An inverse problem for a time-fractional advection equation associated with a nonlinear reaction term

Abstract

Fractional derivative is an important notion in the study of the contemporary mathematics not only because it is more mathematically general than the classical derivative but also it really has applications to understand many physical phenomena. In particular, fractional derivatives are related to long power-law particle jumps, which can be understood as transient anomalous sub-diffusion model (see Sabzikar F, Meerschaert M, Chen J. Tempered fractional calculus. J Comput Phys. 2015;293:14–28; Sokolov IM, Klafter J, Blumen A. Fractional kinetics. Phys Today. 2002;55:48–54; Sokolov IM, Klafter J. Anomalous diffusion spreads its wings. Phys World. 2005;18:19–22; Zhang Y, Meerschaert MM, Packman AI. Linking fluvial bed sediment transport across scales. Geophys Res Lett. 2012;39(20):20404. doi:10.1029/2012GL053476). Based on the models given in Scher H, Montroll EW. Anomalous transit-time dispersion in amorphous solids. Phys Rev B. 1975;12(6):2455–2477 and Zheng GH, Wei T. Spectral regularization method for solving a time-fractional inverse diffusion problem. Appl Math Comput. 2011;218:1972–1990, we study an inverse problem for the advection equation with a nonlinear reaction term in a two-dimensional semi-infinite domain for which we recover the initial distribution from the observation data provided at the final location x = 1. This problem is severely ill-posed in the sense of Hadamard. Thus, we propose a regularization method to construct an approximate solution for the problem. From that, convergence rate of the regularized solution is obtained under some a priori bound assumptions on the exact solution. Eventually, a numerical experiment is given to show the effectiveness of the proposed regularization methods.