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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.