Geodesic
Determination of the unknown source term in a linear parabolic problem from the measured data at the final time
Applications of Mathematics, Tome 59 (2014) no. 6, pp. 715-728 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 $J(F) = \|\mu _T(x)- u(x,T)\|_{0}^2$ 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.
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 $J(F) = \|\mu _T(x)- u(x,T)\|_{0}^2$ 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.
DOI : 10.1007/s10492-014-0081-3
Classification : 35K10, 35R30
Keywords: inverse parabolic problem; unknown source; adjoint problem; Fréchet derivative; Lipschitz continuity 
@article{10_1007_s10492_014_0081_3,
     author = {Kaya, M\"ujdat},
     title = {Determination of the unknown source term in a linear parabolic problem from the measured data at the final time},
     journal = {Applications of Mathematics},
     pages = {715--728},
     year = {2014},
     volume = {59},
     number = {6},
     doi = {10.1007/s10492-014-0081-3},
     mrnumber = {3277735},
     zbl = {06391458},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10492-014-0081-3/}
}
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
SP  - 715
EP  - 728
VL  - 59
IS  - 6
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10492-014-0081-3/
DO  - 10.1007/s10492-014-0081-3
LA  - en
ID  - 10_1007_s10492_014_0081_3
ER  - 
%0 Journal Article
%A Kaya, Müjdat
%T Determination of the unknown source term in a linear parabolic problem from the measured data at the final time
%J Applications of Mathematics
%D 2014
%P 715-728
%V 59
%N 6
%U http://geodesic.mathdoc.fr/articles/10.1007/s10492-014-0081-3/
%R 10.1007/s10492-014-0081-3
%G en
%F 10_1007_s10492_014_0081_3
Kaya, Müjdat. Determination of the unknown source term in a linear parabolic problem from the measured data at the final time. Applications of Mathematics, Tome 59 (2014) no. 6, pp. 715-728. doi: 10.1007/s10492-014-0081-3

Cité par Sources :