Fractional calculus with exponential memory
- 1 March 2021
- journal article
- research article
- Published by AIP Publishing in Chaos: An Interdisciplinary Journal of Nonlinear Science
- Vol. 31 (3), 031103
- https://doi.org/10.1063/5.0043555
Abstract
The standard definition of the Riemann-Liouville integral is revisited. A new fractional integral is proposed with an exponential kernel. Furthermore, some useful properties such as composition relationship of the new fractional integral and Leibniz integral law are provided. Exact solutions of the fractional homogeneous equation and the non-homogeneous equations are given, respectively. Finally, a finite difference scheme is proposed for solving fractional nonlinear differential equations with exponential memory. The results show the efficiency and convenience of the new fractional derivative.Funding Information
- National Natural Science Foundation of China (62076141)
This publication has 18 references indexed in Scilit:
- Tempered fractional Feynman-Kac equation: Theory and examplesPhysical Review E, 2016
- Positive Solutions of the Fractional Relaxation Equation Using Lower and Upper SolutionsVietnam Journal of Mathematics, 2016
- High order schemes for the tempered fractional diffusion equationsAdvances in Computational Mathematics, 2015
- Tempered fractional calculusJournal of Computational Physics, 2015
- Nonlinear dynamics and chaos in fractional-order neural networksNeural Networks, 2012
- Fractional differential approach to detecting textural features of digital image and its fractional differential filter implementationScience China Information Sciences, 2008
- A variable order constitutive relation for viscoelasticityAnnalen der Physik, 2007
- The random walk's guide to anomalous diffusion: a fractional dynamics approachPhysics Reports, 2000
- From continuous time random walks to the fractional Fokker-Planck equationPhysical Review E, 2000
- Deriving fractional Fokker-Planck equations from a generalised master equationEurophysics Letters, 1999