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Fractional differential equations

Usefulness: 🔧
Novelty: 💡
Uncertainty: 🤪 🤪 🤪
Incompleteness: 🚧 🚧 🚧

“Super diffusive” systems, non-Markov processes… 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.

One way you can destroy this is by using fractional derivatives in the formulation of the equation. (Why this choice, as opposed to putting in explicit integrals over the history of the process, I have no idea. Perhaps it leads to more elegant parameterization or solutions?)

I’ll make this precise later, but want to note some evocative similarities to other branching processes which I usually study in discrete index and/or state space.

Popular in modelling Dengue and pharmacokinetics, whatever that is. Connections to to Lévy flights.

To learn: Why not presume a state filter model with hidden states, and learn that? Seems more general and no less tractable.

Refs

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,” 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.”