Recovery of advection coefficient and fractional order in a time-fractional reaction–advection–diffusion-wave equation

Abstract

This paper is concerned with an inverse problem of recovering the space-dependent advection coefficient and the fractional order in a one-dimensional time-fractional reaction–advection–diffusion-wave equation. Based on a transformation, the original equation can be changed into a new form without an advection term. Then we show the uniqueness of recovering the fractional order and the zeroth-order coefficient which contains the information of the “original” advection coefficient by the observation data at two end points. Under the theory of the first-order ordinary differential equation, we obtain the uniqueness result of the advection coefficient. Lastly, we solve the inverse problem numerically from Bayesian perspective by using the iterative regularizing ensemble Kalman method, and numerical examples are presented to show the effectiveness of the proposed method.