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The problem of heat conduction in one-dimensional piecewise homogeneous composite materials is examined by providing an explicit solution of the one-dimensional heat equation in each domain. The location of the interfaces is known, but…
This paper is focused on the study of an inverse problem for a non-self-adjoint hyperbolic equation. More precisely, we attempt to stably recover a first order coefficient appearing in a wave equation from the knowledge of Neumann boundary…
This work is devoted to the strong unique continuation problem for second order parabolic equations with nonsmooth coefficients. Introduction and bibliography have been revised.
The aim of this paper is to establish the framework of the enclosure method for some class of inverse problems whose governing equations are given by parabolic equations with discontinuous coefficients. The framework is given by considering…
An inverse source problem for the heat equation is considered. Extraction formulae for information about the time and location when and where the unknown source of the equation firstly appeared are given from a single lateral boundary…
A problem of diffraction by an elongated body of revolution is studied. The incident wave falls along the axis. The wavelength is small comparatively to the dimensions of the body. The parabolic equation of the diffraction theory is used to…
We investigate a backward anisotropic stochastic parabolic equation with general dynamic boundary conditions, where the drift involves both $\mathbb{L}^2$ and $\mathbb{H}^{-1}$ bulk--surface terms. We first establish the well-posedness of…
We study the diffusion (or heat) equation on a finite 1-dimensional spatial domain, but we replace one of the boundary conditions with a "nonlocal condition", through which we specify a weighted average of the solution over the spatial…
We consider the inverse problem of recovering stationary coefficients in a class of dynamical Schr\"odinger equations with locally analytic nonlinear terms. Upon treating the well-posedness for small initial data and trivial boundary data,…
We establish a link between stability estimates for a hyperbolic inverse problem via the Boundary Control method and the blowup of a constant appearing in the contexts of optimal unique continuation and cost of approximate controllability.
We develop a new variational formulation of the inverse Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundary. We employ optimal control framework,…
This paper investigates the solution of a parabolic inverse problem based upon the convection-diffusion-reaction equation, which can be used to estimate both water and air pollution. We will consider both known and unknown source location:…
In this work, we investigate the estimation of the transient mold-slab heat flux in continuous casting molds given some thermocouples measurements in the mold plates. Mathematically, we can see this problem as the estimation of a Neumann…
We prove an inequality of H\"older type traducing the unique continuation property at one time for the heat equation with a potential and Neumann boundary condition. The main feature of the proof is to overcome the propagation of smallness…
In this paper, we study an inverse coefficients problem for two coupled Schr\"{o}dinger equations with an observation of one component of the solution. The observation is done in a nonempty open subset of the domain where the equations…
This book aims to provide a brief overview of recent advancements in the theory of inverse problems for stochastic partial differential equations. In order to keep the content concise, we will only discuss the inverse problems of two…
For the first time, we develop in this paper the globally convergent convexification numerical method for a Coefficient Inverse Problem for the 3D Helmholtz equation for the case when the backscattering data are generated by a point source…
A broad class of inverse problems deals with determining certain parameters, from measurement data, in models which are associated to certain partial differential equations. In this work we focus on the heat equation on a finite interval…
In this paper, we study the control of the linear heat equation with a space and time dependent coefficient function by the Dirichlet and Neumann boundary control laws. This equation models the heat diffusion and space, time dependent heat…
This paper presents an inverse problem for the nonlinear 1-d Kuramoto-Sivashinsky (K-S) equation. More precisely, we study the nonlinear inverse problem of retrieving the anti-diffusion coefficient from the measurements of the solution on a…