Skip to main content
SpringerLink
Log in

Interval Analysis: Differential Equations

  • Reference work entry
Encyclopedia of Optimization

Article Outline

Keywords

See also

References

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 2,500.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 2,499.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Aberth O (1988) Precise numerical analysis. W.C. Brown, Dubuque, IA

    MATH  Google Scholar 

  2. Aberth O, Schaefer MJ (1992) Precise computation using range arithmetic, via C++. ACM Trans Math Softw 18(4):481–491

    Article  MATH  Google Scholar 

  3. Alefeld G, Frommer A, Lang B (eds) (1996) Scientific computing and validated numerics. Akademie, Berlin

    MATH  Google Scholar 

  4. Caprani O, Madsen K, Rall LB (1981) Integration of interval functions. SIAM J Math Anal 12:321–341

    Article  MathSciNet  MATH  Google Scholar 

  5. Corliss G, Chang YF (1982) Solving ordinary differential equations using Taylor series. ACM Trans Math Softw 8(2):114–144

    Article  MathSciNet  MATH  Google Scholar 

  6. Daniel JW, Moore RE (1970) Computation and theory in ordinary differential equations. Freeman, New York

    MATH  Google Scholar 

  7. Ely JS (1990) Prospects for using variable precision interval software in C++. PhD Thesis Ohio State Univ.

    Google Scholar 

  8. Ely JS, Baker GR (1994) High precision calculations of vortex sheet motion. J Comput Phys 111(2):275–281

    Article  MathSciNet  MATH  Google Scholar 

  9. Fefferman CL, Seco LA (1996) Interval arithmetic in quantum mechanics. In: Kearfott RB, Kreinovich V (eds) Applications of Interval Computations. Kluwer, Dordrecht

    Google Scholar 

  10. Klatte R, Kulisch U, Wiethoff A, Lawo C, Rauch M (1993) C-XSC. Springer, Berlin

    MATH  Google Scholar 

  11. Krückeberg F (1968) Defekterfassung bei gewöhnlichen und partiellen Differentialgleichungen. ISNM, vol 9. Birkhäuser, Basel, pp 69–82

    Google Scholar 

  12. Krückeberg F (1969) Ordinary differential equations. In: Hansen E (ed) Topics in Interval Analysis. Clarendon Press, pp 91–97

    Google Scholar 

  13. Lohner RJ (1987) Enclosing the solutions of ordinary initial and boundary value problems. In: Kaucher E et al (eds) Computer Arithmetic, Scientific Computation and Programming Languages. Teubner, Leipzig, pp 255–286

    Google Scholar 

  14. Lohner RJ (1989) Einschließungen bei Anfangs- und Randwertaufgaben gewöhnlicher Differentialgleichungen. In: Kulisch U (ed) Wissenschaftliches Rechen mit Ergebnisverifikation: Eine Einführung. Viewg, WIesbaden, pp 183–207

    Google Scholar 

  15. Moore RE (1962) Interval arithmetic and automatic error analysis in digital computing. PhD Thesis Stanford Univ.

    Google Scholar 

  16. Moore RE (1965) Automatic local coordinate transformations to reduce the growth of error bounds in interval computation of solutions of ordinary differential equations. In: Rall LB (ed) Error in Digital Computation, vol 2. Wiley, New York, pp 103–140

    Google Scholar 

  17. Moore RE (1966) Interval analysis. Prentice-Hall, Englewood Cliffs, NJ

    MATH  Google Scholar 

  18. Moore RE (1969) Functional analysis for computers. In: Funktionalanalytische Methoden der Numerischen Mathematik, ISNM, vol 12. Birkhäuser, Basel, pp 113–126

    Google Scholar 

  19. Moore RE (1975) Two-sided approximation to solutions of nonlinear operator equations-a comparison of methods from classical analysis, functional analysis and interval analysis. In: Nickel K (ed) Interval Mathematics, Lecture Notes Computer Sci, vol 29. Springer, Berlin, pp 31–47

    Google Scholar 

  20. Moore RE (1978) Bounding sets in function spaces with applications to nonlinear operator equations. SIAM Rev 20:492–512

    Article  MathSciNet  MATH  Google Scholar 

  21. Moore RE (1979) Methods and applications of interval analysis. SIAM, Philadelphia

    MATH  Google Scholar 

  22. Moore RE (1982) A generalization of the method of upper and lower solutions for integral equations. Nonlinear Anal Th Methods Appl 6(8):829–831

    Article  MATH  Google Scholar 

  23. Moore RE (1984) A survey of interval methods for differential equations. Proc. 23rd Conf. Decision and Control, IEEE, pp 1529–1535

    Google Scholar 

  24. Moore RE (1984) Upper and lower bounds on solutions of differential and integral equations without using monotonicity, convexity, or maximum principles. In: Vichnevetsky R, Stepelman RS (eds) Advances in Computer Methods for Partial Differential Equations V. IMACS, pp 458–462

    Google Scholar 

  25. Moore RE (1985) Computational functional analysis. Horwood and Wiley, London

    MATH  Google Scholar 

  26. Moore RE (1985) Simple simultaneous super- and subfunctions. J Integral Eq 8:165–174

    MATH  Google Scholar 

  27. Moore RE (1991) Interval tools for computer aided proofs in analysis. In: Meyer KR, Schmidt DS (eds) Computer Aided Proofs in Analysis. Springer, Berlin, pp 211–216

    Google Scholar 

  28. Moore RE (1994) Numerical solution of differential equations to prescribed accuracy. Comput Math Appl 28(10-12):253–261

    Article  MathSciNet  MATH  Google Scholar 

  29. Moore RE, Shen Z (1983) An interval version of Chebyshev's method for nonlinear operator equations. Nonlinear Anal Th Methods Appl 7(1):21–34

    Article  MathSciNet  MATH  Google Scholar 

  30. Nickel K (1985) How to fight the wrapping effect. In: Nickel K (ed) Interval Mathematics. Lecture Notes Computer Sci, vol 212. Springer, Berlin, pp 121–132

    Google Scholar 

  31. Rall LB (1981) Automatic differentiation: Techniques and applications. Lecture Notes Computer Sci, vol 120. Springer, Berlin

    MATH  Google Scholar 

  32. Stewart NF (1971) A heuristic to reduce the wrapping effect in the numerical solution of X′ = F(T, X)′. BIT 11:328–337

    Article  MATH  Google Scholar 

  33. Website: http://bt.nscl.msu.edu/pub/, work of Berz M and others

Download references

Author information

Authors and Affiliations

  1. Worthington, USA

    Ramon E. Moore

Authors
  1. Ramon E. Moore
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Chemical Engineering, Princeton University, Princeton, NJ, 08544-5263, USA

    Christodoulos A. Floudas

  2. Center for Applied Optimization, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, 32611-6595, USA

    Panos M. Pardalos

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag

About this entry

Cite this entry

Moore, R.E. (2008). Interval Analysis: Differential Equations . In: Floudas, C., Pardalos, P. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-74759-0_294

Download citation

Publish with us

Policies and ethics

Search

Navigation