# Fractional differential equations

March 22, 2016 — September 13, 2021

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

*Physica A: Statistical Mechanics and Its Applications*.

*Journal of Mathematical Biology*.

*Nonlocal Diffusion and Applications*. Lecture Notes of the Unione Matematica Italiana.

*arXiv:1708.06605 [Math]*.

*Journal of Time Series Analysis*.

*International Journal of Forecasting*, Forecasting Long Memory Processes,.

*Journal of Mathematical Analysis and Applications*.

*Journal of Computational Physics*, Fractional PDEs,.

*Journal of Computational Physics*, Fractional PDEs,.

*Communications in Nonlinear Science and Numerical Simulation*.