On the Numerical Analysis and Visualisation of Implicit Ordinary Differential Equations
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 int...
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| Veröffentlicht in: | Mathematics in computer science 2020-06, Vol.14 (2), p.281-293 |
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| creator | Braun, Elishan Seiler, Werner M. Seiß, Matthias |
| description | 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. |
| doi_str_mv | 10.1007/s11786-019-00423-6 |
| format | Article |
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Matlab
is discussed and some concrete examples are presented.</description><subject>Algorithms</subject><subject>Chaos theory</subject><subject>Computer Science</subject><subject>Differential equations</subject><subject>Differential geometry</subject><subject>Dynamic systems theory</subject><subject>Dynamical Systems</subject><subject>Fields (mathematics)</subject><subject>Manifolds (mathematics)</subject><subject>Mathematical analysis</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Numerical analysis</subject><subject>Numerical integration</subject><subject>Ordinary differential equations</subject><subject>System theory</subject><subject>Visualization</subject><issn>1661-8270</issn><issn>1661-8289</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><recordid>eNp9kL1OwzAURi0EEqXwAkyWmBgCduw49liVllaq6MLPaF0Sm7pKk9ZOkPr2uA0qG5OvrHM-3fshdEvJAyUkfwyU5lIkhKqEEJ6yRJyhARWCJjKV6vw05-QSXYWwJkSklNMB-ljWuF0Z_NJtjHcFVHhUQ7UPLmCoS_zuQgeVC9C6psaNxfPNtnKFa_HSl64Gv8dPzlrjTd26KE923REN1-jCQhXMze87RG_Tyet4liyWz_PxaJEULGNtwhgvQSpDobS8tHnKQDFefLKSSs55IXOZpzkAWKWkzajiVkmTUTCMC8sJG6L7PncFld56t4kr6Qacno0W-vBHmOBMEfFNI3vXs1vf7DoTWr1uOh_PDTrlJHbHiMoilfZU4ZsQvLGnWEr0oWzdl62joI9laxEl1kshwvWX8X_R_1g_I4WBHg</recordid><startdate>20200601</startdate><enddate>20200601</enddate><creator>Braun, Elishan</creator><creator>Seiler, Werner M.</creator><creator>Seiß, Matthias</creator><general>Springer International Publishing</general><general>Springer Nature B.V</general><general>Springer</general><scope>AAYXX</scope><scope>CITATION</scope><scope>1XC</scope><orcidid>https://orcid.org/0000-0002-1301-2401</orcidid><orcidid>https://orcid.org/0000-0002-0565-1334</orcidid></search><sort><creationdate>20200601</creationdate><title>On the Numerical Analysis and Visualisation of Implicit Ordinary Differential Equations</title><author>Braun, Elishan ; Seiler, Werner M. ; Seiß, Matthias</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c353t-334da89e1adf4df723a934cb3d18444c878727aaaf998f5194f98e51ae346f403</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Algorithms</topic><topic>Chaos theory</topic><topic>Computer Science</topic><topic>Differential equations</topic><topic>Differential geometry</topic><topic>Dynamic systems theory</topic><topic>Dynamical Systems</topic><topic>Fields (mathematics)</topic><topic>Manifolds (mathematics)</topic><topic>Mathematical analysis</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Numerical analysis</topic><topic>Numerical integration</topic><topic>Ordinary differential equations</topic><topic>System theory</topic><topic>Visualization</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Braun, Elishan</creatorcontrib><creatorcontrib>Seiler, Werner M.</creatorcontrib><creatorcontrib>Seiß, Matthias</creatorcontrib><collection>CrossRef</collection><collection>Hyper Article en Ligne (HAL)</collection><jtitle>Mathematics in computer science</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Braun, Elishan</au><au>Seiler, Werner M.</au><au>Seiß, Matthias</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On the Numerical Analysis and Visualisation of Implicit Ordinary Differential Equations</atitle><jtitle>Mathematics in computer science</jtitle><stitle>Math.Comput.Sci</stitle><date>2020-06-01</date><risdate>2020</risdate><volume>14</volume><issue>2</issue><spage>281</spage><epage>293</epage><pages>281-293</pages><issn>1661-8270</issn><eissn>1661-8289</eissn><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
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| source | SpringerLink Journals - AutoHoldings |
| subjects | Algorithms Chaos theory Computer Science Differential equations Differential geometry Dynamic systems theory Dynamical Systems Fields (mathematics) Manifolds (mathematics) Mathematical analysis Mathematics Mathematics and Statistics Numerical analysis Numerical integration Ordinary differential equations System theory Visualization |
| title | On the Numerical Analysis and Visualisation of Implicit Ordinary Differential Equations |
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