Simultaneous determination of the source terms in a linear hyperbolic problem from the final overdetermination: weak solution approach |
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Authors: | Hasanov Alemdar |
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Institution: |
Department of Mathematics, Kocaeli University, 41380 Umittepe, Izmit–Kocaeli, Turkey
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Abstract: | The problem of determining the pair w:={F(x, t);f(t)} of sourceterms in the hyperbolic equation utt = (k(x)ux)x + F(x, t) andin the Neumann boundary condition k(0)ux(0, t) = f(t) from themeasured data µ(x):=u(x, T) and/or (x):=ut(x, t) at thefinal time t = T is formulated. It is proved that both componentsof the Fréchet gradient of the cost functionals J1(w)= ||u(x, t;w) – µ(x)||02 and J2(w) = ||ut(x, T;w)– (x)||02 can be found via the solutions of correspondingadjoint hyperbolic problems. Lipschitz continuity of the gradientis derived. Unicity of the solution and ill-conditionednessof the inverse problem are analysed. The obtained results permitone to construct a monotone iteration process, as well as toprove the existence of a quasi-solution. |
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Keywords: | inverse source problems hyperbolic equation adjoint problem Fré chet gradient Lipschitz continuity |
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