Fractional differential equations

2016-03-22 — 2021-09-13

Suspiciously similar content

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 relate to fractional Brownian motions?

1 References

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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.

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Ortigueira, and Tenreiro Machado. 2015. “What Is a Fractional Derivative?” Journal of Computational Physics, Fractional PDEs,.

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Sardar, Rana, and Chattopadhyay. 2015. “A Mathematical Model of Dengue Transmission with Memory.” Communications in Nonlinear Science and Numerical Simulation.

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