Quasi-separation of the Biharmonic Partial Differential Equation

In this paper we consider analytical and numerical solutions to the Dirichlet boundary value problem for the biharmonic partial differential equation, on a disk of finite radius in the plane. The physical interpretation of these solutions is that of the harmonic oscillations of a thin, clamped plate. For the linear, fourth-order, biharmonic partial differential equation in the plane, it is well known that the solution method of separation in polar co-ordinates, is not possible, in general. However, in this paper, for circular domains in the plane, it is shown that a method, here called quasi-separation of variables, does lead to solutions of the partial differential equation. These solutions are products of solutions of two ordinary linear differential equations; a fourth-order radial equation, and a second-order angular differential equation. To be expected, without complete separation of the polar variables, there is some restriction on the range of these solutions in comparison with the corresponding separated solutions of the second-order harmonic differential equation in the plane. Notwithstanding these restrictions the quasi-separation method leads to solutions of the Dirichlet boundary value problem on a disk with centre at the origin, with boundary conditions determined by the solution and its inward drawn normal taking the values zero on the edge of the disk. One significant feature for these biharmonic boundary value problems, in general, follows from the form of the biharmonic differential expression when represented in polar co-ordinates. In this form the differential expression has a singularity at the origin, in the radial variable. This singularity translates to a singularity at the origin of the fourth-order radial separated equation; this singularity necessitates the application of a third boundary condition in order to determine a self-adjoint solution to the Dirichlet boundary value problem. The penultimate section of the paper reports on numerical solutions to the Dirichlet boundary value problem; these results are also presented graphically. Two specific cases are studied in detail and numerical values of the eigenvalues are compared with results obtained in earlier studies. Date: 07 August 2008 (File: C:\Swp55\Biharmonic\biharmonic12.tex). 2000 Mathematics Subject Classification. Primary 31A30, 34B05, 74K20; Secondary 31A25, 31B35, 35P10.