On the Property of Linear Autonomy for Symmetries of Fractional Differential Equations and Systems
The problem of finding Lie point symmetries for a certain class of multi-dimensional nonlinear partial fractional differential equations and their systems is studied. It is assumed that considered equations involve fractional derivatives with respect to only one independent variable, and each equati...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-07-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/13/2319 |
| _version_ | 1797442738534219776 |
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| author | Stanislav Yu. Lukashchuk |
| author_facet | Stanislav Yu. Lukashchuk |
| author_sort | Stanislav Yu. Lukashchuk |
| collection | DOAJ |
| description | The problem of finding Lie point symmetries for a certain class of multi-dimensional nonlinear partial fractional differential equations and their systems is studied. It is assumed that considered equations involve fractional derivatives with respect to only one independent variable, and each equation contains a single fractional derivative. The most significant examples of such equations are time-fractional models of processes with memory of power-law type. Two different types of fractional derivatives, namely Riemann–Liouville and Caputo, are used in this study. It is proved that any Lie point symmetry group admitted by equations or systems belonging to considered class consists of only linearly-autonomous point symmetries. Representations for the coordinates of corresponding infinitesimal group generators, as well as simplified determining equations are given in explicit form. The obtained results significantly facilitate finding Lie point symmetries for multi-dimensional time-fractional differential equations and their systems. Three physical examples illustrate this point. |
| first_indexed | 2024-03-09T12:46:20Z |
| format | Article |
| id | doaj.art-f25a8b0611d94d1487de708d0a364f09 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T12:46:20Z |
| publishDate | 2022-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-f25a8b0611d94d1487de708d0a364f092023-11-30T22:12:07ZengMDPI AGMathematics2227-73902022-07-011013231910.3390/math10132319On the Property of Linear Autonomy for Symmetries of Fractional Differential Equations and SystemsStanislav Yu. Lukashchuk0Department of High Performance Computing Technologies and Systems, Ufa State Aviation Technical University, 12 K. Marx Str., Ufa 450008, RussiaThe problem of finding Lie point symmetries for a certain class of multi-dimensional nonlinear partial fractional differential equations and their systems is studied. It is assumed that considered equations involve fractional derivatives with respect to only one independent variable, and each equation contains a single fractional derivative. The most significant examples of such equations are time-fractional models of processes with memory of power-law type. Two different types of fractional derivatives, namely Riemann–Liouville and Caputo, are used in this study. It is proved that any Lie point symmetry group admitted by equations or systems belonging to considered class consists of only linearly-autonomous point symmetries. Representations for the coordinates of corresponding infinitesimal group generators, as well as simplified determining equations are given in explicit form. The obtained results significantly facilitate finding Lie point symmetries for multi-dimensional time-fractional differential equations and their systems. Three physical examples illustrate this point.https://www.mdpi.com/2227-7390/10/13/2319fractional differential equationLie point symmetry grouplinearly autonomous symmetry |
| spellingShingle | Stanislav Yu. Lukashchuk On the Property of Linear Autonomy for Symmetries of Fractional Differential Equations and Systems Mathematics fractional differential equation Lie point symmetry group linearly autonomous symmetry |
| title | On the Property of Linear Autonomy for Symmetries of Fractional Differential Equations and Systems |
| title_full | On the Property of Linear Autonomy for Symmetries of Fractional Differential Equations and Systems |
| title_fullStr | On the Property of Linear Autonomy for Symmetries of Fractional Differential Equations and Systems |
| title_full_unstemmed | On the Property of Linear Autonomy for Symmetries of Fractional Differential Equations and Systems |
| title_short | On the Property of Linear Autonomy for Symmetries of Fractional Differential Equations and Systems |
| title_sort | on the property of linear autonomy for symmetries of fractional differential equations and systems |
| topic | fractional differential equation Lie point symmetry group linearly autonomous symmetry |
| url | https://www.mdpi.com/2227-7390/10/13/2319 |
| work_keys_str_mv | AT stanislavyulukashchuk onthepropertyoflinearautonomyforsymmetriesoffractionaldifferentialequationsandsystems |