A truncated spectral regularization method for a source identification problem
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Abstract inverse source problem of identifying the source function f in the abstract Cauchy problem \(u_t+Au=f(t),\, 0<t<\tau \) with \(u(0)=\phi _0\) when the data, the final value, \(u(\tau )=\phi _\tau \) is noisy is considered, where A is a densly defined self-adjont coercive unbounded operator on a Hilbert space H. This problem is known to be an ill-posed problem. A truncated spectral representation of a mild solution of the above problem is shown to be a regularized approximation, and error analysis is carried out when \(\phi _\tau \) is noisy as well as exact, and stability estimate is given under appropriate parameter choice strategies.
KeywordsParabolic equations Semigroups Ill-posed problems Regularization
Mathematics Subject Classification35K90 47D06 47A52 65F22
Ajoy Jana acknowledges the support received from the University Grant Commission, Government of India, for financial support. Sanction no is Sr. No. F.2-12/2002(SA-I), Ref No: Acad./R3/J.Rpt/2014.