Fractional differential equations

Classically, (stochastic or deterministic) ODEs are β€œmemoryless” 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?


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Bendahmane, Mostafa, Ricardo Ruiz-Baier, and Canrong Tian. 2015. β€œTuring Pattern Dynamics and Adaptive Discretization for a Super-Diffusive Lotka-Volterra Model.” Journal of Mathematical Biology 72 (6): 1441–65.
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Camrud, Evan. 2017. β€œA Novel Approach to Fractional Calculus: Utilizing Fractional Integrals and Derivatives of the Dirac Delta Function.” arXiv:1708.06605 [Math], August.
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Ortigueira, Manuel D., and J. A. Tenreiro Machado. 2015. β€œWhat Is a Fractional Derivative?” Journal of Computational Physics, Fractional PDEs, 293 (July): 4–13.
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Sardar, Tridip, Sourav Rana, and Joydev Chattopadhyay. 2015. β€œA Mathematical Model of Dengue Transmission with Memory.” Communications in Nonlinear Science and Numerical Simulation 22 (1–3): 511–25.
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