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.

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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 $u(x,T)=\mu (x)$ 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 
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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, Tome 59 (2014) no. 6, pp. 715-728. doi : 10.1007/s10492-014-0081-3. http://geodesic.mathdoc.fr/articles/10.1007/s10492-014-0081-3/

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