Determination of the unknown source term in a linear parabolic problem from the measured data at the final time

Müjdat Kaya

Applications of Mathematics (2014)

  • Volume: 59, Issue: 6, page 715-728
  • ISSN: 0862-7940

Abstract

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The problem of determining the source term F ( x , t ) in the linear parabolic equation u t = ( k ( x ) u x ( x , t ) ) x + F ( x , t ) from the measured data at the final time u ( x , T ) = μ ( x ) is formulated. It is proved that the Fréchet derivative of the cost functional J ( F ) = ∥ μ T ( x ) - u ( x , T ) ∥ 0 2 can be formulated via the solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is proved. An existence result for a quasi solution of the considered inverse problem is proved. A monotone iteration scheme is obtained based on the gradient method. Convergence rate is proved.

How to cite

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Kaya, Müjdat. "Determination of the unknown source term in a linear parabolic problem from the measured data at the final time." Applications of Mathematics 59.6 (2014): 715-728. <http://eudml.org/doc/262017>.

@article{Kaya2014,
abstract = {The problem of determining the source term $F(x,t)$ in the linear parabolic equation $u_t=(k(x)u_x(x,t))_x + F(x,t)$ from the measured data at the final time $u(x,T)=\mu (x)$ is formulated. It is proved that the Fréchet derivative of the cost functional $J(F) = \Vert \mu _T(x)- u(x,T)\Vert _\{0\}^2$ can be formulated via the solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is proved. An existence result for a quasi solution of the considered inverse problem is proved. A monotone iteration scheme is obtained based on the gradient method. Convergence rate is proved.},
author = {Kaya, Müjdat},
journal = {Applications of Mathematics},
keywords = {inverse parabolic problem; unknown source; adjoint problem; Fréchet derivative; Lipschitz continuity; inverse parabolic problem; unknown source; Dirichlet boundary conditions; adjoint problem; Fréchet derivative; Lipschitz continuity},
language = {eng},
number = {6},
pages = {715-728},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Determination of the unknown source term in a linear parabolic problem from the measured data at the final time},
url = {http://eudml.org/doc/262017},
volume = {59},
year = {2014},
}

TY - JOUR
AU - Kaya, Müjdat
TI - Determination of the unknown source term in a linear parabolic problem from the measured data at the final time
JO - Applications of Mathematics
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 6
SP - 715
EP - 728
AB - The problem of determining the source term $F(x,t)$ in the linear parabolic equation $u_t=(k(x)u_x(x,t))_x + F(x,t)$ from the measured data at the final time $u(x,T)=\mu (x)$ is formulated. It is proved that the Fréchet derivative of the cost functional $J(F) = \Vert \mu _T(x)- u(x,T)\Vert _{0}^2$ can be formulated via the solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is proved. An existence result for a quasi solution of the considered inverse problem is proved. A monotone iteration scheme is obtained based on the gradient method. Convergence rate is proved.
LA - eng
KW - inverse parabolic problem; unknown source; adjoint problem; Fréchet derivative; Lipschitz continuity; inverse parabolic problem; unknown source; Dirichlet boundary conditions; adjoint problem; Fréchet derivative; Lipschitz continuity
UR - http://eudml.org/doc/262017
ER -

References

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