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.

Adaptive instantaneous frequency estimation: Techniques and algorithms

This thesis deals with the problem of the instantaneous frequency (IF) estimation of sinusoidal signals. This topic plays significant role in signal processing and communications. Depending on the type of the signal, two major approaches are considered. For IF estimation of single-tone or digitally-modulated sinusoidal signals (like frequency shift keying signals) the approach of digital phase-locked loops (DPLLs) is considered, and this is Part-I of this thesis. For FM signals the approach of time-frequency analysis is considered, and this is Part-II of the thesis. In part-I we have utilized sinusoidal DPLLs with non-uniform sampling scheme as this type is widely used in communication systems. The digital tanlock loop (DTL) has introduced significant advantages over other existing DPLLs. In the last 10 years many efforts have been made to improve DTL performance. However, this loop and all of its modifications utilizes Hilbert transformer (HT) to produce a signal-independent 90-degree phase-shifted version of the input signal. Hilbert transformer can be realized approximately using a finite impulse response (FIR) digital filter. This realization introduces further complexity in the loop in addition to approximations and frequency limitations on the input signal. We have tried to avoid practical difficulties associated with the conventional tanlock scheme while keeping its advantages. A time-delay is utilized in the tanlock scheme of DTL to produce a signal-dependent phase shift. This gave rise to the time-delay digital tanlock loop (TDTL). Fixed point theorems are used to analyze the behavior of the new loop. As such TDTL combines the two major approaches in DPLLs: the non-linear approach of sinusoidal DPLL based on fixed point analysis, and the linear tanlock approach based on the arctan phase detection. TDTL preserves the main advantages of the DTL despite its reduced structure. An application of TDTL in FSK demodulation is also considered. This idea of replacing HT by a time-delay may be of interest in other signal processing systems. Hence we have analyzed and compared the behaviors of the HT and the time-delay in the presence of additive Gaussian noise. Based on the above analysis, the behavior of the first and second-order TDTLs has been analyzed in additive Gaussian noise. Since DPLLs need time for locking, they are normally not efficient in tracking the continuously changing frequencies of non-stationary signals, i.e. signals with time-varying spectra. Nonstationary signals are of importance in synthetic and real life applications. An example is the frequency-modulated (FM) signals widely used in communication systems. Part-II of this thesis is dedicated for the IF estimation of non-stationary signals. For such signals the classical spectral techniques break down, due to the time-varying nature of their spectra, and more advanced techniques should be utilized. For the purpose of instantaneous frequency estimation of non-stationary signals there are two major approaches: parametric and non-parametric. We chose the non-parametric approach which is based on time-frequency analysis. This approach is computationally less expensive and more effective in dealing with multicomponent signals, which are the main aim of this part of the thesis. A time-frequency distribution (TFD) of a signal is a two-dimensional transformation of the signal to the time-frequency domain. Multicomponent signals can be identified by multiple energy peaks in the time-frequency domain. Many real life and synthetic signals are of multicomponent nature and there is little in the literature concerning IF estimation of such signals. This is why we have concentrated on multicomponent signals in Part-H. An adaptive algorithm for IF estimation using the quadratic time-frequency distributions has been analyzed. A class of time-frequency distributions that are more suitable for this purpose has been proposed. The kernels of this class are time-only or one-dimensional, rather than the time-lag (two-dimensional) kernels. Hence this class has been named as the T -class. If the parameters of these TFDs are properly chosen, they are more efficient than the existing fixed-kernel TFDs in terms of resolution (energy concentration around the IF) and artifacts reduction. The T-distributions has been used in the IF adaptive algorithm and proved to be efficient in tracking rapidly changing frequencies. They also enables direct amplitude estimation for the components of a multicomponent

Vibrational spectroscopic studies of wool

Fourier transfonn (FT) Raman, Raman microspectroscopy and Fourier transform infrared (FTIR) spectroscopy have been used for the structural analysis and characterisation of untreated and chemically treated wool fibres. For FT -Raman spectroscopy novel methods of sample presentation have been developed and optimised for the analysis of wool. No significant fluorescence was observed and the spectra could be obtained routinely. The stability of wool keratin to the laser source was investigated and the visual and spectroscopic signs of sample damage were established. Wool keratin was found to be extremely robust with no signs of sample degradation observed for laser powers of up to 600 m W and for exposure times of up to seven and half hours. Due to improvements in band resolution and signal-to-noise ratio, several previously unobserved spectral features have become apparent. The assignment of the Raman active vibrational modes of wool have been reviewed and updated to include these features. The infrared spectroscopic techniques of attenuated total reflectance (ATR) and photoacoustic (P A) have been used to examine shrinkproofed and mothproofed wool samples. Shrinkproofing is an oxidative chemical treatment used to selectively modifY the surface of a wool fibre. Mothproofing is a chemical treatment applied to wool for the prevention of insect attack. The ability of PAS and A TR to vary the penetration depth by varying certain instrumental parameters was used to obtain spectra of the near surface regions of these chemically treated samples. These spectra were compared with those taken with a greater penetration depth, which therefore represent more of the bulk wool sample. The PA and ATR spectra demonstrated that oxidation was restricted to the near-surface layer of wool. Extensive curve fitting of ATR spectra of untreated wool indicated that cuticle was composed of a mixed protein conformation, but was predominately that of an a.-helix. The cortex was proposed to be a mixture of both a.helical and ~-pleated sheet protein conformations. These findings were supported by PAS depth profiling results. Raman microspectroscopy was used in an extensive investigation of the molecular structure of the wool fibre. This included determining the orientation of certain functional groups within the wool fibre and the symmetry of particular vibrations. The orientation ofbonds within the wool fibre was investigated by orientating the wool fibre axis parallel and then perpendicular to the plane of polarisation of the electric vector of the incident radiation. It was experimentally determined that the majority of C=O and N-H bonds of the peptide bond of wool lie parallel to the fibre axis. Additionally, a number of the important vibrations associated with the a-helix were also found to lie parallel to the fibre axis. Further investigation into the molecular structure of wool involved determining what effect stretching the wool fibre had on bond orientation. Raman spectra of stretched and unstretched wool fibres indicated that extension altered the orientation ofthe aromatic rings, the CH2 and CH3 groups of the amino acids. Curve fitting results revealed that extension resulted in significant destruction of the a-helix structure a substantial increase in the P-pleated sheet structure. Finally, depolarisation ratios were calculated for Raman spectra. The vibrations associated with the aromatic rings of amino acids had very low ratios which indicated that the vibrations were highly symmetrical.

The transformation of banana with potential virus resistance genes

One approach to reducing the yield losses caused by banana viral diseases is the use of genetic engineering and pathogen-derived resistance strategies to generate resistant cultivars. The development of transgenic virus resistance requires an efficient banana transformation method, particularly for commercially important 'Cavendish' type cultivars such as 'Grand Nain'. Prior to this study, only two examples of the stable transformation of banana had been reported, both of which demonstrated the principle of transformation but did not characterise transgenic plants in terms of the efficiency at which individual transgenic lines were generated, relative activities of promoters in stably transformed plants, and the stability of transgene expression. The aim of this study was to develop more efficient transformation methods for banana, assess the activity of some commonly used and also novel promoters in stably transformed plants, and transform banana with genes that could potentially confer resistance to banana bunchy top nanovirus (BBTV) and banana bract mosaic potyvirus (BBrMV). A regeneration system using immature male flowers as the explant was established. The frequency of somatic embryogenesis in male flower explants was influenced by the season in which the inflorescences were harvested. Further, the media requirements of various banana cultivars in respect to the 2,4-D concentration in the initiation media also differed. Following the optimisation of these and other parameters, embryogenic cell suspensions of several banana (Musa spp.) cultivars including 'Grand Nain' (AAA), 'Williams' (AAA), 'SH-3362' (AA), 'Goldfinger' (AAAB) and 'Bluggoe' (ABB) were successfully generated. Highly efficient transformation methods were developed for both 'Bluggoe' and 'Grand Nain'; this is the first report of microprojectile bombardment transformation of the commercially important 'Grand Nain' cultivar. Following bombardment of embryogenic suspension cells, regeneration was monitored from single transfom1ed cells to whole plants using a reporter gene encoding the green fluorescent protein (gfp). Selection with kanamycin enabled the regeneration of a greater number of plants than with geneticin, while still preventing the regeneration of non-transformed plants. Southern hybridisation confirmed the neomycin phosphotransferase gene (npt II) was stably integrated into the banana genome and that multiple transgenic lines were derived from single bombardments. The activity, stability and tissue specificity of the cauliflower mosaic virus 358 (CaMV 35S) and maize polyubiquitin-1 (Ubi-1) promoters were examined. In stably transformed banana, the Ubi-1 promoter provided approximately six-fold higher p-glucuronidase (GUS) activity than the CaMV 35S promoter, and both promoters remained active in glasshouse grown plants for the six months they were observed. The intergenic regions ofBBTV DNA-I to -6 were isolated and fused to either the uidA (GUS) or gfjJ reporter genes to assess their promoter activities. BBTV promoter activity was detected in banana embryogenic cells using the gfp reporter gene. Promoters derived from BBTV DNA-4 and -5 generated the highest levels of transient activity, which were greater than that generated by the maize Ubi-1 promoter. In transgenic banana plants, the activity of the BBTV DNA-6 promoter (BT6.1) was restricted to the phloem of leaves and roots, stomata and root meristems. The activity of the BT6.1 promoter was enhanced by the inclusion of intron-containing fragments derived from the maize Ubi-1, rice Act-1, and sugarcane rbcS 5' untranslated regions in GUS reporter gene constructs. In transient assays in banana, the rice Act-1 and maize Ubi-1 introns provided the most significant enhancement, increasing expression levels 300-fold and 100-fold, respectively. The sugarcane rbcS intron increased expression about 10-fold. In stably transformed banana plants, the maize Ubi-1 intron enhanced BT6.1 promoter activity to levels similar to that of the CaMV 35S promoter, but did not appear to alter the tissue specificity of the promoter. Both 'Grand Nain' and 'Bluggoe' were transformed with constructs that could potentially confer resistance to BBTV and BBrMV, including constructs containing BBTV DNA-1 major and internal genes, BBTV DNA-5 gene, and the BBrMV coat protein-coding region all under the control of the Ubi-1 promoter, while the BT6 promoter was used to drive the npt II selectable marker gene. At least 30 transgenic lines containing each construct were identified and replicates of each line are currently being generated by micropropagation in preparation for virus challenge.