On a singular boundary value problem arising in the theory of shallow membrane caps

Journal of Mathematical Analysis and Applications

Volumn 332, Issue 1, 2007, Pages 523-541

  Rachůnková, Irena ^a   Koch, Othmar ^b   Pulverer, Gernot ^b   Weinmüller, Ewa ^b  

^a PALACKÝ UNIVERSITY   (Czech Republic)

^b VIENNA UNIVERSITY OF TECHNOLOGY   (Austria)

Author keywords

Collocation method; Lower and upper functions; Positive solution; Shallow membrane; Singular mixed boundary value problem

Indexed keywords

EID: 34147139421     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2006.10.006     Document Type: Article

References (39)

1. Agarwal R.P., and O'Regan D. Singular boundary value problems for superlinear second order ordinary and delay differential equations. J. Differential Equations 130 (1996) 333-355
2. Agarwal R.P., and O'Regan D. Nonlinear superlinear singular and nonsingular second order boundary value problems. J. Differential Equations 143 (1998) 60-95
3. Agarwal R.P., and O'Regan D. Singular problems motivated from classical upper and lower solutions. Acta Math. Hungar. 100 3 (2003) 245-256
4. Agarwal R.P., O'Regan D., and Staněk S. Existence of positive solutions for boundary-value problems with singularities in phase variables. Proc. Edinb. Math. Soc. 47 (2004) 1-13
5. Agarwal R.P., and Staněk S. Nonnegative solutions of singular boundary value problems with sign changing nonlinearities. Comput. Math. Appl. 46 (2003) 1827-1837
6. Ascher U., Mattheij R.M.M., and Russell R.D. Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (1988), Prentice Hall, Englewood Cliffs, New York
7. Auzinger W., Koch O., and Weinmüller E. Efficient collocation schemes for singular boundary value problems. Numer. Algorithms 31 (2002) 5-25
8. Auzinger W., Kneisl G., Koch O., and Weinmüller E. A collocation code for boundary value problems in ordinary differential equations. Numer. Algorithms 33 (2003) 27-39
9. Auzinger W., Koch O., and Weinmüller E. Efficient mesh selection for collocation methods applied to singular BVPs. J. Comput. Appl. Math. 180 (2005) 213-227
10. Auzinger W., Koch O., and Weinmüller E. Analysis of a new error estimate for collocation methods applied to singular boundary value problems. SIAM J. Numer. Anal. 42 (2005) 2366-2386
11. Auzinger W., Koch O., and Weinmüller E. Collocation methods for singular boundary value problems with an essential singularity. In: Lirkov I., Margenov S., Wasniewski J., and Yalamov P. (Eds). Large-Scale Scientific Computing. Lecture Notes in Comput. Sci. vol. 2907 (2004), Springer-Verlag 347-354
12. Baxley J.V. Some singular nonlinear boundary value problems. SIAM J. Math. Anal. 22 (1991) 463-479
13. Baxley J.V., and Gersdorff G.S. Singular reaction-diffusion boundary value problem. J. Differential Equations 115 (1995) 441-457
14. Baxley J.V., and Sorrells K.P. A class of singular nonlinear boundary value problems. Math. Comput. Modelling 32 (2000) 631-641
15. Budd C.J., Koch O., and Weinmüller E. Self-similar blow-up in nonlinear PDEs. AURORA TR-2004-15, Inst. for Anal. and Sci. Comput., Vienna Univ. of Technology, Austria, 2004. Available at. http://www.vcpc.univie.ac.at/aurora/publications/
16. Dickey R.W. Rotationally symmetric solutions for shallow membrane caps. Quart. Appl. Math. 47 (1989) 571-581
17. de Hoog F., and Weiss R. Difference methods for boundary value problems with a singularity of the first kind. SIAM J. Numer. Anal. 13 (1976) 775-813
18. de Hoog F., and Weiss R. The numerical solution of boundary value problems with an essential singularity. SIAM J. Numer. Anal. 16 (1979) 637-669
19. de Hoog F., and Weiss R. The application of Runge-Kutta schemes to singular initial value problems. Math. Comp. 44 (1985) 93-103
20. de Hoog F., and Weiss R. Collocation methods for singular boundary value problems. SIAM J. Numer. Anal. 15 (1978) 198-217
21. Gatica J.A., Oliker V., and Waltman P. Singular nonlinear boundary value problems for second-order ordinary differential equations. J. Differential Equations 79 (1989) 62-78
22. Johnson K.N. Circularly symmetric deformation of shallow elastic membrane caps. Quart. Appl. Math. 55 (1997) 537-550
23. Kannan R., and O'Regan D. Singular and nonsingular boundary value problems with sign changing nonlinearities. J. Inequal. Appl. 5 (2000) 621-637
24. Kelevedjiev P. Nonnegative solutions to some second-order boundary value problems. Nonlinear Anal. 36 (1999) 481-494
25. Kiguradze I.T., and Shekhter B.L. Singular boundary value problems for second order ordinary differential equations. Itogi Nauki Tekh. Ser. Sovrm. Probl. Mat. VINITI 30 (1987) 105-201 (in Russian)
26. G. Kitzhofer, Numerical treatment of implicit singular BVPs, PhD thesis, Inst. for Anal. and Sci. Comput., Vienna Univ. of Technology, Austria, 2005, in preparation
27. Kitzhofer G., Koch O., Lima P., and Weinmüller E. Efficient numerical solution of the density profile equation in hydrodynamics. J. Sci. Comput., in press. Available at. http://www.math.tuwien.ac.at/~ewa
28. Koch O. Asymptotically correct error estimation for collocation methods applied to singular boundary value problems. Numer. Math. 101 (2005) 143-164
29. Koch O., and Weinmüller E. Iterated Defect Correction for the solution of singular initial value problems. SIAM J. Numer. Anal. 38 (2001) 1784-1799
30. Lentini M., and Keller H. Boundary value problems on semi-infinite intervals and their numerical solution. SIAM J. Numer. Anal. 17 (1980) 577-604
31. Markowich P., and Ringhofer C. Collocation methods for boundary value problems on long intervals. Math. Comp. 40 (1983) 123-150
32. O'Regan D. Theory of Singular Boundary Value Problems (1994), World Scientific, Singapore
33. Rachůnková I. Singular mixed boundary value problem. J. Math. Anal. Appl. 320 (2006) 611-618
34. I. Rachůnková, Superlinear mixed BVP with time and space singularities, Conf. Proc. CDDEA 2005, Hindawi, in press
35. Rachůnková I. Strong singularities in mixed boundary value problems. Math. Bohemica 131 (2006)
36. Rachůnková I., Staněk S., and Tvrdý M. Singularities and Laplacians in boundary value problems for nonlinear ordinary differential equations. In: Caňada A., Drábek P., and Fonda A. (Eds). Handbook of Differential Equations. Ordinary Differential Equations vol. 3 (2006), Elsevier 607-723
37. Shin J.Y. A singular nonlinear differential equation arising in the Homann flow. J. Math. Anal. Appl. 212 (1997) 443-451
38. ′ (0) = x (1) = 0. J. Differential Equations 113 (1994) 1-16
39. Weinmüller E. Collocation for singular boundary value problems of second order. SIAM J. Numer. Anal. 23 (1986) 1062-1095