## 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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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

### Keywords

- singular mixed boundary value problem
- positive solution
- shallow membrane
- collocation method
- lower and upper functions