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Analysis and numerical solutions of positive and dead core solutions of singular Sturm-Liouville problems. (English) Zbl 1203.34046

Summary: We investigate the singular Sturm-Liouville problem
\[ u''=\lambda g(u),\;u'(0)=0,\;\beta u'(1)+\alpha u(1)=A, \]
where \(\lambda\) is a nonnegative parameter, \(\beta\geq 0\), \(\alpha >0\), and \(A>0\). We discuss the existence of multiple positive solutions and show that for certain values of \(\lambda\), there also exist solutions that vanish on a subinterval \([0,\rho]\subset [0,1)\), the so-called dead core solutions. The theoretical findings are illustrated by computational experiments for \(g(u)=1/\sqrt{u}\) and for some model problems from the class of singular differential equations \((\phi(u'))'+f(t,u')=\lambda g(t,u,u')\). For the numerical simulation, the collocation method implemented in our MATLAB code bvpsuite has been applied.

MSC:

34B24 Sturm-Liouville theory
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations

Software:

Matlab; Sbvp
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aris R: The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts. Clarendon Press, Oxford, UK; 1975. · Zbl 0315.76051
[2] Baxley JV, Gersdorff GS: Singular reaction-diffusion boundary value problems.Journal of Differential Equations 1995,115(2):441-457. 10.1006/jdeq.1995.1022 · Zbl 0815.35019 · doi:10.1006/jdeq.1995.1022
[3] Agarwal, RP; O’Regan, D.; Staněk, S., Dead core problems for singular equations with [InlineEquation not available: see fulltext.]-Laplacian, No. 2007, 16 (2007)
[4] Staněk S, Pulverer G, Weinmüller EB: Analysis and numerical simulation of positive and dead-core solutions of singular two-point boundary value problems.Computers & Mathematics with Applications 2008,56(7):1820-1837. 10.1016/j.camwa.2008.03.029 · Zbl 1152.34320 · doi:10.1016/j.camwa.2008.03.029
[5] Agarwal RP, O’Regan D, Staněk S:Positive and dead core solutions of singular Dirichlet boundary value problems with[InlineEquation not available: see fulltext.]-Laplacian.Computers & Mathematics with Applications 2007,54(2):255-266. 10.1016/j.camwa.2006.12.026 · Zbl 1134.34010 · doi:10.1016/j.camwa.2006.12.026
[6] Agarwal RP, O’Regan D, Staněk S:Dead cores of singular Dirichlet boundary value problems with[InlineEquation not available: see fulltext.]-Laplacian.Applications of Mathematics 2008,53(4):381-399. 10.1007/s10492-008-0031-z · Zbl 1199.34076 · doi:10.1007/s10492-008-0031-z
[7] Bobisud LE: Asymptotic dead cores for reaction-diffusion equations.Journal of Mathematical Analysis and Applications 1990,147(1):249-262. 10.1016/0022-247X(90)90396-W · Zbl 0706.34052 · doi:10.1016/0022-247X(90)90396-W
[8] Bobisud LE: Behavior of solutions for a Robin problem.Journal of Differential Equations 1990,85(1):91-104. 10.1016/0022-0396(90)90090-C · Zbl 0704.34033 · doi:10.1016/0022-0396(90)90090-C
[9] Bobisud LE, O’Regan D, Royalty WD: Existence and nonexistence for a singular boundary value problem.Applicable Analysis 1988,28(4):245-256. 10.1080/00036818808839765 · Zbl 0628.34025 · doi:10.1080/00036818808839765
[10] Auzinger W, Koch O, Weinmüller E: Efficient collocation schemes for singular boundary value problems.Numerical Algorithms 2002,31(1-4):5-25. 10.1023/A:1021151821275 · Zbl 1021.65038 · doi:10.1023/A:1021151821275
[11] Auzinger W, Kneisl G, Koch O, Weinmüller E: A collocation code for singular boundary value problems in ordinary differential equations.Numerical Algorithms 2003,33(1-4):27-39. 10.1023/A:1025531130904 · Zbl 1030.65089 · doi:10.1023/A:1025531130904
[12] Kitzhofer G: Numerical treatment of implicit singular BVPs, Ph.D. thesis. Institute for Analysis and Scientific Computing, Vienna University of Technology, Vienna, Austria; · Zbl 1152.34320
[13] Budd CJ, Koch O, Weinmüller E: Self-similar blow-up in nonlinear PDEs. Institute for Analysis and Scientific Computing, Vienna University of Technology, Vienna, Austria; 2004.
[14] Budd CJ, Koch O, Weinmüller E: Computation of self-similar solution profiles for the nonlinear Schrödinger equation.Computing 2006,77(4):335-346. 10.1007/s00607-005-0157-8 · Zbl 1122.65066 · doi:10.1007/s00607-005-0157-8
[15] Budd CJ, Koch O, Weinmüller E: Fron nonlinear PDEs to singular ODEs.Applied Numerical Mathematics 2006,56(3-4):413-422. 10.1016/j.apnum.2005.04.012 · Zbl 1089.65104 · doi:10.1016/j.apnum.2005.04.012
[16] Kitzhofer G, Koch O, Lima P, Weinmüller E: Efficient numerical solution of the density profile equation in hydrodynamics.Journal of Scientific Computing 2007,32(3):411-424. 10.1007/s10915-007-9141-0 · Zbl 1178.76280 · doi:10.1007/s10915-007-9141-0
[17] Rachůnková I, Koch O, Pulverer G, Weinmüller E: On a singular boundary value problem arising in the theory of shallow membrane caps.Journal of Mathematical Analysis and Applications 2007,332(1):523-541. 10.1016/j.jmaa.2006.10.006 · Zbl 1118.34013 · doi:10.1016/j.jmaa.2006.10.006
[18] Ascher UM, Mattheij RMM, Russell RD: Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Prentice Hall Series in Computational Mathematics. Prentice Hall, Englewood Cliffs, NJ, USA; 1988:xxiv+595.
[19] Kitzhofer G, Koch O, Weinmüller EB: Pathfollowing for essentially singular boundary value problems with application to the complex Ginzburg-Landau equation.BIT. Numerical Mathematics 2009,49(1):141-160. 10.1007/s10543-008-0208-6 · Zbl 1162.65372 · doi:10.1007/s10543-008-0208-6
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