A unified approach to singular problems arising in the membrane theory

Irena Rachůnková; Gernot Pulverer; Ewa B. Weinmüller

Applications of Mathematics (2010)

  • Volume: 55, Issue: 1, page 47-75
  • ISSN: 0862-7940

Abstract

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We consider the singular boundary value problem ( t n u ' ( t ) ) ' + t n f ( t , u ( t ) ) = 0 , lim t → 0 + t n u ' ( t ) = 0 , 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 , ∞ ) , 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.

How to cite

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Rachůnková, Irena, Pulverer, Gernot, and Weinmüller, Ewa B.. "A unified approach to singular problems arising in the membrane theory." Applications of Mathematics 55.1 (2010): 47-75. <http://eudml.org/doc/37838>.

@article{Rachůnková2010,
abstract = {We consider the singular boundary value problem \[ (t^nu^\{\prime \}(t))^\{\prime \}+ t^nf(t,u(t))=0, \quad \lim \_\{t\rightarrow 0+\}t^nu^\{\prime \}(t)=0, \quad a\_0u(1)+a\_1u^\{\prime \}(1-)=A, \] where $f(t,x)$ is a given continuous function defined on the set $(0,1]\times (0,\infty )$ which can have a time singularity at $t=0$ and a space singularity at $x=0$. Moreover, $n\in \mathbb \{N\}$, $n\ge 2$, and $a_0$, $a_1$, $A$ are real constants such that $a_0\in (0,\infty )$, whereas $a_1,A\in [0,\infty )$. 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.},
author = {Rachůnková, Irena, Pulverer, Gernot, Weinmüller, Ewa B.},
journal = {Applications of Mathematics},
keywords = {singular mixed boundary value problem; positive solution; shallow membrane; collocation method; lower and upper functions; singular mixed boundary value problem; positive solution; shallow membrane; collocation method; lower and upper functions},
language = {eng},
number = {1},
pages = {47-75},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A unified approach to singular problems arising in the membrane theory},
url = {http://eudml.org/doc/37838},
volume = {55},
year = {2010},
}

TY - JOUR
AU - Rachůnková, Irena
AU - Pulverer, Gernot
AU - Weinmüller, Ewa B.
TI - A unified approach to singular problems arising in the membrane theory
JO - Applications of Mathematics
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 55
IS - 1
SP - 47
EP - 75
AB - We consider the singular boundary value problem \[ (t^nu^{\prime }(t))^{\prime }+ t^nf(t,u(t))=0, \quad \lim _{t\rightarrow 0+}t^nu^{\prime }(t)=0, \quad a_0u(1)+a_1u^{\prime }(1-)=A, \] where $f(t,x)$ is a given continuous function defined on the set $(0,1]\times (0,\infty )$ which can have a time singularity at $t=0$ and a space singularity at $x=0$. Moreover, $n\in \mathbb {N}$, $n\ge 2$, and $a_0$, $a_1$, $A$ are real constants such that $a_0\in (0,\infty )$, whereas $a_1,A\in [0,\infty )$. 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.
LA - eng
KW - singular mixed boundary value problem; positive solution; shallow membrane; collocation method; lower and upper functions; singular mixed boundary value problem; positive solution; shallow membrane; collocation method; lower and upper functions
UR - http://eudml.org/doc/37838
ER -

References

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