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IMA Journal of Applied Mathematics Advance Access originally published online on December 18, 2008
IMA Journal of Applied Mathematics 2009 74(1):1-19; doi:10.1093/imamat/hxn042
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© The Author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

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

Alemdar Hasanov{dagger}

Department of Mathematics, Kocaeli University, 41380 Umittepe, Izmit–Kocaeli, Turkey

{dagger} Email: alemdar.hasanoglu{at}gmail.com

Received on June 18, 2007; Revision received August 24, 2008. Accepted on October 31, 2008

The problem of determining the pair w:={F(x, t);f(t)} of source terms in the hyperbolic equation utt = (k(x)ux)x + F(x, t) and in the Neumann boundary condition k(0)ux(0, t) = f(t) from the measured data µ(x):=u(x, T) and/or {nu}(x):=ut(x, t) at the final time t = T is formulated. It is proved that both components of the Fréchet gradient of the cost functionals J1(w) = ||u(x, t;w) – µ(x)||02 and J2(w) = ||ut(x, T;w) – {nu}(x)||02 can be found via the solutions of corresponding adjoint hyperbolic problems. Lipschitz continuity of the gradient is derived. Unicity of the solution and ill-conditionedness of the inverse problem are analysed. The obtained results permit one to construct a monotone iteration process, as well as to prove the existence of a quasi-solution.

Keywords: inverse source problems; hyperbolic equation; adjoint problem; Fréchet gradient; Lipschitz continuity.


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