Mean value and Taylor-type results for tempered fractional derivatives. (English) Zbl 1541.26028
Summary: In this paper, we deal with the mean value theorem for tempered fractional operators, some properties with monotone functions and the Taylor theorem for Caputo tempered fractional derivative. Furthermore, we present a geometric interpretation of the Riemann-Liouville tempered fractional integral as a shadow on the wall.
MSC:
| 26A33 | Fractional derivatives and integrals |
Keywords:
tempered fractional calculus; mean value theorem; Taylor theorem; tempered stable Lévy processReferences:
| [1] | Almeida, R.; Luísa Morgado, M., Analysis and numerical approximation of tempered fractional calculus of variations problems, J. Comput. Appl. Math., 361, 1-12, 2019 · Zbl 1425.49015 · doi:10.1016/j.cam.2019.04.010 |
| [2] | Andrić, M., Pečarić, P., Perić, I.: Inequalities of Opial and Jensen. Element, Zagreb (2016) · Zbl 1357.26004 |
| [3] | Buschman, R., Decomposition of an integral operator by use of Mikusinski calculus, SIAM J. Math. Anal., 3, 83-85, 1972 · Zbl 0233.45019 · doi:10.1137/0503010 |
| [4] | Bullock, GL, A geometric interpretation of the Riemann-Stieltjes, Am. Math. Mon., 95, 5, 448-455, 1988 · Zbl 0651.26009 · doi:10.1080/00029890.1988.11972030 |
| [5] | Chen, M.; Deng, W., Discretized fractional substantial calculus, ESAIM: Math. Model. Numer. Anal., 49, 373-394, 2015 · Zbl 1314.26007 |
| [6] | del-Castillo-Negrete, D., Truncation effects in superdiffusive front propagation with Lévy flights, Phys. Rev. E, 79, 031120, 2009 · doi:10.1103/PhysRevE.79.031120 |
| [7] | Diethelm, K., Fractional Differential Equations, 2010, Berlin: Springer, Berlin · Zbl 1215.34001 · doi:10.1007/978-3-642-14574-2 |
| [8] | Diethelm, K., Monotonicity of functions and sing changes of their Caputo derivatives, Fract. Calc. Appl. Anal., 19, 2, 561-566, 2016 · Zbl 1337.26009 · doi:10.1515/fca-2016-0029 |
| [9] | Evangelista, L.; Lenzi, E., An Introduction to Anomalous Diffusion and Relaxation, 2023, Cham: Springer, Cham · Zbl 1515.60011 · doi:10.1007/978-3-031-18150-4 |
| [10] | Fernandez, A.; Ustaoǧlu, C., On some analytic properties of tempered fractional calculus, J. Comput. Appl. Math., 366, 112400, 2020 · Zbl 1428.26012 · doi:10.1016/j.cam.2019.112400 |
| [11] | Fernadez, A.; Fahad, H., On the importance of conjugation relations in fractional calculus, Comput. Appl. Math., 41, 246, 2022 · Zbl 1513.26012 · doi:10.1007/s40314-022-01925-z |
| [12] | Gajda, J.; Magdziarz, M., Fractional Fokker-Planck equation with tempered \(\alpha \)-stable waiting times: Langevin picture and computer simulation, Phys. Rev. E, 82, 011117, 2010 · doi:10.1103/PhysRevE.82.011117 |
| [13] | Kullberg, A.; del-Castillo-Negrete, D., Transport in the spatially tempered, fractional Fokker-Planck equation, J. Phys. A Math. Theor., 45, 255101, 2012 · Zbl 1250.82032 · doi:10.1088/1751-8113/45/25/255101 |
| [14] | Meerschaert, M.; Sikorskii, A., Stochastic Models for Fractional Calculus, 2012, Berlin/Boston: Walter de Gruyter GmbH & Co. KG, Berlin/Boston · Zbl 1247.60003 |
| [15] | Meerschaert, M.; Zhang, Y.; Baeumer, B., Tempered anomalous diffusion in heterogeneous systems, Geophys. Res. Lett., 35, L17403, 2008 · doi:10.1029/2008GL034899 |
| [16] | Li, C.; Deng, W.; Zhao, L., Well-posedness and numerical algorithm for the tempered fractional differential equations, Discrete Contin Dyn Syst Ser B, 24, 4, 1989-2015, 2019 · Zbl 1414.34005 |
| [17] | Obeidat, N.; Bentil, D., New theories and applications of tempered fractional differential equations, Nonlinear Dyn., 105, 1689-1702, 2021 · Zbl 1537.34017 · doi:10.1007/s11071-021-06628-4 |
| [18] | Podlubny, I., Geometric and physical interpretation of fractional integration and fractional differentiation, FCAA, 5, 4, 367-386, 2002 · Zbl 1042.26003 |
| [19] | Sabzikar, F.; Meerschaert, M.; Chen, J., Tempered fractional calculus, J. Comput. Phys., 293, 14-28, 2015 · Zbl 1349.26017 · doi:10.1016/j.jcp.2014.04.024 |
| [20] | Salim, A.; Lazreg, J.; Benchohra, M., A Novel study on tempered \((k, \psi )\)-Hilfer fractional operators, Res. Sq., 2023 · doi:10.21203/rs.3.rs-3316718/v1 |
| [21] | Stanislavsky, A.; Weron, K.; Weron, A., Diffusion and relaxation controlled by tempered \(\alpha \)-stable processes, Phys. Rev. E, 78, 051106, 2008 · doi:10.1103/PhysRevE.78.051106 |
| [22] | Torres Ledesma, C.; Cuti Gutierrez, H.; Ávalos Rodríguez, J.; Zubiaga Vera, W., Some boundedness results for Riemann-Liouville tempered fractional integrals, Fract. Calc. Appl. Anal., 2024 · doi:10.1007/s13540-024-00247-7 |
| [23] | Torres Ledesma, C.; Cuti Gutierrez, H.; Ávalos Rodríguez, J.; Montalvo Bonilla, M., Boundary value problem with tempered fractional derivatives and oscillating term, J. Pseudo-Differ. Oper. Appl., 14, 62, 2023 · Zbl 07762276 · doi:10.1007/s11868-023-00558-y |
| [24] | Torres Ledesma, C., Tempered fractional differential equation: variational approach, Math. Meth. Appl. Sci., 40, 4962-4973, 2017 · Zbl 1380.34023 |
| [25] | Torres Ledesma, C.; Montalvo Bonilla, M., Fractional Sobolev space with Riemann-Liouville fractional derivative and application to a fractional concave-convex problem, Adv. Oper. Theory, 6, 65, 2021 · Zbl 1490.26006 · doi:10.1007/s43036-021-00159-w |
| [26] | Zayernouri, M.; Ainsworth, M.; Em Karniadakis, G., Tempered fractional Sturm-Liouville eigenproblems, SIAM J. Sci. Comput., 37, 4, A1777-A1800, 2015 · Zbl 1323.34012 · doi:10.1137/140985536 |
| [27] | Zhang, Y., Moments for tempered fractional advection-diffusion equations, J. Stat. Phys., 139, 915-939, 2010 · Zbl 1301.82044 · doi:10.1007/s10955-010-9965-0 |
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