Download PDF - Enclosures and semi-analytic discretization of boundary value problemsAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Enclosures and semi-analytic discretization of boundary value problems

Authors 1

Affiliations

  1. Department of Mathematics, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait

Bibliography

  1. H. Abhari and C. Grossmann, Approximate bases for boundary value problems and applications to MID-techniques, Z. Angew. Math. Mech. 72 (1992), 83-91.
  2. H. Abhari and M. Al-Zanaidi, Computational aspects of monotone iteration discretization algorithm, preprint, TU Dresden, 07-01-88.
  3. R. A. Adams, Sobolev Spaces, Academic Press, New York 1975.
  4. E. Adams, Invers-Monotonie, direkte und indirekte Intervallmethoden, Forschungszentrum Graz, Ber. 185 (1982).
  5. E. Adams, R. Ansorge, C. Grossmann and H.-G. Roos (eds.), Discretization in Differential Equations and Enclosures, Akademie-Verlag, Berlin 1987.
  6. E. Adams, D. Cordes and H. Keppler, Enclosure methods as applied to linear periodic ODEs and matrices, Z. Angew. Math. Mech. 70 (1990), 565-578.
  7. E. Adams und H. Spreuer, Konvergente numerische Schrankenkonstruktionen mit Spline-Funktionen für nichtlineare gewöhnliche bzw. parabolische Randwertaufgaben, in: K. Nickel (ed.), Interval Mathematics, Springer, Berlin 1975.
  8. M. Al-Zanaidi and C. Grossmann, Monotone discretization in boundary value problems using PASCAL-SC, in: I. Marek (ed.), Proc. ISNA 87, Teubner, Leipzig 1988, 91-96.
  9. M. Al-Zanaidi and C. Grossmann, Monotone iteration discretization algorithm for BVP's, Computing 31 (1989), 59-74.
  10. M. Al-Zanaidi and C. Grossmann, 1D-grid generation by monotone iteration discretization, ibid. 34 (1990), 377-390.
  11. N. Anderson and A. M. Arthurs, Variational solutions of the Thomas-Fermi equations, Quart. Appl. Math. 39 (1981), 127-129.
  12. S. Carl, The monotone iterative technique for a parabolic boundary value problem with discontinuous nonlinearity, Nonlinear Anal. 13 (1989), 1399-1407.
  13. S. Carl and C. Grossmann, Iterative spline bounds for systems of boundary value problems, in: J. W. Schmidt and H. Späth (eds.), Splines in Numerical Analysis, Akademie-Verlag, Berlin 1989, 19-30.
  14. S. Carl and C. Grossmann, Monotone enclosure for elliptic and parabolic systems with nonmonotone nonlinearities, J. Math. Anal. Appl. 151 (1990), 190-202.
  15. S. Carl and S. Heikkilä, On extremal solutions of an elliptic boundary value problem involving discontinuous nonlinearities, preprint, Univ. Oulu, 1990; to appear in Differential Integral Equations.
  16. C. Y. Chan and Y. C. Hon, A constructive solution for a generalized Thomas-Fermi theory of ionized atoms, Quart. Appl. Math. 45 (1987), 591-599.
  17. R. C. Duggan and C. Goodmann, Pointwise bounds for a nonlinear heat conduction model of the human head, Bull. Math. Biol. 48 (1986), 229-236.
  18. A. Felgenhauer, Monotone discretization of the Poisson equation in the plane, to appear.
  19. R. C. Flagg, C. D. Luning and W. L. Perry, Implementation of new iterative techniques for solutions of Thomas-Fermi and Emden-Fowler equations, J. Comput. Phys. 38 (1980), 396-405.
  20. E. C. Gartland, On the uniform convergence of the Scharfetter-Gummel discretization, preprint, Kent State Univ., Feb. 1991.
  21. E. C. Gartland and C. Grossmann, Semianalytic solution of boundary value problems by using approximated operators, Z. Angew. Math. Mech. 72 (1992), 615-619.
  22. C. Grossmann, Monotone Einschließung höherer Ordnung für 2-Punkt-Randwertauf- gaben, Z. Angew. Math. Mech. 67 (1987), T475-T477.
  23. C. Grossmann, Monotone discretization of two-point boundary value problems and related numerical methods, in [5], 99-122.
  24. C. Grossmann, Semianalytic discretization of weakly nonlinear boundary value problems with variable coefficients. Z. Anal. Anwendungen 10 (1991), 513-523.
  25. C. Grossmann, Enclosures of the solution of the Thomas-Fermi equation by monotone discretization, J. Comput. Phys. (1992), 26-38.
  26. C. Grossmann und M. Krätzschmar, Monotone Diskretisierung und adaptive Gittergenerierung für Zwei-Punkt-Randwertaufgaben, Z. Angew. Math. Mech. 65 (1985), T264-T266.
  27. C. Grossmann, M. Krätzschmar und H.-G. Roos, Gleichmäßig einschließende Diskretisierungsverfahren für schwach nichtlineare Randwertaufgaben, Numer. Math. 49 (1986), 95-110.
  28. C. Grossmann and H.-G. Roos, Feedback grid generation via monotone discretization for two-point boundary-value problems, IMA J. Numer. Anal. 6 (1986), 421-432.
  29. C. Grossmann and H.-G. Roos, Uniform enclosure of high order for boundary value problems by monotone discretization, Math. Comp. 53 (1989), 609-617.
  30. C. Grossmann and H.-G. Roos, Convergence analysis of higher order monotone discretization, Wiss. Z. Tech. Univ. Dresden 38 (1989), 155-168.
  31. C. Grossmann and H.-G. Roos, Enclosing discretization for singular and singularly perturbed boundary value problems, in: H.-G. Roos, A. Felgenhauer and L. Angermann (eds.), Numerical Methods in Singular Perturbed Problems, TU Dresden, 1991, 71-82.
  32. G. Koeppe, H.-G. Roos and L. Tobiska, An enclosure generating modification of the method of discretization in time, Comment. Math. Univ. Carolin. 28 (1987), 447-453.
  33. M. Krätzschmar, Iterationsverfahren zur Lösung schwach nichtlinearer elliptischer Randwertaufgaben mit monotoner Lösungseinschließung, Dissertation, TU Dresden, 1983.
  34. U. Kulisch (ed.), PASCAL-SC, Pascal Extension for Scientific Computation, Wiley, New York and Teubner, Stuttgart 1987.
  35. G. S. Ladde, V. Lakshmikantham and A. S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, 1985.
  36. R. Lohner, Einschließung der Lösung gewöhnlicher Anfangs- und Randwertaufgaben und Anwendungen, Dissertation, Univ. Karlsruhe, 1988.
  37. J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York 1970.
  38. M. Plum, Numerical existence proofs and explicit bounds for solutions of nonlinear elliptic boundary value problems, to appear.
  39. H.-G. Roos, The Rothe method and monotone discretization for parabolic equations, in [5], 155-166.
  40. H.-G. Roos, Uniformly enclosing discretization methods and grid generation for semilinear boundary value problems with first order terms, Apl. Mat. 34 (1989), 274-284.
  41. H.-G. Roos, A uniformly convergent discretization method for singularly perturbed boundary value problems of the fourth order, Zb. Rad. Prirod.-Mat. Fak. Novi Sad Ser. Mat. 19 (1989), 51-64.
  42. H.-G. Roos, Global uniformly convergent schemes for a singularly perturbed boundary-value problem using patched base spline functions, J. Comput. Appl. Math. 29 (1990), 69-77.
  43. H.-G. Roos, Uniform enclosing discretization methods for semilinear boundary value problems, in: Nonlinear Analysis and Mathematical Modelling, Banach Center Publ. 24, PWN, Warszawa 1990, 257-268.
  44. J. W. Schmidt, Ein Einschließungsverfahren für Lösungen fehlerbehafteter linearer Gleichungen, Period. Math. Hungar. 13 (1982), 29-37.
  45. J. Schröder, Operator Inequalities, Academic Press, New York 1980.
  46. J. Schröder, A method for producing verified results for two-point boundary value problems, in: Comput. Suppl. 6, Springer, Vienna 1988, 9-22.
  47. J. Schröder, Operator inequalities and applications, in: E. W. Norrie (ed.), Inequalities, Marcel Dekker, New York 1991, 163-210.
  48. J. Sprekels and H. Voss, Pointwise inclusions of fixed points by finite dimensional iteration schemes, Numer. Math. 32 (1979), 381-392.
  49. G. Vanden Berghe and H. De Meyer, Accurate computation of higher Sturm-Liouville eigenvalues, to appear.
  50. R. Voller, Enclosure of solutions of weakly nonlinear elliptic boundary value problems and their computation, Computing 42 (1989), 245-258.
  51. W. Walter, Differential and Integral Inequalities, Springer, Berlin 1970.
  52. J. Weissinger, A kind of difference methods for enclosing solutions of ordinary linear boundary value problems, in: Comput. Suppl. 6, Springer, Vienna 1988, 23-32.
Banach Center Publications
1994/29/1
Export to PBN, Opens in new tab
Pages:
283-304
Main language of publication
English
Published
1994
Exact and natural sciences