An undetermined time-dependent coefficient in a fractional diffusion equation

Abstract

In this work, we consider a FDE (fractional diffusion equation) $${}^C{D}_t^{\alpha }u(x,t)-a(t)\mathcal{L}u(x,t)=F(x,t)$$ with a time-dependent diffusion coefficient $a(t)$. For the direct problem, given an $a(t),$ we establish the existence, uniqueness and some regularity properties with a more general domain $\Omega$ and right-hand side $F(x,t)$. For the inverse problem--recovering $a(t),$ we introduce an operator $K$ one of whose fixed points is $a(t)$ and show its monotonicity, uniqueness and existence of its fixed points. With these properties, a reconstruction algorithm for $a(t)$ is created and some numerical results are provided to illustrate the theories.

References in corpus (2)

• Fractional Calculus: Integral and Differential Equations of Fractional Order
• An analysis of the L1 Scheme for the subdiffusion equation with nonsmooth data

Cited by in corpus (3)

• Inverse problems for heat equation and space-time fractional diffusion equation with one measurement
• Reconstruction of the time-dependent source term in a stochastic fractional diffusion equation
• Recovery of the time-dependent source term in the stochastic fractional diffusion equation with heterogeneous medium