Using the Picard Contraction Mapping to Solve Inverse Problems in Ordinary Differential Equations
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.
Unable to display preview. Download preview PDF.
- 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