Existence in the Large for Caputo Fractional Multi-Order Systems with Initial Conditions
One of the key applications of the Caputo fractional derivative is that the fractional order of the derivative can be utilized as a parameter to improve the mathematical model by comparing it to real data. To do so, we must first establish that the solution to the fractional dynamic equations exists...
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| Format: | Article |
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MDPI AG
2023-05-01
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| Series: | Foundations |
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| Online Access: | https://www.mdpi.com/2673-9321/3/2/21 |
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| author | Zachary Denton Aghalaya S. Vatsala |
| author_facet | Zachary Denton Aghalaya S. Vatsala |
| author_sort | Zachary Denton |
| collection | DOAJ |
| description | One of the key applications of the Caputo fractional derivative is that the fractional order of the derivative can be utilized as a parameter to improve the mathematical model by comparing it to real data. To do so, we must first establish that the solution to the fractional dynamic equations exists and is unique on its interval of existence. The vast majority of existence and uniqueness results available in the literature, including Picard’s method, for ordinary and/or fractional dynamic equations will result in only local existence results. In this work, we generalize Picard’s method to obtain the existence and uniqueness of the solution of the nonlinear multi-order Caputo derivative system with initial conditions, on the interval where the solution is bounded. The challenge presented to establish our main result is in developing a generalized form of the Mittag–Leffler function that will cooperate with all the different fractional derivative orders involved in the multi-order nonlinear Caputo fractional differential system. In our work, we have developed the generalized Mittag–Leffler function that suffices to establish the generalized Picard’s method for the nonlinear multi-order system. As a result, we have obtained the existence and uniqueness of the nonlinear multi-order Caputo derivative system with initial conditions in the large. In short, the solution exists and is unique on the interval where the norm of the solution is bounded. The generalized Picard’s method we have developed is both a theoretical and a computational method of computing the unique solution on the interval of its existence. |
| first_indexed | 2024-03-11T02:27:56Z |
| format | Article |
| id | doaj.art-ae8bd34de4424a53a8476899f7ca1c54 |
| institution | Directory Open Access Journal |
| issn | 2673-9321 |
| language | English |
| last_indexed | 2024-03-11T02:27:56Z |
| publishDate | 2023-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Foundations |
| spelling | doaj.art-ae8bd34de4424a53a8476899f7ca1c542023-11-18T10:28:55ZengMDPI AGFoundations2673-93212023-05-013226027410.3390/foundations3020021Existence in the Large for Caputo Fractional Multi-Order Systems with Initial ConditionsZachary Denton0Aghalaya S. Vatsala1Department of Mathematics, North Carolina A&T State University, Greensboro, NC 27411, USADepartment of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504, USAOne of the key applications of the Caputo fractional derivative is that the fractional order of the derivative can be utilized as a parameter to improve the mathematical model by comparing it to real data. To do so, we must first establish that the solution to the fractional dynamic equations exists and is unique on its interval of existence. The vast majority of existence and uniqueness results available in the literature, including Picard’s method, for ordinary and/or fractional dynamic equations will result in only local existence results. In this work, we generalize Picard’s method to obtain the existence and uniqueness of the solution of the nonlinear multi-order Caputo derivative system with initial conditions, on the interval where the solution is bounded. The challenge presented to establish our main result is in developing a generalized form of the Mittag–Leffler function that will cooperate with all the different fractional derivative orders involved in the multi-order nonlinear Caputo fractional differential system. In our work, we have developed the generalized Mittag–Leffler function that suffices to establish the generalized Picard’s method for the nonlinear multi-order system. As a result, we have obtained the existence and uniqueness of the nonlinear multi-order Caputo derivative system with initial conditions in the large. In short, the solution exists and is unique on the interval where the norm of the solution is bounded. The generalized Picard’s method we have developed is both a theoretical and a computational method of computing the unique solution on the interval of its existence.https://www.mdpi.com/2673-9321/3/2/21caputo fractional differential systemsmulti-order systemsexistence in the large |
| spellingShingle | Zachary Denton Aghalaya S. Vatsala Existence in the Large for Caputo Fractional Multi-Order Systems with Initial Conditions Foundations caputo fractional differential systems multi-order systems existence in the large |
| title | Existence in the Large for Caputo Fractional Multi-Order Systems with Initial Conditions |
| title_full | Existence in the Large for Caputo Fractional Multi-Order Systems with Initial Conditions |
| title_fullStr | Existence in the Large for Caputo Fractional Multi-Order Systems with Initial Conditions |
| title_full_unstemmed | Existence in the Large for Caputo Fractional Multi-Order Systems with Initial Conditions |
| title_short | Existence in the Large for Caputo Fractional Multi-Order Systems with Initial Conditions |
| title_sort | existence in the large for caputo fractional multi order systems with initial conditions |
| topic | caputo fractional differential systems multi-order systems existence in the large |
| url | https://www.mdpi.com/2673-9321/3/2/21 |
| work_keys_str_mv | AT zacharydenton existenceinthelargeforcaputofractionalmultiordersystemswithinitialconditions AT aghalayasvatsala existenceinthelargeforcaputofractionalmultiordersystemswithinitialconditions |