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In this paper we propose two numerical algorithms to derive the extremalprincipal eigenvalue of the bi-Laplacian operator under Navier boundaryconditions or Dirichlet boundary conditions. Consider a non-homogeneous hingedor clamped plate $\Omega$, the algorithms converge to the density functions on$\Omega$ which they yield the maximum or minimum basic frequency of the plate.

Extremal basic frequency of non-homogeneous plates