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A unified approach to singular problems arising in the membrane theory

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Abstract

We consider the singular boundary value problem \(({t^n}u't))' + {t^n}f(t,u(t)) = 0,{\rm{ }}\mathop {\lim }\limits_{t \to 0 + } {t^n}u'(t) = 0,{\rm{ }}{a_0}u(1) + {a_1}u'(1 - ) = A,\) 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 a 0, a 1, A are real constants such that a 0 ∈ (0,1), whereas a 1,A ∈ [0,∞). The main aim of this paper is to discuss the existence of solutions to the above problem and apply the 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.

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Authors and Affiliations

  1. Department of Mathematical Analysis, Faculty of Science, Palacký University, tř 17. listopadu 12, 771 46, Olomouc, Czech Republic

    Irena Rachůnková

  2. Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstrasse 6-10, A-1040, Vienna, Austria

    Gernot Pulverer & Ewa B. Weinmüller

Authors
  1. Irena Rachůnková
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  2. Gernot Pulverer
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  3. Ewa B. Weinmüller
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Correspondence to Irena Rachůnková.

Additional information

The first author was supported by the grant No. A100190703 of the Grant Agency of the Academy of Sciences of the Czech Republic and by the Council of Czech Government MSM 6198959214; the second and the third author were supported by the Austrian Science Fund Project P17253.

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Rachůnková, I., Pulverer, G. & Weinmüller, E.B. A unified approach to singular problems arising in the membrane theory. Appl Math 55, 47–75 (2010). https://doi.org/10.1007/s10492-010-0002-z

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  • DOI: https://doi.org/10.1007/s10492-010-0002-z

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