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


Hasanov A.

IMA JOURNAL OF APPLIED MATHEMATICS, cilt.74, ss.1-19, 2009 (SCI İndekslerine Giren Dergi) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 74 Konu: 1
  • Basım Tarihi: 2009
  • Doi Numarası: 10.1093/imamat/hxn042
  • Dergi Adı: IMA JOURNAL OF APPLIED MATHEMATICS
  • Sayfa Sayıları: ss.1-19

Özet

The problem of determining the pair w := {F(x, t); f(t)} of source terms in the hyperbolic equation u(tt) = (k(x)u(x))(x) + F(x, t) and in the Neumann boundary condition k(0)u(x)(0, t) = f (t) from the measured data mu(x) := u(x, T) and/or nu(x) := u(t)(x, t) at the final time t = T is formulated. It is proved that both components of the Frechet gradient of the cost functionals J(1)(w) = ||u(x, t; w) - mu(x)||(0)(2) and J(2)(w) = ||u(t)(x, T; w) - nu(x)||(2)(0) 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.