Using the Picard Contraction Mapping to Solve Inverse Problems in Ordinary Differential Equations

  • H. E. Kunze
  • E. R. Vrscay
Conference paper
Part of the The IMA Volumes in Mathematics and its Application book series (IMA, volume 132)

Abstract

The essential ingredients in fractal-based methods are Banach’s Fixed Point Theorem and the related Collage Theorem. In this paper, this class of inverse problems for ODEs is considered from a fractal-based perspective: given a solution curve x(t) (which may be an interpolation of data points) for t ∈ [0,1], find an ODE ẋ = f(x, t) that admits x(t) as a solution as closely as desired, where f may be restricted to a prescribed class of functions, perhaps affine or quadratic in x. A Collage Theorem for this setting is developed, and an algorithm for solving such inverse problems is presented. Several examples are considered.

Keywords

Dition Verse 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M.F. Barnsley, Fractals Everywhere, Academic Press, New York, 1988.MATHGoogle Scholar
  2. [2]
    M.F. Barnsley, V. Ervin, D. Hardin, and J. Lancaster, Solution of an inverse problem for fractals and other sets, Proc. Nat. Acad. Sci. USA, 83: 1975–1977 (1985).MathSciNetCrossRefGoogle Scholar
  3. [3]
    P. Centore and E.R. Vrscay, Continuity of attractors and invariant measures for Iterated Function Systems, Canad. Math. Bull., 37(3): 315–329 (1994).MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    E.A. CODDINGTON AND N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.MATHGoogle Scholar
  5. [5]
    J. Crutchfield and B. McNamara, Equations of Motion from a Data Series, Complex Systems, 1: 417–452 (1987).MathSciNetMATHGoogle Scholar
  6. [6]
    Y. Fisher, Fractal Image Compression, Theory and Application, Springer-Verlag, New York, 1995.CrossRefGoogle Scholar
  7. [7]
    B. Forte and E.R. Vrscay, Inverse Problem Methods for Generalized Fractal Transforms, in Fractal Image Encoding and Analysis, NATO ASI Series F, Vol. 159, ed. Y. Fisher, Springer Verlag, Heidelberg, 1998.Google Scholar
  8. [8]
    N. Lu, Fractal Imaging, Academic Press, New York, 1997.MATHGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 2002

Authors and Affiliations

  • H. E. Kunze
    • 1
  • E. R. Vrscay
    • 2
  1. 1.Department of Mathematics and StatisticsUniversity of GuelphGuelphCanada
  2. 2.Department of Applied MathematicsUniversity of WaterlooWaterlooCanada

Personalised recommendations