Nonlocal Symmetries for Time-Dependent Order Differential Equations
A new type of ordinary differential equation is introduced and discussed: time-dependent order ordinary differential equations. These equations are solved via fractional calculus by transforming them into Volterra integral equations of second kind with singular integrable kernel. The solutions of th...
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| Format: | Article |
| Language: | English |
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MDPI AG
2018-12-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/10/12/771 |
| _version_ | 1811186366641340416 |
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| author | Andrei Ludu |
| author_facet | Andrei Ludu |
| author_sort | Andrei Ludu |
| collection | DOAJ |
| description | A new type of ordinary differential equation is introduced and discussed: time-dependent order ordinary differential equations. These equations are solved via fractional calculus by transforming them into Volterra integral equations of second kind with singular integrable kernel. The solutions of the time-dependent order differential equation represent deformations of the solutions of the classical (integer order) differential equations, mapping them into one-another as limiting cases. This equation can also move, remove or generate singularities without involving variable coefficients. An interesting symmetry of the solution in relation to the Riemann zeta function and Harmonic numbers is observed. |
| first_indexed | 2024-04-11T13:44:56Z |
| format | Article |
| id | doaj.art-aa14106cf8fb4ea09cb426d4889b0e09 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-11T13:44:56Z |
| publishDate | 2018-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-aa14106cf8fb4ea09cb426d4889b0e092022-12-22T04:21:07ZengMDPI AGSymmetry2073-89942018-12-01101277110.3390/sym10120771sym10120771Nonlocal Symmetries for Time-Dependent Order Differential EquationsAndrei Ludu0Department of Mathematics, Embry-Riddle Aeronautical University, Daytona Beach, FL 32114, USAA new type of ordinary differential equation is introduced and discussed: time-dependent order ordinary differential equations. These equations are solved via fractional calculus by transforming them into Volterra integral equations of second kind with singular integrable kernel. The solutions of the time-dependent order differential equation represent deformations of the solutions of the classical (integer order) differential equations, mapping them into one-another as limiting cases. This equation can also move, remove or generate singularities without involving variable coefficients. An interesting symmetry of the solution in relation to the Riemann zeta function and Harmonic numbers is observed.https://www.mdpi.com/2073-8994/10/12/771variable order derivativefractional differential equationVoltera equationsingular integrable kernel |
| spellingShingle | Andrei Ludu Nonlocal Symmetries for Time-Dependent Order Differential Equations Symmetry variable order derivative fractional differential equation Voltera equation singular integrable kernel |
| title | Nonlocal Symmetries for Time-Dependent Order Differential Equations |
| title_full | Nonlocal Symmetries for Time-Dependent Order Differential Equations |
| title_fullStr | Nonlocal Symmetries for Time-Dependent Order Differential Equations |
| title_full_unstemmed | Nonlocal Symmetries for Time-Dependent Order Differential Equations |
| title_short | Nonlocal Symmetries for Time-Dependent Order Differential Equations |
| title_sort | nonlocal symmetries for time dependent order differential equations |
| topic | variable order derivative fractional differential equation Voltera equation singular integrable kernel |
| url | https://www.mdpi.com/2073-8994/10/12/771 |
| work_keys_str_mv | AT andreiludu nonlocalsymmetriesfortimedependentorderdifferentialequations |