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A general coupled equation approach for solving the biharmonic boundary value problem. (English) Zbl 0237.65067

Summary: The biharmonic boundary value problem with Dirichlet boundary conditions is reduced to a coupled system of Poisson equations, which depends upon an arbitrary, positive coupling constant \(c\). Since each of the Poisson equations is well-posed, the system may be solved by iteration. We show that the iterates may be represented as a linear combination of the eigenfunctions of the “Dirichlet eigenvalue problem” (a fourth order boundary value problem with the eigenvalue in the boundary condition). Convergence of the iterative scheme occurs when \(0 < c < 2v_1\), where \(v_1\) is the smallest eigenvalue. By making use of an averaging scheme, convergence may be produced for any positive \(c\). With the proper choice of \(c\), the rate of convergence may be increased. This coupled equation approach includes the finite difference approach as a special case.

MSC:

65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
31A25 Boundary value and inverse problems for harmonic functions in two dimensions
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