Higher-Order Symmetries of a Time-Fractional Anomalous Diffusion Equation
Higher-order symmetries are constructed for a linear anomalous diffusion equation with the Riemann–Liouville time-fractional derivative of order <inline-formula><math display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo>...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/3/216 |
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| author | Rafail K. Gazizov Stanislav Yu. Lukashchuk |
| author_facet | Rafail K. Gazizov Stanislav Yu. Lukashchuk |
| author_sort | Rafail K. Gazizov |
| collection | DOAJ |
| description | Higher-order symmetries are constructed for a linear anomalous diffusion equation with the Riemann–Liouville time-fractional derivative of order <inline-formula><math display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>∪</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>. It is proved that the equation in question has infinite sequences of nontrivial higher-order symmetries that are generated by two local recursion operators. It is also shown that some of the obtained higher-order symmetries can be rewritten as fractional-order symmetries, and corresponding fractional-order recursion operators are presented. The proposed approach for finding higher-order symmetries is applicable for a wide class of linear fractional differential equations. |
| first_indexed | 2024-03-09T04:02:07Z |
| format | Article |
| id | doaj.art-07e30ef030184031b710c7c03fee680f |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T04:02:07Z |
| publishDate | 2021-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-07e30ef030184031b710c7c03fee680f2023-12-03T14:10:42ZengMDPI AGMathematics2227-73902021-01-019321610.3390/math9030216Higher-Order Symmetries of a Time-Fractional Anomalous Diffusion EquationRafail K. Gazizov0Stanislav Yu. Lukashchuk1RN-BashNIPIneft LLC, 3/1 Bekhtereva Str., 450103 Ufa, RussiaLaboratory “Group Analysis of Mathemaical Models in Natural and Engineering Sciences”, Ufa State Aviation Technical University, 12 K. Marx Str., 450008 Ufa, RussiaHigher-order symmetries are constructed for a linear anomalous diffusion equation with the Riemann–Liouville time-fractional derivative of order <inline-formula><math display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>∪</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>. It is proved that the equation in question has infinite sequences of nontrivial higher-order symmetries that are generated by two local recursion operators. It is also shown that some of the obtained higher-order symmetries can be rewritten as fractional-order symmetries, and corresponding fractional-order recursion operators are presented. The proposed approach for finding higher-order symmetries is applicable for a wide class of linear fractional differential equations.https://www.mdpi.com/2227-7390/9/3/216anomalous diffusionRiemann–Liouville fractional derivativeLie–Bäcklund transformationhigher-order symmetryrecursion operator |
| spellingShingle | Rafail K. Gazizov Stanislav Yu. Lukashchuk Higher-Order Symmetries of a Time-Fractional Anomalous Diffusion Equation Mathematics anomalous diffusion Riemann–Liouville fractional derivative Lie–Bäcklund transformation higher-order symmetry recursion operator |
| title | Higher-Order Symmetries of a Time-Fractional Anomalous Diffusion Equation |
| title_full | Higher-Order Symmetries of a Time-Fractional Anomalous Diffusion Equation |
| title_fullStr | Higher-Order Symmetries of a Time-Fractional Anomalous Diffusion Equation |
| title_full_unstemmed | Higher-Order Symmetries of a Time-Fractional Anomalous Diffusion Equation |
| title_short | Higher-Order Symmetries of a Time-Fractional Anomalous Diffusion Equation |
| title_sort | higher order symmetries of a time fractional anomalous diffusion equation |
| topic | anomalous diffusion Riemann–Liouville fractional derivative Lie–Bäcklund transformation higher-order symmetry recursion operator |
| url | https://www.mdpi.com/2227-7390/9/3/216 |
| work_keys_str_mv | AT rafailkgazizov higherordersymmetriesofatimefractionalanomalousdiffusionequation AT stanislavyulukashchuk higherordersymmetriesofatimefractionalanomalousdiffusionequation |