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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 ∇4U = f, for which we determine a particular solution, U, given f, via use of Sinc convolution; and (2) The boundary value problem ∇4V = 0 for which we determine V given V = g and normal derivative Vn = 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 Sinc-based 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 ∇4W = f in B, given f in B and given W and its normal derivative, Wn on the boundary of B. Given any C > 0, the time complexity of sequential computation of an approximation of Wε to W to within a uniform error of ε in B, i.e., such that sup(x,y)∈B | W(x, y) - Wε(x, y) | < ε, is O((log(ε))6).

 

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