Fractional differential equations



Classically, (stochastic or deterministic) ODEs are “memoryless” in the sense that the current state (and not the history) of the system determines the future states/distribution of states. In the stochastic case, they are Markov.

One way you can destroy this locality/memorylessness is by using fractional derivatives in the formulation of the equation. These use the Laplace-transform representation to do something like differentiating to a non-integer order.

This is not the only way we could introduce memory; for example we could put some explicit integrals over the history of the process into the defining equations; but that is a different notebook, or a hidden state. But this option fits into certain ODEs elegantly, which is an attraction.

Note some evocative similarities to branching processes which I usually study in discrete index and/or state space, and a connection I have been told exists but do not understand, to fractals.

Popular in modelling Dengue and pharmacokinetics, whatever that is. Keywords that pop up in the vicinity: Super diffusive systems

How do these related to fractional Brownian motions?

References

Ahmed, E., and A. S. Elgazzar. 2007. “On Fractional Order Differential Equations Model for Nonlocal Epidemics.” Physica A: Statistical Mechanics and Its Applications 379 (2): 607–14. https://doi.org/10.1016/j.physa.2007.01.010.
Bendahmane, Mostafa, Ricardo Ruiz-Baier, and Canrong Tian. 2015. “Turing Pattern Dynamics and Adaptive Discretization for a Super-Diffusive Lotka-Volterra Model.” Journal of Mathematical Biology 72 (6): 1441–65. https://doi.org/10.1007/s00285-015-0917-9.
Bucur, Claudia, and Enrico Valdinoci. 2016. Nonlocal Diffusion and Applications. Vol. 20. Lecture Notes of the Unione Matematica Italiana. New York, NY: Springer International Publishing. https://doi.org/10.1007/978-3-319-28739-3.
Camrud, Evan. 2017. “A Novel Approach to Fractional Calculus: Utilizing Fractional Integrals and Derivatives of the Dirac Delta Function.” arXiv:1708.06605 [math], August. https://doi.org/10.18576/pfda/040402.
Granger, C. W. J., and Roselyne Joyeux. 1980. “An Introduction to Long-Memory Time Series Models and Fractional Differencing.” Journal of Time Series Analysis 1 (1): 15–29. https://doi.org/10.1111/j.1467-9892.1980.tb00297.x.
Hurvich, Clifford M. 2002. “Multistep Forecasting of Long Memory Series Using Fractional Exponential Models.” International Journal of Forecasting, Forecasting Long Memory Processes, 18 (2): 167–79. https://doi.org/10.1016/S0169-2070(01)00151-0.
Nguyen Tien, Dung. 2013. “Fractional Stochastic Differential Equations with Applications to Finance.” Journal of Mathematical Analysis and Applications 397 (1): 334–48. https://doi.org/10.1016/j.jmaa.2012.07.062.
Ortigueira, Manuel D., and J. A. Tenreiro Machado. 2015. “What Is a Fractional Derivative?” Journal of Computational Physics, Fractional PDEs, 293 (July): 4–13. https://doi.org/10.1016/j.jcp.2014.07.019.
Sabzikar, Farzad, Mark M. Meerschaert, and Jinghua Chen. 2015. “Tempered Fractional Calculus.” Journal of Computational Physics, Fractional PDEs, 293 (July): 14–28. https://doi.org/10.1016/j.jcp.2014.04.024.
Sardar, Tridip, Sourav Rana, and Joydev Chattopadhyay. 2015. “A Mathematical Model of Dengue Transmission with Memory.” Communications in Nonlinear Science and Numerical Simulation 22 (1–3): 511–25. https://doi.org/10.1016/j.cnsns.2014.08.009.
Williams, Paul. 2007. “Fractional Calculus of Schwartz Distributions.”

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