A note on function space and boundedness of the general fractional integral in continuous time random walk. (English) Zbl 1486.26009
Summary: The general fractional calculus becomes popular in continuous time random walk recently. However, the boundedness condition of the general fractional integral is one of the fundamental problems. It wasn’t given yet. In this short communication, the classical norm space is used, and a general boundedness theorem is presented. Finally, various long-tailed waiting time probability density functions are suggested in continuous time random walk since the general fractional integral is well defined.
MSC:
| 26A33 | Fractional derivatives and integrals |
| 60K50 | Anomalous diffusion models (subdiffusion, superdiffusion, continuous-time random walks, etc.) |
References:
| [1] | Osler, TJ, Leibniz rule for fractional derivatives generalized and an application to infinite series, SIAM J. Appl. Math., 18, 658-674 (1970) · Zbl 0201.44102 · doi:10.1137/0118059 |
| [2] | Samko, SG; Kilbas, AA; Marichev, OI, Fractional integrals and derivatives: theory and applications (1993), CRC Press · Zbl 0818.26003 |
| [3] | Kilbas, AA; Srivastava, HM; Trujillo, JJ, Theory and Applications of Fractional Differential Equations (2006), Amsterdam: Elsevier Science B. V, Amsterdam · Zbl 1092.45003 |
| [4] | Almeida, R., A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simulat., 44, 460-481 (2017) · Zbl 1465.26005 · doi:10.1016/j.cnsns.2016.09.006 |
| [5] | Fahad, HM; Rehman, MU; Fernandez, A., On Laplace transforms with respect to functions and their applications to fractional differential equations, Math. Method. Appl. Sci. (2021) · Zbl 1541.44004 · doi:10.1002/mma.7772 |
| [6] | Farad, J.; Abdeljawad, T., Generalized fractional derivatives and Laplace transform, Discrete Cont. Dyn-S, 13, 709-722 (2020) · Zbl 1444.44002 |
| [7] | Restrepo, JE; Ruzhansky, M.; Suragan, D., Explicit solutions for linear variable-coefficient fractional differential equations with respect to functions, Appl. Math. Comput., 403, 126177 (2021) · Zbl 1510.34022 |
| [8] | Fahad, HM; Fernandez, A., Operational calculus for Caputo fractional calculus with respect to functions and the associated fractional differential equations, Appl. Math. Comput., 409, 126400 (2021) · Zbl 1510.44008 |
| [9] | Wu, G.C., Kong, H., Luo, M., Huang, L.L.: Unified predictor-corrector method for fractional differential equations with general kernel functions (2021) (submitted) |
| [10] | Metzler, R.; Barkai, E.; Klafter, J., Anomalous diffusion and relaxation close to thermal equilibrium: A fractional Fokker-Planck equation approach, Phys. Rev. Lett., 82, 3563 (1999) · doi:10.1103/PhysRevLett.82.3563 |
| [11] | Barkai, E.; Metzler, R.; Klafter, J., From continuous time random walks to the fractional Fokker-Planck equation, Phys. Rev. E, 61, 132 (2000) · doi:10.1103/PhysRevE.61.132 |
| [12] | Fu, H.; Wu, GC; Yang, G.; Huang, LL, Continuous time random walk to a general fractional Fokker-Planck equation on fractal media, Eur. Phys. J. Spec. Top. (2021) · doi:10.1140/epjs/s11734-021-00323-6 |
| [13] | Fahad, HM; Fernandez, A.; Rehman, MU; Siddiqi, M., Tempered and Hadamard-type fractional calculus with respect to functions, Mediterr. J. Math., 18, 143 (2021) · Zbl 1470.26009 · doi:10.1007/s00009-021-01783-9 |
| [14] | Katugampola, UN, New apporach to a generalized fractional integral, Appl. Math. Comput., 218, 860-865 (2011) · Zbl 1231.26008 |
| [15] | Garra, R.; Giusti, A.; Mainardi, F., The fractional Dodson diffusion equation: a new approach, Ricerche Mat., 67, 899-909 (2018) · Zbl 1403.35314 · doi:10.1007/s11587-018-0354-3 |
| [16] | Fu, H.; Wu, GC; Yang, G.; Huang, LL, Fractional calculus with exponential memory, Chaos, 31, 031103 (2021) · Zbl 1459.26010 · doi:10.1063/5.0043555 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
