No Arabic abstract
Given $(M,g)$, a compact connected Riemannian manifold of dimension $d geq 2$, with boundary $partial M$, we consider an initial boundary value problem for a fractional diffusion equation on $(0,T) times M$, $T>0$, with time-fractional Caputo derivative of order $alpha in (0,1) cup (1,2)$. We prove uniqueness in the inverse problem of determining the smooth manifold $(M,g)$ (up to an isometry), and various time-independent smooth coefficients appearing in this equation, from measurements of the solution on a subset of $partial M$ at fixed time. In the flat case where $M$ is a compact subset of $mathbb R^d$, two out the three coefficients $rho$ (weight), $a$ (conductivity) and $q$ (potential) appearing in the equation $rho partial_t^alpha u-textrm{div}(a abla u)+ q u=0$ on $(0,T)times Omega$ are recovered simultaneously.
In this paper, we discuss the uniqueness for solution to time-fractional diffusion equation $partial_t^alpha (u-u_0) + Au=0$ with the homogeneous Dirichlet boundary condition, where an elliptic operator $-A$ is not necessarily symmetric. We prove that the solution is identically zero if its normal derivative with respect to the operator $A$ vanishes on an arbitrary small part of the spatial domain over a time interval. The proof is based on the Laplace transform and the spectral decomposition, and is valid for more general time-fractional partial differential equations, including those involving non symmetric operators.
This paper is concerned with the inverse problem on determining an orbit of the moving source in a fractional diffusion(-wave) equations in a connected bounded domain of $mathbb R^d$ or in the whole space $mathbb R^d$. Based on a newly established fractional Duhamels principle, we derive a Lipschitz stability estimate in the case of a localized moving source by the observation data at $d$ interior points. The uniqueness for the general non-localized moving source is verified with additional data of more interior observations.
We examine initial-boundary value problems for diffusion equations with distributed order time-fractional derivatives. We prove existence and uniqueness results for the weak solution to these systems, together with its continuous dependency on initial value and source term. Moreover, under suitable assumption on the source term, we establish that the solution is analytic in time.
This article is concerned with two inverse problems on determining moving source profile functions in evolution equations with a derivative order $alphain(0,2]$ in time. In the first problem, the sources are supposed to move along known straight lines, and we suitably choose partial interior observation data in finite time. Reducing the problems to the determination of initial values, we prove the unique determination of one and two moving source profiles for $0<alphale1$ and $1<alphale2$, respectively. In the second problem, the orbits of moving sources are assumed to be known, and we consider the full lateral Cauchy data. At the cost of infinite observation time, we prove the unique determination of one moving source profile by constructing test functions.
In this paper we prove uniqueness for an inverse boundary value problem (IBVP) arising in electrodynamics. We assume that the electromagnetic properties of the medium, namely the magnetic permeability, the electric permittivity and the conductivity, are described by continuously differentiable functions.