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Mathematics in Computer Science

Erschienen in:

12.11.2019

On the Numerical Analysis and Visualisation of Implicit Ordinary Differential Equations

verfasst von: Elishan Braun, Werner M. Seiler, Matthias Seiß

Erschienen in: Mathematics in Computer Science | Ausgabe 2/2020

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Abstract

We discuss how the geometric theory of differential equations can be used for the numerical integration and visualisation of implicit ordinary differential equations, in particular around singularities of the equation. The Vessiot theory automatically transforms an implicit differential equation into a vector field distribution on a manifold and thus reduces its analysis to standard problems in dynamical systems theory like the integration of a vector field and the determination of invariant manifolds. For the visualisation of low-dimensional situations we adapt the streamlines algorithm of Jobard and Lefer to 2.5 and 3 dimensions. A concrete implementation in Matlab is discussed and some concrete examples are presented.

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It is well-known that the fibration \(\pi ^{q}_{q-1}:J_{q}\pi \rightarrow J_{q-1}\pi \) defines an affine bundle. A differential equation \(\mathcal {R}_{q}\subset J_{q}\pi \) is quasi-linear, if it is an affine subbundle.

https://www.paraview.org.

Note that, strictly speaking, we are dealing here with a vector field on a two-dimensional manifold. If we had a nice parametrisation of the manifold, we could express the vector field X in these parameters and would obtain a \(2\times 2\) Jacobian. As it is in general difficult to find such parametrisations, we use instead the three coordinates of the ambient space \(J_{1}\pi \). Consequently, we obtain a too large Jacobian and must see which two eigenvalues are the right ones. This is easily decided by checking whether the corresponding (generalised) eigenvectors are tangential to \(\mathcal {R}_{1}\).

In [5] it is shown that for these parameter values there are infinitely many solutions reaching the “tip” \(\rho \).

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Titel: On the Numerical Analysis and Visualisation of Implicit Ordinary Differential Equations
verfasst von: Elishan Braun
Werner M. Seiler
Matthias Seiß
Publikationsdatum: 12.11.2019
Verlag: Springer International Publishing
Erschienen in: Mathematics in Computer Science / Ausgabe 2/2020
Print ISSN: 1661-8270
Elektronische ISSN: 1661-8289
DOI: https://doi.org/10.1007/s11786-019-00423-6

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