scholarly journals On the determination of the principal coefficient from boundary measurements in a KdV equation

2014 ◽  
Vol 22 (6) ◽  
Author(s):  
Lucie Baudouin ◽  
Eduardo Cerpa ◽  
Emmanuelle Crépeau ◽  
Alberto Mercado
Keyword(s):  
Inverse Problem ◽  
Kdv Equation ◽  
Carleman Estimate ◽  
Lipschitz Stability ◽  
Single Solution ◽  
De Vries ◽  
Boundary Measurements ◽  
Global Carleman Estimate

AbstractThis paper concerns the inverse problem of retrieving the principal coefficient in a Korteweg–de Vries (KdV) equation from boundary measurements of a single solution. The Lipschitz stability of this inverse problem is obtained using a new global Carleman estimate for the linearized KdV equation. The proof is based on the Bukhgeĭm–Klibanov method.

Lipschitz stability for a semi-linear inverse stochastic transport problem

2020 ◽  
Vol 28 (2) ◽  
pp. 185-193
Author(s):  
Zhongqi Yin
Keyword(s):  
Inverse Problem ◽  
Main Tool ◽  
Terminal State ◽  
Lipschitz Stability ◽  
Unknown Parameters ◽  
Initial Displacement ◽  
Stochastic Transport ◽  
Random Term ◽  
Global Carleman Estimate ◽  
Stochastic Transport Equation

AbstractThis paper is addressed to a semi-linear stochastic transport equation with three unknown parameters. It is proved that the initial displacement, the terminal state and the random term in diffusion are uniquely determined by the state on partial boundary and a Lipschitz stability of the inverse problem is established. The main tool we employ is a global Carleman estimate for stochastic transport equations.

scholarly journals Global Carleman estimate on a network for the wave equation and application to an inverse problem

2011 ◽  
Vol 1 (3) ◽  
pp. 307-330 ◽  
Author(s):  
Lucie Baudouin ◽  
◽  
Emmanuelle Crépeau ◽  
Julie Valein ◽  
◽  
...  
Keyword(s):  
Inverse Problem ◽  
Wave Equation ◽  
Carleman Estimate ◽  
Global Carleman Estimate

A NEW UNIQUE CONTINUATION PROPERTY FOR THE KORTEWEG–DE VRIES EQUATION

2014 ◽  
Vol 90 (1) ◽  
pp. 90-98 ◽  
Author(s):  
MO CHEN ◽  
PENG GAO
Keyword(s):  
Vries Equation ◽  
Finite Interval ◽  
Unique Continuation ◽  
Carleman Estimate ◽  
Unique Continuation Property ◽  
De Vries ◽  
Continuation Property ◽  
Global Carleman Estimate

AbstractThe aim of this paper is to obtain a new unique continuation property (UCP) for the Korteweg–de Vries equation posed on a finite interval. Compared with the previous UCP, we need fewer conditions on the solution. For this purpose, we have to establish a global Carleman estimate for the Korteweg–de Vries equation.

scholarly journals Inverse Problem Related to Boundary Shape Identification for a Hyperbolic Differential Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Fagueye Ndiaye ◽  
Idrissa Ly
Keyword(s):  
Inverse Problem ◽  
Dirichlet Boundary ◽  
Hyperbolic Equations ◽  
Normal Derivative ◽  
Uniqueness Result ◽  
Lipschitz Stability ◽  
Shape Identification ◽  
Boundary Part ◽  
Principle Theory

In this paper, we are interested in the inverse problem of the determination of the unknown part ∂ Ω , Γ 0 of the boundary of a uniformly Lipschitzian domain Ω included in ℝ N from the measurement of the normal derivative ∂ n v on suitable part Γ 0 of its boundary, where v is the solution of the wave equation ∂ t t v x , t − Δ v x , t + p x v x = 0 in Ω × 0 , T and given Dirichlet boundary data. We use shape optimization tools to retrieve the boundary part Γ of ∂ Ω . From necessary conditions, we estimate a Lagrange multiplier k Ω which appears by derivation with respect to the domain. By maximum principle theory for hyperbolic equations and under geometrical assumptions, we prove a uniqueness result of our inverse problem. The Lipschitz stability is established by increasing of the energy of the system. Some numerical simulations are made to illustrate the optimal shape.

scholarly journals On Inverse Problems for Characteristic Sources in Helmholtz Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-16 ◽  
Author(s):  
Carlos J. S. Alves ◽  
Roberto Mamud ◽  
Nuno F. M. Martins ◽  
Nilson C. Roberty
Keyword(s):  
Inverse Problem ◽  
Direct Problem ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
External Boundary ◽  
Equivalent Formulation ◽  
Helmholtz Equations ◽  
Functional Formulation ◽  
Boundary Measurements

We consider the inverse problem that consists in the determination of characteristic sources, in the modified and classical Helmholtz equations, based on external boundary measurements. We identify the location of the barycenter establishing a simple formula for symmetric shapes, which also holds for the determination of a single source point. We use this for the reconstruction of the characteristic source, based on the Method of Fundamental Solutions (MFS). The MFS is also applied as a solver for the direct problem, using an equivalent formulation as a jump or transmission problem. As a solver for the inverse problem, we may apply minimization using an equivalent reciprocity functional formulation. Numerical experiments with the barycenter and the boundary reconstructions are presented.

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