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.

Numerical modelling of industrial microwave heating

The numerical modelling of electromagnetic waves has been the focus of many research areas in the past. Some specific applications of electromagnetic wave scattering are in the fields of Microwave Heating and Radar Communication Systems. The equations that govern the fundamental behaviour of electromagnetic wave propagation in waveguides and cavities are the Maxwell's equations. In the literature, a number of methods have been employed to solve these equations. Of these methods, the classical Finite-Difference Time-Domain scheme, which uses a staggered time and space discretisation, is the most well known and widely used. However, it is complicated to implement this method on an irregular computational domain using an unstructured mesh. In this work, a coupled method is introduced for the solution of Maxwell's equations. It is proposed that the free-space component of the solution is computed in the time domain, whilst the load is resolved using the frequency dependent electric field Helmholtz equation. This methodology results in a timefrequency domain hybrid scheme. For the Helmholtz equation, boundary conditions are generated from the time dependent free-space solutions. The boundary information is mapped into the frequency domain using the Discrete Fourier Transform. The solution for the electric field components is obtained by solving a sparse-complex system of linear equations. The hybrid method has been tested for both waveguide and cavity configurations. Numerical tests performed on waveguides and cavities for inhomogeneous lossy materials highlight the accuracy and computational efficiency of the newly proposed hybrid computational electromagnetic strategy.

Changes in Level of Virus Accumulation and Incidence of Infection are Critical in the Characterization of Rice Tungro Bacilliform Virus (RTBV) Resistance in Rice

Analysis by enzyme-linked immunosorbent assay showed that Rice tungro bacilliform virus (RTBV) accumulated in a cyclic pattern from early to late stages of infection in tungro-susceptible variety, Taichung Native 1 (TN1), and resistant variety, Balimau Putih, singly infected with RTBV or co-infected with RTBV+Rice tungro spherical virus (RTSV). These changes in virus accumulation resulted in differences in RTBV levels and incidence of infection. The virus levels were expressed relative to those of the susceptible variety and the incidence of infection was assessed at different weeks after inoculation. At a particular time point, RTBV levels in TN1 or Balimau Putih singly infected with RTBV were not significantly different from the virus level in plants co-infected with RTBV+RTSV. The relative RTBV levels in Balimau Putih either singly infected with RTBV or co-infected with RTBV+RTSV were significantly lower than those in TN1. The incidence of RTBV infection varied at different times in Balimau Putih but not in TN1, and to determine the actual infection, the number of plants that became infected at least once anytime during the 4wk observation period was considered. Considering the changes in RTBV accumulation, new parameters for analyzing RTBV resistance were established. Based on these parameters, Balimau Putih was characterized having resistance to virus accumulation although the actual incidence of infection was >75%.

Fabrication using a rapid prototyping system and in vitro characterization of PEG-PCL-PLA scaffolds for tissue engineering

n the field of tissue engineering new polymers are needed to fabricate scaffolds with specific properties depending on the targeted tissue. This work aimed at designing and developing a 3D scaffold with variable mechanical strength, fully interconnected porous network, controllable hydrophilicity and degradability. For this, a desktop-robot-based melt-extrusion rapid prototyping technique was applied to a novel tri-block co-polymer, namely poly(ethylene glycol)-block-poly(epsi-caprolactone)-block-poly(DL-lactide), PEG-PCL-P(DL)LA. This co-polymer was melted by electrical heating and directly extruded out using computer-controlled rapid prototyping by means of compressed purified air to build porous scaffolds. Various lay-down patterns (0/30/60/90/120/150°, 0/45/90/135°, 0/60/120° and 0/90°) were produced by using appropriate positioning of the robotic control system. Scanning electron microscopy and micro-computed tomography were used to show that 3D scaffold architectures were honeycomb-like with completely interconnected and controlled channel characteristics. Compression tests were performed and the data obtained agreed well with the typical behavior of a porous material undergoing deformation. Preliminary cell response to the as-fabricated scaffolds has been studied with primary human fibroblasts. The results demonstrated the suitability of the process and the cell biocompatibility of the polymer, two important properties among the many required for effective clinical use and efficient tissue-engineering scaffolding.