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.
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Kunze, H.E., Vrscay, E.R. (2002). Using the Picard Contraction Mapping to Solve Inverse Problems in Ordinary Differential Equations. In: Barnsley, M.F., Saupe, D., Vrscay, E.R. (eds) Fractals in Multimedia. The IMA Volumes in Mathematics and its Application, vol 132. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-9244-6_8
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DOI: https://doi.org/10.1007/978-1-4684-9244-6_8
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