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)


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.


Vector Field Inverse Problem Solution Curve Target Solution Contraction Factor 
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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

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