Simultaneous determination of source terms in a linear parabolic problem from the final overdetermination: Weak solution approach

By Alemdar Hasanov. 2007

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The problem of determining the pair w:={F(x,t);T0(t)} of source terms in the parabolic equation ut=(k(x)ux)x+F(x,t) and Robin boundary condition −k(l)ux(l,t)=v[u(l,t)−T0(t)] from the measured final data μT(x)=u(x,T) is formulated. It is proved that both components of the Frechet gradient of the cost functional J(w)=‖μT(x)−u(x,T:w)‖02 can be found via the same solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is derived. The obtained results permit one to prove existence of a quasi-solution of the considered inverse problem, as well as to construct a monotone iteration scheme based on a gradient method.

Subfields: Biomedical Engineering, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fb2518bd55b7ddf6ec3bd331570fb7fc46ed1208