About Some Possible Implementations of the Fractional Calculus
| dc.contributor.author | Velasco, María Pilar | |
| dc.contributor.author | Usero Mainer, David | |
| dc.contributor.author | Jiménez, Salvador | |
| dc.contributor.author | Vázquez Martínez, Luis | |
| dc.contributor.author | Vázquez Poletti, José Luis | |
| dc.contributor.author | Mortazavi, Mina | |
| dc.date.accessioned | 2023-06-17T08:55:56Z | |
| dc.date.available | 2023-06-17T08:55:56Z | |
| dc.date.issued | 2020 | |
| dc.description.abstract | We present a partial panoramic view of possible contexts and applications of the fractional calculus. In this context, we show some different applications of fractional calculus to different models in ordinary differential equation (ODE) and partial differential equation (PDE) formulations ranging from the basic equations of mechanics to diffusion and Dirac equations. | |
| dc.description.faculty | Instituto de Matemática Interdisciplinar (IMI) | |
| dc.description.refereed | TRUE | |
| dc.description.sponsorship | Unión Europea. Horizonte 2020 | |
| dc.description.sponsorship | Ministerio de Economía y Competitividad (MINECO) | |
| dc.description.status | pub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/63195 | |
| dc.identifier.doi | 10.3390/math8060893 | |
| dc.identifier.issn | 2227-7390 | |
| dc.identifier.officialurl | https://doi.org/10.3390/math8060893 | |
| dc.identifier.relatedurl | https://www.mdpi.com/2227-7390/8/6/893 | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/7549 | |
| dc.issue.number | 6 | |
| dc.journal.title | Mathematics | |
| dc.language.iso | eng | |
| dc.page.initial | 893 | |
| dc.publisher | https://mdpi.com | |
| dc.relation.projectID | IN-TIME (823934) | |
| dc.relation.projectID | (ESP2016-79135-R) | |
| dc.rights | Atribución 3.0 España | |
| dc.rights.accessRights | open access | |
| dc.rights.uri | https://creativecommons.org/licenses/by/3.0/es/ | |
| dc.subject.cdu | 517.9 | |
| dc.subject.keyword | Fractional calculus | |
| dc.subject.keyword | Fractional differential equations | |
| dc.subject.keyword | Nonlocal effects | |
| dc.subject.keyword | Ecuaciones diferenciales fraccionadas | |
| dc.subject.ucm | Matemáticas (Matemáticas) | |
| dc.subject.ucm | Ecuaciones diferenciales | |
| dc.subject.unesco | 12 Matemáticas | |
| dc.subject.unesco | 1202.07 Ecuaciones en Diferencias | |
| dc.title | About Some Possible Implementations of the Fractional Calculus | |
| dc.type | journal article | |
| dc.volume.number | 8 | |
| dcterms.references | 1. Samko, S.G.; Kilbas, A.A.; Marichev, O.I. Fractional Integrals and Derivatives: Theory and Applications; Taylor & Francis: London, UK, 2002 2. Podlubny, I. Fractional Differential Equations; Academic Press: San Diego, CA, USA, 1999. 3. Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; Elsevier: Amsterdam, The Netherlands, 2006. 4. Baleanu, D.; Fernández, A. On fractional operators and their classifications. Mathematics 2019, 7, 830. 5. Ortigueira, M.D.; Machado, J.A.T. Which Derivative? Fractal Fract. 2017, 1, 3. 6. Ortigueira, M.D.; Machado, J.A.T. On fractional vectorial calculus. Bull. Pol. Acad. Tech. 2018, 66, 389–402. 7. Ortigueira, M.D.; Machado, J.A.T. Fractional Derivatives: The Perspective of System Theory. Mathematics 2019, 7, 150. 8. Atangana, A.; Baleanu, D. New fractional derivatives with non-local and non-singular kernel. Theory and application to heat transfer model. Therm. Sci. 2016, 20, 763–769. 9. Baleanu, D.; Fernandez, A. On some new properties of fractional derivatives with Mittag-Leffler kernel. Commun. Nonlinear Sci. Numer. Simul. 2018, 59, 444–462. 10. Sopasakis, P.; Sarimveis, H.; Macheras, P.; Dokoumetzidis, A. Fractional calculus in pharmacokinetics. J. Pharmacokinet. Pharmacodyn. 2018, 45, 107–125. 11. Ionescu, C.M. A computationally efficient Hill curve adaptation strategy during continuous monitoring of dose-effect relation in anaesthesia. Nonlinear Dyn. 2018, 92, 843–852. 12. Turchetti, G.; Usero, D.; Vázquez, L. Hamiltonian systems with fractional time derivative. Tamsui Oxf. J. Math. Sci. 2002, 18, 31–44. 13. Usero, D. Propagación de Ondas no Lineales en Medios Heterogéneos. Ph.D. Thesis, Universidad Complutense de Madrid, Madrid, Spain, 2004. 14. Usero, D.; Vázquez, L. Fractional derivative: A new formulation for damped systems. Localization Energy Transf. Nonlinear Syst. 2003, 296–303. 15. Riewe, F. Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 1996, 53, 1890–1899. 16. Riewe, F. Mechanics with fractional derivatives. Phys. Rev. E 1997, 55, 3581–3592. 17. Agrawall, O.P. Formulation of Euler–Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 2001, 272, 368–379. 18. Muslih, S.I.; Baleanu, D. Hamiltonian formulation of systems with linear velocities within Riemann–Liouville fractional derivatives. J. Math. Anal. Appl. 2005, 304, 599–606. 19. Rabei, E.M.; Nawafleh, K.H.I.; Hijjawi, R.S.; Muslih, S.I.; Baleanu, D. The Hamilton formalism with fractional derivatives. J. Math. Anal. Appl. 2007, 327, 891–897. 20. Dattoli, G.; Cesarano, C.; Ricci, P.E.; Vázquez, L. Fractional operators, integral representations and special polynomials. Int. J. Appl. Math. 2003, 10, 131–139. 21. Dattoli, G.; Cesarano, C.; Ricci, P.E.; Vázquez, L. Special polynomials and fractional calculus. Math. Comput. Model. 2003, 37, 729–733. 22. Rodríguez-Germá, L.; Trujillo, J.J.; Velasco, M.P. Fractional calculus framework to avoid singularities of differential equations. Fract. Cal. Appl. Anal. 2008, 11, 431–441. 23. Rivero, M.; Rodríguez-Germá, L.; Trujillo, J.J.; Velasco, M.P. Fractional operators and some special functions. Comput. Math. Appl. 2010, 59, 1822–1834. 24. Magin, R.L. Fractional Calculus in Bioengineering; Begell House Publishers: Danbury, CT, USA, 2006. 25. Rocco, A.; West, B.J. Fractional calculus and the evoluction of fractal phenomena. Physica A 1999, 265, 535–546. 26. West, B.J.; Bologna, M.; Grigolini, P. Physics of Fractal Operators; Springer Verlag: New York, NY, USA, 2003. 27. Ionescu, C.; Lopes, A.; Copot, D.; Machado, J.A.T.; Bates, J.H.T. The role of fractional calculus in modeling biological phenomena: A review. Commun. Nonlinear Sci. Numer. Simul. 2017, 51, 141–159. 28. Larsson, M.; Balatsky, A. Paul Dirac and the Nobel Prize in Physics. Phys. Today 2019, 72, 46–52. 29. Vázquez, L. Fractional Diffusion Equations with Internal Degrees of Freedom. J. Comp. Math. 2003, 21, 491–494. 30. Pierantozzi, T.; Vázquez, L. An Interpolation between the Wave and Diffusion Equations through the Fractional Evolution Equations Dirac Like. J. Math. Phys. 2005, 46, 113521. 31. Vázquez, L.; Trujillo, J.J.; Velasco, M.P. Fractional heat equation and the second law of thermodynamics. Fract. Cal. Appl. Anal. 2011, 14, 334–342. 32. Angstrom, A. On the atmospheric transmission of sun radiation and on dust in the air. Geogr. Ann. 1929, 11, 156–166 . 33. Córdoba-Jabonero, C.; Vázquez, L. Characterization of atmospheric aerosols by an in-situ photometris technique in planetary environments. SPIE 2005, 4878, 54–58. 34. Jiménez, S.; Usero, D.; Vázquez, L.; Velasco, M.P. Fractional diffusion models for the atmosphere of Mars. Fractal Fract. 2018, 2, 1. 35. Vázquez, L.; Velasco, M.P.; Vázquez-Poletti, J.L.; Llorente, I.M.; Usero, D.; Jiménez, S. Modeling and simulation of the atmospheric dust dynamic: Fractional Calculus and Cloud Computing. Int. J. Numer. Anal. Model. 2018, 15, 74–85. 36. Vázquez-Poletti, J.L.; Llorente, I.M.; Velasco, M.P.; Vicente-Retortillo, A.; Aguirre, C.; Caro-Carretero, R.; Valero, F.; Vázquez, L. Martian Computing Clouds: A Two Use Case Study. In Proceedings of the Seventh Moscow Solar System Symposium (7M-S3), Moscow, Russia, 10–14 October 2016. 37. De Lucas, E.; Miguel, M.J.; Mozos, D.; Vázquez, L. Martian dust devils detector over FPGA. Geosci. Instrum. Method Data Syst. 2012, 1, 23–31. 38. Aguirre, C.; Franzese, G.; Esposito, F.; Vázquez, L.; Caro-Carretero, R.; Vilela-Mendes, R.; Ramírez-Nicolás, M.; Cozzolino, F.; Popa, C.I. Signal-adapted tomography as a tool for dust devil detection. Aeolian Res. 2017, 29, 12–22. 39. Gómez-Elvira, J.; Armiens, C.; Castañer, L.; Domínguez, M.; Genzer, M.; Gómez, F.; Haberle, R.; Harri, A.M.; Jiménez, V.; Kahanpaa, H.; et al. REMS: The Environmental Sensor Suite for the Mars Science Laboratory Rover. Space Sci. Rev. 2012, 170, 583–640. 40. Domínguez-Pumar, M.; Pérez, E.; Ramón, M.; Jiménez, V.; Bermejo, S.; Pons-Nin, J. Acceleration of the Measurement Time of Thermopiles Using Sigma-delta Control. Sensors 2019, 19, 3159. 41. Guckenheimer, J.; Holmes, P.H. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields; Springer-Verlag: New York, NY, USA, 1986. 42. Gao, X.; Yu, J. Chaos in the fractional order periodically forced complex Duffing’s oscillators. Chaos Solitons Fractals 2005, 24, 1097–1104. 43. Sheu, L.J.; Chen, H.K.; Chen, J.H.; Tam, L.M. Chaotic dynamics of the fractionally damped Duffing equation. Chaos Solitons Fractals 2007, 32, 1459–1468. 44. Jiménez, S.; González, J.A.; Vázquez, L. Fractional Duffing’s equation and geometrical resonance. Int. J. Bifurcation Chaos 2013, 23, 1350089-1–1350089-13. 45. Jiménez, S.; Zufiria, P.J. Chaos in a fractional Duffing’s equation. Dyn. Syst. Differ. Equ. Appl. 2015, 10, 660–669. | |
| dspace.entity.type | Publication | |
| relation.isAuthorOfPublication | 254fa329-0c04-4bdf-a7bb-068605fece1a | |
| relation.isAuthorOfPublication | 1b01aaca-9afe-42d7-81ef-a86f477ac820 | |
| relation.isAuthorOfPublication | d3c2b5a8-3672-4a45-b84e-cbd3ba076155 | |
| relation.isAuthorOfPublication.latestForDiscovery | 1b01aaca-9afe-42d7-81ef-a86f477ac820 |
Download
Original bundle
1 - 1 of 1
Loading...
- Name:
- mathematics-08-00893-v3.pdf
- Size:
- 416.07 KB
- Format:
- Adobe Portable Document Format


