Derivation of fractional-order models from a stochastic process

Abstract

Fractional calculus has a long history, almost as old as calculus itself, dating back to the late seventeenth century. There has been a great deal of mathematical interest in this area by pure mathematicians but it is only in recent decades that the applications of fractional calculus have been systematically explored. The physical interest in fractional calculus is due to it’s nonlocal nature which introduces a history dependence into the system.

Differential equations are the mainstay of mathematical models that describe and predict the evolution of systems in time. It is intuitive to replace some of the integer order time derivatives with fractional order time derivatives to provide a model that incorporates a history dependence. However, including fractional derivatives in this way can lead to problems in reconciling the dimensions of parameters in the systems. In this thesis we have developed a modelling approach, to include fractional derivatives and a history dependence, which is based on a well defined stochastic process. The resulting fractional order models and their parameters are well posed.

The thesis begins with a discussion of the history of fractional calculus, leading to the application to partial differential equations (PDEs), derived from continuous time random walks (CTRWs). We provide a brief overview of CTRWs and their role in deriving fractional order ordinary differential equations (ODEs) and PDEs. Some of the fundamental tools of fractional calculus are introduced. A discrete time analogue of a CTRW is also introduced.

After the introductory material, the remainder of the thesis is a compilation of original published work that I have co-authored. This material is separated into three parts. Part I, consisting of Chapters 2 - 7, features the derivation of fractional order ODE models and their discretisations. Part II, consisting of Chapters 8 - 10, is focused on the derivation of fractional PDE models. Part III, consisting of Chapters 11 - 12, presents novel numerical approaches for solving fractional-order ODEs and PDEs via piecewise approximations. This thesis includes the results drawn from nine published papers produced over the course of my PhD candidature.