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Reconstruction of kernel depending also on space variable
Author(s) -
Giorgi Claudio,
Guidetti Davide
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4914
Subject(s) - mathematics , kernel (algebra) , mathematical analysis , variable (mathematics) , viscoelasticity , boundary (topology) , domain (mathematical analysis) , convolution (computer science) , dirichlet boundary condition , dirichlet problem , type (biology) , boundary value problem , geometry , bounded function , pure mathematics , physics , ecology , machine learning , biology , artificial neural network , computer science , thermodynamics
The paper deals with the reconstruction of the convolution kernel, together with the solution, in a mixed linear evolution system of hyperbolic type. This problem describes uniaxial deformations u of a cylindrical domain (0, π ) × Ω, which is filled with a linear viscoelastic solid whose material properties are supposed to be uniform on Ω‐sections perpendicular to the x axis. Various types of boundary conditions in [0, T ] × {0, π } × Ω are prescribed, whereas Dirichlet conditions are assumed in [0, T ] × (0, π ) × ∂ Ω. To reconstruct both u and k , we suppose of knowing for any time t and any x ∈ (0, π ) the flux of the viscoelastic stress vector through the boundary of the Ω‐section. The main novelty is that the unknown kernel k is allowed to depend, not only on the time variable t but also on the space variable x .