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

top
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

top

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

top
  1. Agarwal, R. P., O'Regan, D., 10.1016/S0020-7462(03)00041-6, Int. J. Non-Linear Mech. 39 (2004), 779-784. (2004) MR2036912DOI10.1016/S0020-7462(03)00041-6
  2. Agarwal, R. P., O'Regan, D., Singular problems arising in circular membrane theory, Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 10 (2003), 965-972. (2003) Zbl1056.34029MR2008758
  3. Agarwal, R. P., Staněk, S., 10.1016/S0898-1221(03)90239-2, Comput. Math. Appl. 46 (2003), 1827-1837. (2003) MR2018768DOI10.1016/S0898-1221(03)90239-2
  4. Auzinger, W., Koch, O., Weinmüller, E., 10.1023/A:1021151821275, Numer. Algorithms 31 (2002), 5-25. (2002) MR1950909DOI10.1023/A:1021151821275
  5. Auzinger, W., Kneisl, G., Koch, O., Weinmüller, E., 10.1023/A:1025531130904, Numer. Algorithms 33 (2003), 27-39. (2003) MR2005549DOI10.1023/A:1025531130904
  6. Auzinger, W., Koch, O., Weinmüller, E., 10.1137/S0036142902418928, SIAM J. Numer. Anal. 42 (2005), 2366-2386. (2005) MR2139397DOI10.1137/S0036142902418928
  7. Auzinger, W., Koch, O., Weinmüller, E., 10.1016/j.cam.2004.10.013, J. Comput. Appl. Math. 180 (2005), 213-227. (2005) MR2141496DOI10.1016/j.cam.2004.10.013
  8. Baxley, J. V., Robinson, S. B., 10.1016/S0377-0427(97)00216-1, J. Comput. Appl. Math. 88 (1998), 203-224. (1998) Zbl0928.74054MR1609149DOI10.1016/S0377-0427(97)00216-1
  9. Budd, C. J., Koch, O., Weinmüller, E., Self-Similar Blow-Up in Nonlinear {PDE}s. AURORA TR-2004-15, Institute for Analysis and Scientific Computing, Vienna Univ. of Technology, Austria (2004), available at http://www.vcpc.univie.ac.at/aurora/publications/. (2004) 
  10. Budd, C. J., Koch, O., Weinmüller, E., 10.1007/s00607-005-0157-8, Computing 77 (2006), 335-346. (2006) MR2244946DOI10.1007/s00607-005-0157-8
  11. Budd, C. J., Koch, O., Weinmüller, E., 10.1016/j.apnum.2005.04.012, Appl. Numer. Math. 56 (2006), 413-422. (2006) MR2207599DOI10.1016/j.apnum.2005.04.012
  12. Coster, C. De, Habets, P., 10.1016/S1874-5725(00)80004-8, Handbook of Differential Equations, Ordinary Differential Equations, Vol. I A. Caňada, P. Drábek, A. Fonda Elsevier/North Holland Amsterdam (2004), 69-161. (2004) MR2166490DOI10.1016/S1874-5725(00)80004-8
  13. Hoog, F. de, Weiss, R., 10.1137/0715013, SIAM J. Numer. Anal. 15 (1978), 198-217. (1978) Zbl0398.65051MR0468203DOI10.1137/0715013
  14. Dickey, R. W., 10.1090/qam/1012280, Q. Appl. Math. 47 (1989), 571-581. (1989) Zbl0683.73022MR1012280DOI10.1090/qam/1012280
  15. Johnson, K. N., 10.1090/qam/1466147, Q. Appl. Math. 55 (1997), 537-550. (1997) Zbl0885.73027MR1466147DOI10.1090/qam/1466147
  16. Kannan, R., O'Regan, D., Singular and nonsingular boundary value problems with sign changing nonlinearities, J. Inequal. Appl. 5 (2000), 621-637. (2000) Zbl0976.34017MR1812574
  17. Kiguradze, I. T., Shekhter, B. L., Singular boundary value problems for second order ordinary differential equations, Itogi Nauki Tekh., Ser. Sovrm. Probl. Mat. 30 (1987), 105-201 Russian. (1987) Zbl0631.34021MR0925830
  18. Kitzhofer, G., Numerical treatment of implicit singular {BVP}s, PhD. Thesis Institute for Analysis and Scientific Computing, Vienna Univ. of Technology, Austria. In preparation. 
  19. Kitzhofer, G., Koch, O., Weinmüller, E., Collocation methods for the computation of bubble-type solutions of a singular boundary value problem in hydrodynamics, J. Sci. Comput (to appear). Available at http://www.math.tuwien.ac.at/ {ewa}. MR2335788
  20. Koch, O., 10.1007/s00211-005-0617-2, Numer. Math. 101 (2005), 143-164. (2005) Zbl1076.65073MR2194722DOI10.1007/s00211-005-0617-2
  21. Rachůnková, I., Koch, O., Pulverer, G., Weinmüller, E., 10.1016/j.jmaa.2006.10.006, J. Math. Anal. Appl. 332 (2007), 523-541. (2007) MR2319681DOI10.1016/j.jmaa.2006.10.006
  22. Rachůnková, I., Staněk, S., Tvrdý, M., Singularities and Laplacians in Boundary Value Problems for Nonlinear Ordinary Differential Equations. Handbook of Differential Equations. Ordinary Differential Equations, Vol. 3, A. Caňada, P. Drábek, A. Fonda Elsevier Amsterdam (2006). (2006) MR2457638
  23. Ascher, U. M., Mattheij, R. M. M., Russell, R. D., Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Prentice-Hall Englewood Cliffs (1988). (1988) Zbl0671.65063MR1000177
  24. Weinmüller, E., 10.1137/0723074, SIAM J. Numer. Anal. 23 (1986), 1062-1095. (1986) MR0859018DOI10.1137/0723074

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.