Derivatives and integrals of fractional orders are used to describe objects that can be characterized by power-law nonlocality,[2] power-law long-range dependence or fractal properties.
[1] A general dynamical system of fractional order can be written in the form[3] where
A common special case of this is the linear time-invariant (LTI) system in one variable: The orders
are in general complex quantities, but two interesting cases are when the orders are commensurate and when they are also rational: When
, the derivatives are of integer order and the system becomes an ordinary differential equation.
By applying a Laplace transform to the LTI system above, the transfer function becomes For general orders
Non-rational transfer functions cannot be written as an expansion in a finite number of terms (e.g., a binomial expansion would have an infinite number of terms) and in this sense fractional orders systems can be said to have the potential for unlimited memory.
[3] Exponential laws are a classical approach to study dynamics of population densities, but there are many systems where dynamics undergo faster or slower-than-exponential laws.
In such case the anomalous changes in dynamics may be best described by Mittag-Leffler functions.
[4] Anomalous diffusion is one more dynamic system where fractional-order systems play significant role to describe the anomalous flow in the diffusion process.
In case of real materials the relationship between stress and strain given by Hooke's law and Newton's law both have obvious disadvances.
So G. W. Scott Blair introduced a new relationship between stress and strain given by In chaos theory, it has been observed that chaos occurs in dynamical systems of order 3 or more.
[5] In neuroscience, it has been found that single rat neocortical pyramidal neurons adapt with a time scale that depends on the time scale of changes in stimulus statistics.
This multiple time scale adaptation is consistent with fractional order differentiation, such that the neuron's firing rate is a fractional derivative of slowly varying stimulus parameters.
For numerical simulation of solution of the above equations, Kai Diethelm has suggested fractional linear multistep Adams–Bashforth method or quadrature methods.