An inverse problem in time-fractional diffusion equations with nonlinear boundary condition

Abstract

This paper deals with the 1D time-fractional diffusion equations with nonlinear boundary condition. We first give an integral equation of the solution via the theta-function and eigenfunction expansion and establish the short time asymptotic behavior of the solution. We then verify the uniqueness of the inverse problem in determining the fractional order by the one endpoint observation whether the nonlinear boundary condition is known or not. Furthermore, in the case when the nonlinear boundary condition is known, we can establish the Lipschitz stability of the fractional order with respect to the measured data at the one endpoint by using the Lipschitz continuity of the solution with respect to the fractional order.