On the Method of Differential Invariants for Solving Higher Order Ordinary Differential Equations
There are many routines developed for solving ordinary differential Equations (ODEs) of different types. In the case of an <i>n</i>th-order ODE that admits an <i>r</i>-parameter Lie group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" disp...
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MDPI AG
2022-10-01
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| Online Access: | https://www.mdpi.com/2075-1680/11/10/555 |
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| author | Winter Sinkala Molahlehi Charles Kakuli |
| author_facet | Winter Sinkala Molahlehi Charles Kakuli |
| author_sort | Winter Sinkala |
| collection | DOAJ |
| description | There are many routines developed for solving ordinary differential Equations (ODEs) of different types. In the case of an <i>n</i>th-order ODE that admits an <i>r</i>-parameter Lie group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>3</mn><mo>≤</mo><mi>r</mi><mo>≤</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula>, there is a powerful method of Lie symmetry analysis by which the ODE is reduced to an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mi>r</mi><mo>)</mo></mrow></semantics></math></inline-formula>th-order ODE plus <i>r</i> quadratures provided that the Lie algebra formed by the infinitesimal generators of the group is solvable. It would seem this method is not widely appreciated and/or used as it is not mentioned in many related articles centred around integration of higher order ODEs. In the interest of mainstreaming the method, we describe the method in detail and provide four illustrative examples. We use the case of a third-order ODE that admits a three-dimensional solvable Lie algebra to present the gist of the integration algorithm. |
| first_indexed | 2024-03-09T20:42:46Z |
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| id | doaj.art-6842a70f5ce344ea993c99cc60a209ee |
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| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-09T20:42:46Z |
| publishDate | 2022-10-01 |
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| spelling | doaj.art-6842a70f5ce344ea993c99cc60a209ee2023-11-23T22:54:15ZengMDPI AGAxioms2075-16802022-10-01111055510.3390/axioms11100555On the Method of Differential Invariants for Solving Higher Order Ordinary Differential EquationsWinter Sinkala0Molahlehi Charles Kakuli1Department of Mathematical Sciences and Computing, Faculty of Natural Sciences, Walter Sisulu University, Private Bag X1, Mthatha 5117, South AfricaDepartment of Mathematical Sciences and Computing, Faculty of Natural Sciences, Walter Sisulu University, Private Bag X1, Mthatha 5117, South AfricaThere are many routines developed for solving ordinary differential Equations (ODEs) of different types. In the case of an <i>n</i>th-order ODE that admits an <i>r</i>-parameter Lie group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>3</mn><mo>≤</mo><mi>r</mi><mo>≤</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula>, there is a powerful method of Lie symmetry analysis by which the ODE is reduced to an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mi>r</mi><mo>)</mo></mrow></semantics></math></inline-formula>th-order ODE plus <i>r</i> quadratures provided that the Lie algebra formed by the infinitesimal generators of the group is solvable. It would seem this method is not widely appreciated and/or used as it is not mentioned in many related articles centred around integration of higher order ODEs. In the interest of mainstreaming the method, we describe the method in detail and provide four illustrative examples. We use the case of a third-order ODE that admits a three-dimensional solvable Lie algebra to present the gist of the integration algorithm.https://www.mdpi.com/2075-1680/11/10/555ordinary differential equationlie symmetry analysissolvable lie algebradifferential invariantreduction of order |
| spellingShingle | Winter Sinkala Molahlehi Charles Kakuli On the Method of Differential Invariants for Solving Higher Order Ordinary Differential Equations Axioms ordinary differential equation lie symmetry analysis solvable lie algebra differential invariant reduction of order |
| title | On the Method of Differential Invariants for Solving Higher Order Ordinary Differential Equations |
| title_full | On the Method of Differential Invariants for Solving Higher Order Ordinary Differential Equations |
| title_fullStr | On the Method of Differential Invariants for Solving Higher Order Ordinary Differential Equations |
| title_full_unstemmed | On the Method of Differential Invariants for Solving Higher Order Ordinary Differential Equations |
| title_short | On the Method of Differential Invariants for Solving Higher Order Ordinary Differential Equations |
| title_sort | on the method of differential invariants for solving higher order ordinary differential equations |
| topic | ordinary differential equation lie symmetry analysis solvable lie algebra differential invariant reduction of order |
| url | https://www.mdpi.com/2075-1680/11/10/555 |
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