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.

Development and evaluation of hydrocarbon sorbent materials with the aid of chemometrics

Hydrocarbon spills on roads are a major safety concern for the driving public and can have severe cost impacts both on pavement maintenance and to the economy through disruption to services. The time taken to clean-up spills and re-open roads in a safe driving condition is an issue of increasing concern given traffic levels on major urban arterials. Thus, the primary aim of the research was to develop a sorbent material that facilitates rapid clean-up of road spills. The methodology involved extensive research into a range of materials (organic, inorganic and synthetic sorbents), comprehensive testing in the laboratory, scale-up and field, and product design (i.e. concept to prototype). The study also applied chemometrics to provide consistent, comparative methods of sorbent evaluation and performance. In addition, sorbent materials at every stage were compared against a commercial benchmark. For the first time, the impact of diesel on asphalt pavement has been quantified and assessed in a systematic way. Contrary to conventional thinking and anecdotal observations, the study determined that the action of diesel on asphalt was quite rapid (i.e. hours rather than weeks or months). This significant finding demonstrates the need to minimise the impact of hydrocarbon spills and the potential application of the sorbent option. To better understand the adsorption phenomenon, surface characterisation techniques were applied to selected sorbent materials (i.e. sand, organo-clay and cotton fibre). Brunauer Emmett Teller (BET) and thermal analysis indicated that the main adsorption mechanism for the sorbents occurred on the external surface of the material in the diffusion region (sand and organo-clay) and/or capillaries (cotton fibre). Using environmental scanning electron microscopy (ESEM), it was observed that adsorption by the interfibre capillaries contributed to the high uptake of hydrocarbons by the cotton fibre. Understanding the adsorption mechanism for these sorbents provided some guidance and scientific basis for the selection of materials. The study determined that non-woven cotton mats were ideal sorbent materials for clean-up of hydrocarbon spills. The prototype sorbent was found to perform significantly better than the commercial benchmark, displaying the following key properties: • superior hydrocarbon pick-up from the road pavement; • high hydrocarbon retention capacity under an applied load; • adequate field skid resistance post treatment; • functional and easy to use in the field (e.g. routine handling, transportation, application and recovery); • relatively inexpensive to produce due to the use of raw cotton fibre and simple production process; • environmentally friendly (e.g. renewable materials, non-toxic to environment and operators, and biodegradable); and • rapid response time (e.g. two minutes total clean-up time compared with thirty minutes for reference sorbents). The major outcomes of the research project include: a) development of a specifically designed sorbent material suitable for cleaning up hydrocarbon spills on roads; b) submission of patent application (serial number AU2005905850) for the prototype product; and c) preparation of Commercialisation Strategy to advance the sorbent product to the next phase (i.e. R&D to product commercialisation).

Modelling of some biological materials using continuum mechanics

Continuum mechanics provides a mathematical framework for modelling the physical stresses experienced by a material. Recent studies show that physical stresses play an important role in a wide variety of biological processes, including dermal wound healing, soft tissue growth and morphogenesis. Thus, continuum mechanics is a useful mathematical tool for modelling a range of biological phenomena. Unfortunately, classical continuum mechanics is of limited use in biomechanical problems. As cells refashion the �bres that make up a soft tissue, they sometimes alter the tissue's fundamental mechanical structure. Advanced mathematical techniques are needed in order to accurately describe this sort of biological `plasticity'. A number of such techniques have been proposed by previous researchers. However, models that incorporate biological plasticity tend to be very complicated. Furthermore, these models are often di�cult to apply and/or interpret, making them of limited practical use. One alternative approach is to ignore biological plasticity and use classical continuum mechanics. For example, most mechanochemical models of dermal wound healing assume that the skin behaves as a linear viscoelastic solid. Our analysis indicates that this assumption leads to physically unrealistic results. In this thesis we present a novel and practical approach to modelling biological plasticity. Our principal aim is to combine the simplicity of classical linear models with the sophistication of plasticity theory. To achieve this, we perform a careful mathematical analysis of the concept of a `zero stress state'. This leads us to a formal de�nition of strain that is appropriate for materials that undergo internal remodelling. Next, we consider the evolution of the zero stress state over time. We develop a novel theory of `morphoelasticity' that can be used to describe how the zero stress state changes in response to growth and remodelling. Importantly, our work yields an intuitive and internally consistent way of modelling anisotropic growth. Furthermore, we are able to use our theory of morphoelasticity to develop evolution equations for elastic strain. We also present some applications of our theory. For example, we show that morphoelasticity can be used to obtain a constitutive law for a Maxwell viscoelastic uid that is valid at large deformation gradients. Similarly, we analyse a morphoelastic model of the stress-dependent growth of a tumour spheroid. This work leads to the prediction that a tumour spheroid will always be in a state of radial compression and circumferential tension. Finally, we conclude by presenting a novel mechanochemical model of dermal wound healing that takes into account the plasticity of the healing skin.