Volume 12, Number 3, Fall 2004
SINC SOLUTION OF BIHARMONIC PROBLEMS
FRANK STENGER, THOMAS COOK AND ROBERT M. KIRBY
Abstract. In this paper we solve two biharmonic problems
over a square, B = (1, 1) × (1, 1). (1) The problem
∇^{4}U = f, for which we determine a particular solution, U,
given f, via use of Sinc convolution; and (2) The boundary
value problem ∇^{4}V = 0 for which we determine V given V = g
and normal derivative V_{n} = h on ∂B, the boundary of B. The
solution to this problem is carried out based on the identity
V = ℜ { (z  c) E + F } = (x  a) u + (y  b) v + Φ,
where E = u + i v and F = Φ + iψ are functions analytic in
B, and where c = a + i b is an arbitrary constant. We thus
determine approximations to the harmonic functions u, v and
Φ on ∂B, via use of Sinc quadrature, and Sinc approximation
of derivatives. We then use a special, explicit Sincbased analytic
continuation procedure to extend the functions u, v and
Φ to the interior of B. These procedures enable us to determine
functions W which solve a boundary problem of the form
∇^{4}W = f in B, given f in B and given W and its normal
derivative, W_{n} on the boundary of B.
Given any C > 0, the time complexity of sequential computation
of an approximation of W_{ε} to W to within a uniform error
of ε in B, i.e., such that sup_{(x,y)∈B}  W(x, y)  W_{ε}(x, y)  < ε,
is O((log(ε))^{6}).
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