Novel analytical and numerical methods for solving fractional dynamical systems

During the past three decades, the subject of fractional calculus (that is, calculus of integrals and derivatives of arbitrary order) has gained considerable popularity and importance, mainly due to its demonstrated applications in numerous diverse and widespread fields in science and engineering. For example, fractional calculus has been successfully applied to problems in system biology, physics, chemistry and biochemistry, hydrology, medicine, and finance. In many cases these new fractional-order models are more adequate than the previously used integer-order models, because fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes that are governed by anomalous diffusion. Hence, there is a growing need to find the solution behaviour of these fractional differential equations. However, the analytic solutions of most fractional differential equations generally cannot be obtained. As a consequence, approximate and numerical techniques are playing an important role in identifying the solution behaviour of such fractional equations and exploring their applications. The main objective of this thesis is to develop new effective numerical methods and supporting analysis, based on the finite difference and finite element methods, for solving time, space and time-space fractional dynamical systems involving fractional derivatives in one and two spatial dimensions. A series of five published papers and one manuscript in preparation will be presented on the solution of the space fractional diffusion equation, space fractional advectiondispersion equation, time and space fractional diffusion equation, time and space fractional Fokker-Planck equation with a linear or non-linear source term, and fractional cable equation involving two time fractional derivatives, respectively. One important contribution of this thesis is the demonstration of how to choose different approximation techniques for different fractional derivatives. Special attention has been paid to the Riesz space fractional derivative, due to its important application in the field of groundwater flow, system biology and finance. We present three numerical methods to approximate the Riesz space fractional derivative, namely the L1/ L2-approximation method, the standard/shifted Gr¨unwald method, and the matrix transform method (MTM). The first two methods are based on the finite difference method, while the MTM allows discretisation in space using either the finite difference or finite element methods. Furthermore, we prove the equivalence of the Riesz fractional derivative and the fractional Laplacian operator under homogeneous Dirichlet boundary conditions – a result that had not previously been established. This result justifies the aforementioned use of the MTM to approximate the Riesz fractional derivative. After spatial discretisation, the time-space fractional partial differential equation is transformed into a system of fractional-in-time differential equations. We then investigate numerical methods to handle time fractional derivatives, be they Caputo type or Riemann-Liouville type. This leads to new methods utilising either finite difference strategies or the Laplace transform method for advancing the solution in time. The stability and convergence of our proposed numerical methods are also investigated. Numerical experiments are carried out in support of our theoretical analysis. We also emphasise that the numerical methods we develop are applicable for many other types of fractional partial differential equations.

Dangerous places: A novel

Dangerous Places is a novel about the gap between mythological (or 'dreamed') constructions of reality and actual life. The story centres on V en, a married woman with two young children. Her love for her children is fiercely protective and encompassing, but she feels alienated from her husband and to a certain extent her society; so when her first love, Yanni, re-enters her life,she is strongly tempted to resume her affair with him. She is however seduced more by the memories she has 'mythologized' about him than by his physical reality; in the course of the novel she is forced to come to terms with her own delusions. The subplot of the novel involves other characters who are caught between illusion and reality as well, and who deal with 'truth' in differing ways. The themes of the book are explored using a number of structures which underlie and support the surface story. The Greek myths of Adonis/ Aphrodite and Hades/Persephone are framing agents for the plot, and the setting in contemporary Brisbane and North Stradbroke Island is symbolic.

Re-visioning a "novel concept" : beyond vision and revision to advanced editing strategies

This thesis is a work of creative practice-led research comprising two components. The first component is a speculative thriller novel, entitled Diamond Eyes. (Contracted for publication in 2009 by Harper Collins: Voyager as the first in a trilogy, under the name AA Bell.) The second component is an exegesis exploring the notion of re-visioning a novel. Re-visioning, not to be confused with revision, refers to advance editing strategies required when the original vision of a novel changes during development.