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I. Rachunkova, G. Pulverer, E. Weinmüller:
"A unified approach to singular problems arising in the membrane theory";
in: "ASC Report 26/2007", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2007, ISBN: 978-3-902627-00-1.

We consider the singular boundary value problem
(tnu′(t))′ + tnf(t, u(t)) = 0, lim
t→0+
tnu′(t) = 0, a0u(1) + a1u′(1−) = A,
where f(t, x) is a given continuous function defined on the set (0, 1] × (0,∞)
which can have a time singularity at t = 0 and a space singularity at x = 0.
Moreover, n ∈ N, n ≥ 2, and a0, a1,A are real constants such that a0 ∈ (0,∞),
whereas a1, A ∈ [0,∞). The main aim of this paper is to discuss the existence of
solutions to the above problem and apply these general results to cover certain
classes of singular problems arising in the theory of shallow membrane caps, where
we are especially interested in characterizing positive solutions. We illustrate the
analytical findings by numerical simulations based on polynomial collocation.

Singular mixed boundary value problem, positive solution, shallow

http://www.asc.tuwien.ac.at/preprint/2007/asc26x2007.pdf