Advertisement

On Methods for the Construction of the Boundaries of Sets of Solutions for Differential Equations or Finite-Dimensional Approximations with Input Sets

  • E. Adams
Part of the Computing Supplementum book series (COMPUTING, volume 2)

Abstract

Collections of linear or nonlinear operator equations Au = f are considered which may represent (i) differential or integral equations or (ii) finite-dimensional approximations. Input sets of coefficients a or data f are admitted. The envelope of the set of solutions is to be constructed where this boundary refers (i) to the ränge of values of the solutions or (ii) to a finite-dimensional space. The construction employs either topological boundary mapping or truncated Taylor expansions. Estimates of the local procedural errors are due to suitable a priori sets and interval mathematics. The relation between local and global error estimates is due to boundary mapping or an auxiliary inverse-monotone operator B. The operator B is constructed for the case of arbitrary linear ordinary differential equations with boundary or initial conditions, provided the admitted A satisfy a mild condition.

Keywords

Boundary Mapping Operator Equation Remainder Term Discrete Analogy Outer Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Adams, E., Spreuer, H.: Uniqueness and stability for boundary value problems with weakly coupled systems of nonlinear integro-differential equations and application to chemical reactions. JMAA 49, 393–410 (1975).MATHMathSciNetGoogle Scholar
  2. [2]
    Adams, E., Spreuer, H.: On the construction of an interval containing the set of solutions of non- inverse isotone linear problems with intervals admitted for both data and coefficients. Report CAM 14 (1978). (Center for Appl. Math., The Univ. of Georgia, Athens, Georgia, U.S.A.)Google Scholar
  3. [3]
    Alefeld, G. Herzberger, J.: Einführung in die Intervallrechnung. Mannheim-Wien-Zürich: Bibliographisches Institut 1974.MATHGoogle Scholar
  4. [4]
    Beeck, H.: Über Struktur und Abschätzungen der Lösungsmenge von linearen Gleichungssystemen mit Intervallkoeffizienten. Comp. 10, 231–244 (1972).CrossRefMATHMathSciNetGoogle Scholar
  5. [5]
    Beeck, H.: Zur Problematik der Hüllenbestimmung von Intervallgleichungssystemen, in: Lecture Notes in Computer Science, Vol. 29. Berlin-Heidelberg-New York: Springer 1975.Google Scholar
  6. [6]
    Blum, K. E.: Numerical Analysis and Computation Theory and Practice. Reading, Mass.-Menlo Park, Calif.-London-Don Mills, Ont.: Addison-Wesley 1972.Google Scholar
  7. [7]
    Bogoljubow, N. N., Mitropolski, J. A.: Asymptotische Methoden in der Theorie der nichtlinearen Schwingungen. Berlin: Akademie-Verlag 1965. (Translation from the Russian.)Google Scholar
  8. [8]
    Deimling, K.: Nichtlineare Gleichungen und Abbildungsgrade. Berlin-Heidelberg-New York: Springer 1974.MATHGoogle Scholar
  9. [9]
    Fichtenholz, G. M.: Differential- und Integralrechnung, Vol. II, 6. Aufl. Berlin: VEB Deutscher Verlag der Wissenschaften 1974. (Translation from the Russian.)Google Scholar
  10. [10]
    Föllinger, O.: Laplace- und Fourier-Transformation. Berlin: Elitera-Verlag 1977.MATHGoogle Scholar
  11. [11]
    Kamke, E.: Differentialgleichungen, Vol. I, 6. Aufl. Leipzig: Akad. Verlagsgesellschaft Geest & Portig K.G. 1969.MATHGoogle Scholar
  12. [12]
    Kasriel, R. H.: Undergraduate Topology. Philadelphia-London-Toronto: W. B. Saunders Co. 1971.MATHGoogle Scholar
  13. [13]
    Lakshmikantham, V., Leela, S.: Differential and Integral Inequalities, Theory and Applications, Vol. I and II. New York-London: Academic Press 1969.MATHGoogle Scholar
  14. [14]
    Lohner, R., Adams, E.: On initial value problems in R N with intervals for both initial data and a Parameter in the differential equation. Report CAM 8 (1978). (Center for Appl. Math., The Univ. of Georgia, Athens, Georgia, U.S.A.)Google Scholar
  15. [15]
    Lohner, R.: Anfangswertaufgaben im [RM mit kompakten Mengen für Anfangswerte und Parameter. Diplomarbeit, Karlsruhe, 1978.Google Scholar
  16. [16]
    Moore, R. E.: Interval Analysis. Englewood Cliffs, N. J.: Prentice-Hall 1966.MATHGoogle Scholar
  17. [17]
    Nickel, K.: Die Überschätzung des Wertebereichs einer Funktion in der Intervallrechnung mit Anwendung auf lineare Gleichungssysteme. Comp. 18, 15–36 (1977).CrossRefMATHMathSciNetGoogle Scholar
  18. [18]
    Nickel, K.: Schranken für die Lösungsmenge von Funktional-Differentialgleichungen. Freiburger Intervall-Berichte 79/4 (1979).Google Scholar
  19. [19]
    Ortega, J. M., Rheinboldt, W. C.: Iterative Solution of Nonlinear Equations in Several Variables. New York-San Francisco-London: Academic Press 1970.MATHGoogle Scholar
  20. [20]
    Spreuer, H.: Konvergente numerische Schranken für partielle Randwertaufgaben von monotoner Art, in: Lecture Notes in Computer Science, Vol. 29. Berlin-Heidelberg-New York: Springer 1975.Google Scholar
  21. [21]
    Stetter, H. J.: Analysis of Discretization Methods for Ordinary Differential Equations. Berlin- Heidelberg-New York: Springer 1973.CrossRefMATHGoogle Scholar
  22. [22]
    Stoer, J.: Einführung in die Numerische Mathematik, Vol. I. Berlin-Heidelberg-New York: Springer 1972.MATHGoogle Scholar
  23. [23]
    Walter, W.: Differential and Integral Inequalities. Berlin-Heidelberg-New York: Springer 1970.MATHGoogle Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • E. Adams
    • 1
  1. 1.Institut für Angewandte MathematikUniversität KarlsruheKarlsruheFederal Republic of Germany

Personalised recommendations