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) = µ(x) is formulated. It is proved that the Fréchet derivative of the cost functional J(F) = ‖µ T (x) − u(x, T)‖ 20 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.
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Kaya, M. Determination of the unknown source term in a linear parabolic problem from the measured data at the final time. Appl Math 59, 715–728 (2014). https://doi.org/10.1007/s10492-014-0081-3
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DOI: https://doi.org/10.1007/s10492-014-0081-3