An inverse time-dependent source problem for a time-fractional diffusion equation (Q2820671)

Let \(\Omega\) be a bounded domain in \(\mathbb R^d\), \(d\geq 1\) with sufficiently smooth boundary \(\partial \Omega\). The authors consider the time-fractional diffusion equation \(\partial^{\alpha}_{0+} u(x,t)=\Delta u(x,t)+f(x) p(t)\), \(x\in \Omega\), \(t\in (0,T]\); \(\alpha\in (0,1)\). Here, \(\partial^{\alpha}_{0+}\) denotes the Caputo fractional left derivative of order \(\alpha\). The initial and boundary conditions are \(u(x,0)=\psi(x)\), \(x\in \Omega\) and \(\partial u/\partial n (x,t)=0\), \(x\in \partial \Omega\), \(t\in (0,T]\), and \(n\) is the unit outward normal to \(\partial \Omega\). The inverse problem under investigation is to determine the source term \(p(t)\) from the additional lateral data \(u(x,t)=g(x,t)\), \(x\in \Gamma\), \(t\in (0,T]\), where \(\Gamma\) is a nonempty part of \(\partial \Omega\). The authors establish the uniqueness and the stability estimate for the inverse problem. Numerical examples with the Tikhonov regularization method in 1D and 2D problems are discussed.