Weighted Generalized Fractional Integration by Parts and the Euler–Lagrange Equation
Integration by parts plays a crucial role in mathematical analysis, e.g., during the proof of necessary optimality conditions in the calculus of variations and optimal control. Motivated by this fact, we construct a new, right-weighted generalized fractional derivative in the Riemann–Liouville sense...
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| Format: | Article |
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MDPI AG
2022-04-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/11/4/178 |
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| author | Houssine Zine El Mehdi Lotfi Delfim F. M. Torres Noura Yousfi |
| author_facet | Houssine Zine El Mehdi Lotfi Delfim F. M. Torres Noura Yousfi |
| author_sort | Houssine Zine |
| collection | DOAJ |
| description | Integration by parts plays a crucial role in mathematical analysis, e.g., during the proof of necessary optimality conditions in the calculus of variations and optimal control. Motivated by this fact, we construct a new, right-weighted generalized fractional derivative in the Riemann–Liouville sense with its associated integral for the recently introduced weighted generalized fractional derivative with Mittag–Leffler kernel. We rewrite these operators equivalently in effective series, proving some interesting properties relating to the left and the right fractional operators. These results permit us to obtain the corresponding integration by parts formula. With the new general formula, we obtain an appropriate weighted Euler–Lagrange equation for dynamic optimization, extending those existing in the literature. We end with the application of an optimization variational problem to the quantum mechanics framework. |
| first_indexed | 2024-03-09T11:09:20Z |
| format | Article |
| id | doaj.art-b4e3e9d740d244ffbea83a9027caa5fc |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-09T11:09:20Z |
| publishDate | 2022-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-b4e3e9d740d244ffbea83a9027caa5fc2023-12-01T00:48:40ZengMDPI AGAxioms2075-16802022-04-0111417810.3390/axioms11040178Weighted Generalized Fractional Integration by Parts and the Euler–Lagrange EquationHoussine Zine0El Mehdi Lotfi1Delfim F. M. Torres2Noura Yousfi3Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, PortugalLaboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’sik, Hassan II University of Casablanca, P.O. Box 7955, Sidi Othman, Casablanca 20000, MoroccoCenter for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, PortugalLaboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’sik, Hassan II University of Casablanca, P.O. Box 7955, Sidi Othman, Casablanca 20000, MoroccoIntegration by parts plays a crucial role in mathematical analysis, e.g., during the proof of necessary optimality conditions in the calculus of variations and optimal control. Motivated by this fact, we construct a new, right-weighted generalized fractional derivative in the Riemann–Liouville sense with its associated integral for the recently introduced weighted generalized fractional derivative with Mittag–Leffler kernel. We rewrite these operators equivalently in effective series, proving some interesting properties relating to the left and the right fractional operators. These results permit us to obtain the corresponding integration by parts formula. With the new general formula, we obtain an appropriate weighted Euler–Lagrange equation for dynamic optimization, extending those existing in the literature. We end with the application of an optimization variational problem to the quantum mechanics framework.https://www.mdpi.com/2075-1680/11/4/178weighted generalized fractional calculusintegration by parts formulaEuler–Lagrange equationquantum mechanicscalculus of variations |
| spellingShingle | Houssine Zine El Mehdi Lotfi Delfim F. M. Torres Noura Yousfi Weighted Generalized Fractional Integration by Parts and the Euler–Lagrange Equation Axioms weighted generalized fractional calculus integration by parts formula Euler–Lagrange equation quantum mechanics calculus of variations |
| title | Weighted Generalized Fractional Integration by Parts and the Euler–Lagrange Equation |
| title_full | Weighted Generalized Fractional Integration by Parts and the Euler–Lagrange Equation |
| title_fullStr | Weighted Generalized Fractional Integration by Parts and the Euler–Lagrange Equation |
| title_full_unstemmed | Weighted Generalized Fractional Integration by Parts and the Euler–Lagrange Equation |
| title_short | Weighted Generalized Fractional Integration by Parts and the Euler–Lagrange Equation |
| title_sort | weighted generalized fractional integration by parts and the euler lagrange equation |
| topic | weighted generalized fractional calculus integration by parts formula Euler–Lagrange equation quantum mechanics calculus of variations |
| url | https://www.mdpi.com/2075-1680/11/4/178 |
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