Fractional differential equations

March 22, 2016 — September 13, 2021

Figure 1

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?

1 References

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Bendahmane, Ruiz-Baier, and Tian. 2015. Turing Pattern Dynamics and Adaptive Discretization for a Super-Diffusive Lotka-Volterra Model.” Journal of Mathematical Biology.
Bucur, and Valdinoci. 2016. Nonlocal Diffusion and Applications. Lecture Notes of the Unione Matematica Italiana.
Camrud. 2017. A Novel Approach to Fractional Calculus: Utilizing Fractional Integrals and Derivatives of the Dirac Delta Function.” arXiv:1708.06605 [Math].
Granger, and Joyeux. 1980. An Introduction to Long-Memory Time Series Models and Fractional Differencing.” Journal of Time Series Analysis.
Hurvich. 2002. Multistep Forecasting of Long Memory Series Using Fractional Exponential Models.” International Journal of Forecasting, Forecasting Long Memory Processes,.
Nguyen Tien. 2013. Fractional Stochastic Differential Equations with Applications to Finance.” Journal of Mathematical Analysis and Applications.
Ortigueira, and Tenreiro Machado. 2015. What Is a Fractional Derivative? Journal of Computational Physics, Fractional PDEs,.
Sabzikar, Meerschaert, and Chen. 2015. Tempered Fractional Calculus.” Journal of Computational Physics, Fractional PDEs,.
Sardar, Rana, and Chattopadhyay. 2015. A Mathematical Model of Dengue Transmission with Memory.” Communications in Nonlinear Science and Numerical Simulation.
Williams. 2007. “Fractional Calculus of Schwartz Distributions.”