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Inverse Problem Related to Boundary Shape Identification for a Hyperbolic Differential Equation

Author

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  • Fagueye Ndiaye
  • Idrissa Ly
  • Remi Léandre

Abstract

In this paper, we are interested in the inverse problem of the determination of the unknown part ∂Ω,Γ0 of the boundary of a uniformly Lipschitzian domain Ω included in ℠N from the measurement of the normal derivative ∂nv on suitable part Γ0 of its boundary, where v is the solution of the wave equation ∂ttvx,t−Δvx,t+pxvx=0 in Ω×0,T and given Dirichlet boundary data. We use shape optimization tools to retrieve the boundary part Γ of ∂Ω. From necessary conditions, we estimate a Lagrange multiplier kΩ which appears by derivation with respect to the domain. By maximum principle theory for hyperbolic equations and under geometrical assumptions, we prove a uniqueness result of our inverse problem. The Lipschitz stability is established by increasing of the energy of the system. Some numerical simulations are made to illustrate the optimal shape.

Suggested Citation

  • Fagueye Ndiaye & Idrissa Ly & Remi Léandre, 2021. "Inverse Problem Related to Boundary Shape Identification for a Hyperbolic Differential Equation," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2021, pages 1-12, October.
    • Handle: RePEc:hin:jijmms:1716027
    DOI: 10.1155/2021/1716027
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    Cited by:

    1. Roman Chapko & Leonidas Mindrinos, 2022. "On the Numerical Solution of a Hyperbolic Inverse Boundary Value Problem in Bounded Domains," Mathematics, MDPI, vol. 10(5), pages 1-11, February.

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