Abstract
This contribution describes a new approach to formulation of ODE and PDE critical points using implicit formulation as t-variant scalar function using the Taylor expansion. A general condition for the critical points is derived and specified for t invariant case. It is expected, that the given new formulae lead to more reliable detection of critical points especially for large 3D fluid flow data acquisition, which enable high 3D vector compression and their representation using radial basis functions (RBF).
In the case of vector field visualization, e.g. fluid flow, electromagnetic fields, etc., the critical points of ODE are critical for physical phenomena behavior.
Research supported by the University of West Bohemia - Institutional research support.
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Notes
- 1.
It should be noted that in the case of the PDE the coordinates\(\mathbf {x}\) are as sumed to be time independent.
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Acknowledgments
The author thanks to students and colleagues at the University of West Bohemia, Plzen and VSB-Technical University, Ostrava for their critical comments, discussions and especially to Michal Smolik for producing some images. Thanks belong also to the anonymous reviewers, as their comments and hints helped to improve this paper significantly.
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Skala, V. (2022). Critical Points Properties of Ordinary Differential Equations as a Projection of Implicit Functions Using Spatio-temporal Taylor Expansion. In: Gervasi, O., Murgante, B., Hendrix, E.M.T., Taniar, D., Apduhan, B.O. (eds) Computational Science and Its Applications – ICCSA 2022. ICCSA 2022. Lecture Notes in Computer Science, vol 13376. Springer, Cham. https://doi.org/10.1007/978-3-031-10450-3_15
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