Mathematics

Generalising orders of differentiation and integration is very popular nowadays, but some fundamental issues need to be addressed when extending operators and differential equations to fractional orders. If an nth-order differential equation requires n initial conditions, then how many initial conditions does a fractional-order differential equation need, and of what type? If a fractional derivative is defined using an integral from the initial point to the independent variable, how is it possible to choose any initial condition at all? Real-world models should be dimensionally consistent, but what happens to the dimensions when fractional derivatives are involved? All of these questions and more will be addressed in this overview: starting from the basics of fractional calculus, proceeding to some fresh new ideas, and leaving open some questions concerning stochastic properties and applications.