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This paper concerns the inverse source problem for the time-harmonic wave equation in a one dimensional domain. The goal is to determine the source function from the boundary measurements. The problem is challenging due to complexity of the…

Analysis of PDEs · Mathematics 2020-04-08 Elham Sohrabi

We address the inverse problem of identifying a time-dependent source coefficient in a one-dimensional heat equation with a fractional Laplacian subject to Dirichlet boundary conditions and an integral nonlocal data. An a priori estimate is…

Numerical Analysis · Mathematics 2025-11-21 Arshyn Altybay , Niyaz Tokmagambetov , Gulzat Nalzhupbayeva

The classical Calder\'on problem with partial data is known to be log-log stable in some special cases, but even the uniqueness problem is open in general. We study the partial data stability of an analogous inverse fractional conductivity…

Analysis of PDEs · Mathematics 2025-05-27 Giovanni Covi , Antti Kujanpää , Jesse Railo

This paper concerns the stability on the inverse source scattering problem for the one-dimensional Helmholtz equation in a two-layered medium. We show that the increasing stability can be achieved by using multi-frequency wave field at the…

Analysis of PDEs · Mathematics 2017-09-13 Yue Zhao , Peijun Li

We consider a parabolic equation in a bounded domain $\OOO$ over a time interval $(0,T)$ with the homogeneous Neumann boundary condition. We arbitrarily choose a subboundary $\Gamma \subset \ppp\OOO$. Then, we discuss an inverse problem of…

Analysis of PDEs · Mathematics 2022-11-23 O. Imanuvilov , M. Yamamoto

In this paper, a boundary integral method is used to solve an inverse linear heat conduction problem in two-dimensional bounded domain. An inverse problem of measuring the heat flux from partial (on part of the boundary) dynamic boundary…

Mathematical Physics · Physics 2008-05-06 Daveau Christian , Khelifi Abdessatar , Shamma M. Nour

In this work we study the inverse boundary value problem of determining the refractive index in the acoustic equation. It is known that this inverse problem is ill-posed. Nonetheless, we show that the ill-posedness decreases when we…

Analysis of PDEs · Mathematics 2011-10-25 Sei Nagayasu , Gunther Uhlmann , Jenn-Nan Wang

In this article, we consider inverse problems of determining a source term and a coefficient of a first-order partial differential equation and prove conditional stability estimates with minimum boundary observation data and relaxed…

Analysis of PDEs · Mathematics 2015-09-02 Fikret Gölgeleyen , Masahiro Yamamoto

We consider an inverse problem arising in corrosion detection. We prove a stability result of logarithmic type for the determination of the corroded portion of the boundary and impedance by two measurements on the accessible portion of the…

Analysis of PDEs · Mathematics 2010-04-14 Eva Sincich

We consider inverse problems of determining coefficients or time independent factors of source terms in radiative transport equations by means of Carleman estimate. We establish global Lipschitz stability results with an additional…

Analysis of PDEs · Mathematics 2020-09-10 Manabu Machida , Masahiro Yamamoto

We study the inverse boundary value problems of determining a potential in the Helmholtz type equation for the perturbed biharmonic operator from the knowledge of the partial Cauchy data set. Our geometric setting is that of a domain whose…

Analysis of PDEs · Mathematics 2020-07-13 Boya Liu

This second version of the manuscript includes, in the appendices, an erratum that points out an error on the published version and offers alternative results for the Lipschitz stability analysis of the backward heat propagation problem and…

Analysis of PDEs · Mathematics 2023-03-20 Pablo Arratia , Matias Courdurier , Evelyn Cueva , Axel Osses , Benjamin Palacios

We study the inverse boundary value problem for the Helmholtz equation using the Dirichlet-to-Neumann map at selected frequency as the data. We develop an explicit reconstruction of the wavespeed using a multi-level nonlinear projected…

Numerical Analysis · Mathematics 2014-06-11 Elena Beretta , Maarten V. de Hoop , Lingyun Qiu , Otmar Scherzer

We establish a logarithmic stability inequality for the inverse problem of determining the non linear term, appearing in a semilinear BVP, from the corresponding Dirichlet-to-Neumann map (abbreviated to DtN map in the rest of this text).…

Analysis of PDEs · Mathematics 2020-09-08 Mourad Choulli , Guanghui Hu , Masahiro Yamamoto

We establish a double logarithmic stability inequality for the problem of determining the initial data in an IBVP for the wave equation outside a non-trapping obstacle from two localized measurements.

Analysis of PDEs · Mathematics 2024-01-05 Mourad Choulli

In this paper, we consider a variational formulation for the Dirichlet problem of the wave equation with zero boundary and initial conditions, where we use integration by parts in space and time. To prove unique solvability in a subspace of…

Numerical Analysis · Mathematics 2021-01-19 Olaf Steinbach , Marco Zank

We study the stability in the inverse problem of determining the time dependent zeroth-order coefficient $q(t,x)$ arising in the wave equation, from boundary observations. We derive, in dimension $n\geq 2$, a log-type stability estimate in…

Analysis of PDEs · Mathematics 2015-12-09 Ibtissem Ben Aïcha

For an initial-boundary value problem for a parabolic equation in the spatial variable $x=(x_1,.., x_n)$ and time $t$, we consider an inverse problem of determining a coefficient which is independent of one spatial component $x_n$ by extra…

Analysis of PDEs · Mathematics 2020-09-22 Oleg Yu. Imanuvilov , Yavar Kian , Masahiro Yamamoto

This paper is concerned with the inverse problem to recover the scalar, complex-valued refractive index of a medium from measurements of scattered time-harmonic electromagnetic waves at a fixed frequency. The main results are two…

Numerical Analysis · Mathematics 2017-02-09 Frederic Weidling , Thorsten Hohage

A 3D coefficient inverse problem for a hyperbolic equation with non-overdetermined data is considered. The forward problem is the Cauchy problems with the initial condition the delta function concentrated at a single plane (i.e. the plane…

Analysis of PDEs · Mathematics 2022-03-23 Michael V. Klibanov , Vladimir G. Romanov
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