On the Reconstruction of Convection Coefficient in a Semilinear Anomalous Diffusion System

Abstract

In the paper, we study an inverse problem of recovering a time-dependent convection coefficient from the measured data at an interior/boundary point in a one-dimensional nonlinear subdiffusion model with non-homogeneous boundary conditions. Due to the nonlinearity and non-homogeneous boundary conditions of the system, such an inverse problem is novel and important. We first investigate the unique existence and some regularities of the solution to forward problem by using the transposition method and the fixed point theorem. Then a conditional stability of the inverse problem is obtained based on the regularity of solution for the direct problem and some generalized Gronwall's inequalities. Finally, we transform the inverse problem into a variational problem. The existence and convergence of the regularization solution for the variational problem are proved and we use a modified Levenberg–Marquardt method to find an approximate convection coefficient function. The efficiency and accuracy of the algorithm are illustrated with two numerical examples.