Analysis of a stochastic SIRS model with interval parameters
Comparing motion of curves and hypersurfaces in $ \mathbb{R}^m $
Jiří Minarčík 1,, , Masato Kimura 2, and Michal Beneš 1,
Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, Prague, 12000, Czech Republic
Faculty of Mathematics and Physics, Kanazawa University, Kakuma, Kanazawa, 920-1192, Japan
* Corresponding author: Jiří Minarčík
Received January 2018 Revised June 2018 Published September 2019 Early access February 2019
Fund Project: The first and third author were partly supported by the project No. 15-27178A of Ministry of Health of the Czech Republic, the first author was partly supported by the Foundation Nadání Josefa, Marie a Zdeňky Hlávkových, by the Study at KU Scholarship from Kanazawa University and by the project of the Student Grant Agency of the Czech Technical University in Prague No. SGS17/194/OHK4/3T/14, and the second author was partly supported by JSPS KAKENHI Grant Numbers 15H03632 and 16H03953.
Figure(4) / Table(2)
This article aims to contribute to the understanding of the curvature flow of curves in a higher-dimensional space. Evolution of curves in $ \mathbb{R}^m $ by their curvature is compared to the motion of hypersurfaces with constrained normal velocity. The special case of shrinking hyperspheres is further analyzed both theoretically and numerically by means of a semi-discrete scheme with discretization based on osculating circles. Computational examples of evolving spherical curves are provided along with the measurement of the experimental order of convergence.
Keywords: Curvature flow, moving hypersurfaces, comparison principle, spherical curves, signed distance function.
Mathematics Subject Classification: Primary: 53C44, 14Q05, 14J70; Secondary: 34K12, 35B51.
Citation: Jiří Minarčík, Masato Kimura, Michal Beneš. Comparing motion of curves and hypersurfaces in $ \mathbb{R}^m $. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4815-4826. doi: 10.3934/dcdsb.2019032
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Figure 1. Visualization for the definition of $ \mathbf{y}(u,t) $ and $ \bar{u}(t) $
Figure 2. Visualization of the geometric quantities defined in the osculating plane
Figure 3. Results of the numerical simulation from Example 1. The discretized curve is visualized at four different time levels along with the corresponding sphere
Table 1. Results of the numerical computation from Example 1. The error measurements were taken during time interval $ [0,0.45] $
$N$ $\mathcal{E}_{\infty}(N)$ EOC$_{\infty}$ $\mathcal{E}_{1}(N)$ EOC1 $\mathcal{E}_{2}(N)$ EOC2
100 $1.0205 \cdot 10^{-2}$ 2.0283 $9.3770 \cdot 10^{-3}$ 2.0540 $6.4629 \cdot 10^{-3}$ 2.0794
1600 $5.0943 \cdot 10^{-5}$ $3.5888 \cdot 10^{-5}$ $2.4118 \cdot 10^{-5}$
Table Options
$N$ $\mathcal{E}_{\infty}(N)$ EOC$_{\infty}$ $\mathcal{E}_{1}(N)$ EOC$_1$ $\mathcal{E}_{2}(N)$ EOC$_2$
1200 $1.2998 \cdot 10^{-2}$ 2.0218 $1.2996 \cdot 10^{-2}$ 2.0222 $9.4608 \cdot 10^{-3}$ 2.0547
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Jiří Minarčík Masato Kimura Michal Beneš