High Precision Solutions of Two Fourth Order Eigenvalue Problems Eigenvalue Problems

摘要

We solve the biharmonic eigenvalue problem Δ~2=λu and the bucking plate problem　Δ~u=-λΔu on the unit square using a highly accurate spectral Legendre-Galerkin method. We study the nodal lines for the first eigenfunction near a comer for the two problens. Five sign changes are comprted and the results show that the eigenfunction exhibits a self similar pattern as one approaches the corner. The amplitudes of the extremal values and the coordinates of their location as measured from the corner are reduced by constant factors. These tesuits are compared with the known asymptotic expansion of the solution near a corner. This comparison shows that the asymptotic expansion is highly accurate already from the first sign change as we have complete agreement between the numerical and the analytical results. Thus, we have an accurate description of the eigenfunction in the entire domain.