A general dynamical system of fractional order can be written in the form[3]
where and are functions of the fractional derivative operator of orders and and and are functions of time. A common special case of this is the linear time-invariant (LTI) system in one variable:
The orders and 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. Thus by increasing specialization, LTI systems can be of general order, commensurate order, rational order or integer order.
For general orders and this is a non-rational transfer function. 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]
Motivation to study fractional-order systems
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.
Viscoelasticity is the property of material in which the material exhibits its nature between purely elastic and pure fluid. 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. With the introduction of fractional-order systems, some researchers study chaos in the system of total order less than 3.[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.[6]
Here, under the continuity condition on function f, one can convert the above equation into corresponding integral equation.
One can construct a solution space and define, by that equation, a continuous self-map on the solution space, then apply a fixed-point theorem, to get a fixed-point, which is the solution of above equation.
Tarasov, V.E. (2010). Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer. ISBN978-3-642-14003-7.
Miller, Kenneth S. (1993). Ross, Bertram (ed.). An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley. ISBN0-471-58884-9.
Oldham, Keith B.; Spanier, Jerome (1974). The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order. Mathematics in Science and Engineering. Vol. V. Academic Press. ISBN0-12-525550-0.