r, t) ∈ ∂D × [0, T ].…”

A Fractional Tikhonov Regularisation Method for Finding Source Terms in a Time-Fractional Radial Heat Equation

“…The inverse source problem has been vigorously studied. Thus Wang et al [30] applied reproducing kernel space method to solve an inverse space-dependent source problem, Wei and Wang [31] used a modified quasi-boundary value method, Zhang and Xu [37] employed the Cauchy data at one end, Tatar et al [26,27] considered it for a spacetime fractional diffusion equation and investigated a nonlocal inverse source problem, Cheng et al [4] used a spectral method to determine an unknown heat source term from the final temperature history in the radial domain and provided logarithmic-type error estimates for regularised solutions, Xiong and Ma [33] discussed a backward ill-posed problem for an axis-symmetric fractional diffusion equation. For other relevant results, the reader is referred to [3,9,12,14,28,[34][35][36].…”