Optimal regularization for an unknown source of space-fractional diffusion equation

Abstract

We study an inverse problem of finding the source Q(x, t) for diffusion equation with space-fractional Laplacian ut(x,t)=−μ(−Δ)γu(x,t)+Q(x,t)(0<γ≤1,μ>0,x∈R) from the initial and the final data. Determining the unknown source is to identify the nature of diffusion process and that is one of the most important and well-studied problems in many branches of engineering sciences. This problem must be solved since its result will be used in scientific and technical applications. We limit the severely ill-posed problem to the one of finding Q(x,t)=Φ(t)S(x) with Φ being known. The cases that the variable x is in bounded domains of Rn or the differential operator has the divergence form (−∇·(μ∇u))γ with μ=μ(x,t) or μ=μ(x,t,u) are still not investigated in this paper. The uniqueness and the conditional stability for S are obtained. We apply the Fourier truncation method to construct stable approximation problem and we show the asymptotically optimal estimates for the worst case error of the method. This optimality cannot deduce directly from Tautenhaln’s results. We also establish a general method of a posteriori regularization parameter choice rule which has an error estimate independent of the smoothing parameter of S and, moreover, we prove that the choice rule gives in some cases the best error order ϵ. The μ-depended results show the incremental influent of the diffusivity μ on the optimal choice rules when μ is close to zero.