Solutions of Fractional Diffusion-Wave Equations in Terms of H-functions

MSC 2010: 35R11, 42A38, 26A33, 33E12

Calculus of Variations with Classical and Fractional Derivatives

MSC 2010: 49K05, 26A33

A new low temperature synthesis route of fraipontite (Zn,Al)3(Si,Al)2O5(OH)4

A low temperature synthesis method based on the decomposition of urea at 90°C in water has been developed to synthesise fraipontite. This material is characterised by a basal reflection 001 at 7.44 Å. The trioctahedral nature of the fraipontite is shown by the presence of a 06l band around 1.54 Å, while a minor band around 1.51 Å indicates some cation ordering between Zn and Al resulting in Al-rich areas with a more dioctahedral nature. TEM and IR indicate that no separate kaolinite phase is present. An increase in the Al content however, did result in the formation of some SiO2 in the form of quartz. Minor impurities of carbonate salts were observed during the synthesis caused by to the formation of CO32- during the decomposition of urea.

Implicit difference approximation for the time fractional diffusion equation

In this paper, we consider a time fractional diffusion equation on a finite domain. The equation is obtained from the standard diffusion equation by replacing the first-order time derivative by a fractional derivative (of order $0<\alpha<1$ ). We propose a computationally effective implicit difference approximation to solve the time fractional diffusion equation. Stability and convergence of the method are discussed. We prove that the implicit difference approximation (IDA) is unconditionally stable, and the IDA is convergent with $O(\tau+h^2)$, where $\tau$ and $h$ are time and space steps, respectively. Some numerical examples are presented to show the application of the present technique.

Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term

In this paper, we consider a variable-order fractional advection-diffusion equation with a nonlinear source term on a finite domain. Explicit and implicit Euler approximations for the equation are proposed. Stability and convergence of the methods are discussed. Moreover, we also present a fractional method of lines, a matrix transfer technique, and an extrapolation method for the equation. Some numerical examples are given, and the results demonstrate the effectiveness of theoretical analysis.

Numerical methods for fractional partial differential equations with Riesz space fractional derivatives

In this paper, we consider the numerical solution of a fractional partial differential equation with Riesz space fractional derivatives (FPDE-RSFD) on a finite domain. Two types of FPDE-RSFD are considered: the Riesz fractional diffusion equation (RFDE) and the Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second-order space derivative with the Riesz fractional derivative of order αset membership, variant(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order βset membership, variant(0,1) and of order αset membership, variant(1,2], respectively. Firstly, analytic solutions of both the RFDE and RFADE are derived. Secondly, three numerical methods are provided to deal with the Riesz space fractional derivatives, namely, the L1/L2-approximation method, the standard/shifted Grünwald method, and the matrix transform method (MTM). Thirdly, the RFDE and RFADE are transformed into a system of ordinary differential equations, which is then solved by the method of lines. Finally, numerical results are given, which demonstrate the effectiveness and convergence of the three numerical methods.

Fundamental solution and discrete random walk model for time-space fractional diffusion equation

In this paper, we consider a time-space fractional diffusion equation of distributed order (TSFDEDO). The TSFDEDO is obtained from the standard advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α∈(0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of orders β 1∈(0,1) and β 2∈(1,2], respectively. We derive the fundamental solution for the TSFDEDO with an initial condition (TSFDEDO-IC). The fundamental solution can be interpreted as a spatial probability density function evolving in time. We also investigate a discrete random walk model based on an explicit finite difference approximation for the TSFDEDO-IC.

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.

Nitrous oxide emissions from irrigated cotton soils of northern Australia

An automated gas sampling methodology has been used to estimate nitrous oxide (N2O) emissions from heavy black clay soil in northern Australia where split applications of urea were applied to furrow irrigated cotton. Nitrous oxide emissions from the beds were 643 g N/ha over the 188 day measurement period (after planting), whilst the N2O emissions from the furrows were significantly higher at 967 g N/ha. The DNDC model was used to develop a full season simulation of N2O and N2 emissions. Seasonal N2O emissions were equivalent to 0.83% of applied N, with total gaseous N losses (excluding NH3) estimated to be 16% of the applied N.

Diffusion tensor of water in model articular cartilage

We used Monte Carlo simulations of Brownian dynamics of water to study anisotropic water diffusion in an idealised model of articular cartilage. The main aim was to use the simulations as a tool for translation of the fractional anisotropy of the water diffusion tensor in cartilage into quantitative characteristics of its collagen fibre network. The key finding was a linear empirical relationship between the collagen volume fraction and the fractional anisotropy of the diffusion tensor. Fractional anisotropy of the diffusion tensor is potentially a robust indicator of the microstructure of the tissue because, in the first approximation, it is invariant to the inclusion of proteoglycans or chemical exchange between free and collagen-bound water in the model. We discuss potential applications of Monte Carlo diffusion-tensor simulations for quantitative biophysical interpretation of MRI diffusion-tensor images of cartilage. Extension of the model to include collagen fibre disorder is also discussed.

Computationally efficient numerical methods for time- and space-fractional Fokker–Planck equations

Fractional Fokker–Planck equations have been used to model several physical situations that present anomalous diffusion. In this paper, a class of time- and space-fractional Fokker–Planck equations (TSFFPE), which involve the Riemann–Liouville time-fractional derivative of order 1-α (α(0, 1)) and the Riesz space-fractional derivative (RSFD) of order μ(1, 2), are considered. The solution of TSFFPE is important for describing the competition between subdiffusion and Lévy flights. However, effective numerical methods for solving TSFFPE are still in their infancy. We present three computationally efficient numerical methods to deal with the RSFD, and approximate the Riemann–Liouville time-fractional derivative using the Grünwald method. The TSFFPE is then transformed into a system of ordinary differential equations (ODE), which is solved by the fractional implicit trapezoidal method (FITM). Finally, numerical results are given to demonstrate the effectiveness of these methods. These techniques can also be applied to solve other types of fractional partial differential equations.

Development of novel vaccine strategies to prevent genital tract chlamydial infections

Chlamydia trachomatis is the most prevalent bacterial sexually transmitted infection in the developed world and the leading cause of preventable blindness worldwide. As reported by the World Health Organization in 2001, there are approximately 92 million new infections detected annually, costing health systems billions of dollars to treat not only the acute infection, but also to treat infection-associated sequelae. The majority of genital infections are asymptomatic, with 50-70% going undetected. Genital tract infections can be easily treated with antibiotics when detected. Lack of treatment can lead to the development of pelvic inflammatory disease, ectopic pregnancies and tubal factor infertility in women and epididymitis and prostatitis in men. With infection rates on the continual rise and the large number of infections going undetected, there is a need to develop an efficacious vaccine which prevents not only infection, but also the development of infection-associated pathology. Before a vaccine can be developed and administered, the pathogenesis of chlamydial infections needs to be fully understood. This includes the kinetics of ascending infection and the effects of inoculating dose on ascension and development of pathology. The first aim in this study was to examine these factors in a murine model. Female BALB/c mice were infected intravaginally with varying doses of C. muridarum, the mouse variant of human C. trachomatis, and the ascension of infection along the reproductive tract and the time-course of infection-associated pathology development, including inflammatory cell infiltration, pyosalpinx and hydrosalpinx, were determined. It was found that while the inoculating dose did affect the rate and degree of infection, it did not affect any of the pathological parameters examined. This highlighted that the sexual transmission dose may have minimal effect on the development of reproductive sequelae. The results of the first section enabled further studies presented here to use an optimal inoculating dose that would ascend the reproductive tract and cause pathology development, so that vaccine efficacy could be determined. There has been a large amount of research into the development of an efficacious vaccine against genital tract chlamydial infections, with little success. However, there have been no studies examining the effects of the timing of vaccination, including the effects of vaccination during an active genital infection, or after clearance of a previous infection. These are important factors that need to be examined, as it is not yet known whether immunization will enhance not only the individual's immune response, but also pathology development. It is also unknown whether any enhancement of the immune responses will cause the Chlamydia to enter a dormant, persistent state, and possibly further enhance any pathology development. The second section of this study aimed to determine if vaccination during an active genital tract infection, or after clearance of a primary infection, enhanced the murine immune responses and whether any enhanced or reduced pathology occurred. Naïve, actively infected, or previously infected animals were immunized intranasally or transcutaneously with the adjuvants cholera toxin and CpG-ODN in combination with either the major outer membrane protein (MOMP) of C. muridarum, or MOMP and ribonucleotide reductase small chain protein (NrdB) of C. muridarum. It was found that the systemic immune responses in actively or previously infected mice were altered in comparison to animals immunized naïve with the same combinations, however mucosal antibodies were not enhanced. It was also found that there was no difference in pathology development between any of the groups. This suggests that immunization of individuals who may have an asymptomatic infection, or may have been previously exposed to a genital infection, may not benefit from vaccination in terms of enhanced immune responses against re-exposure. The final section of this study aimed to determine if the vaccination regimes mentioned above caused in vivo persistence of C. muridarum in the upper reproductive tracts of mice. As there has been no characterization of C. muridarum persistence in vitro, either ultrastructurally or via transcriptome analysis, this was the first aim of this section. Once it had been shown that C. muridarum could be induced into a persistent state, the gene transcriptional profiles of the selected persistent marker genes were used to determine if persistent infections were indeed present in the upper reproductive tracts of the mice. We found that intranasal immunization during an active infection induced persistent infections in the oviducts, but not the uterine horns, and that intranasal immunization after clearance of infection, caused persistent infections in both the uterine horns and the oviducts of the mice. This is a significant finding, not only because it is the first time that C. muridarum persistence has been characterized in vitro, but also due to the fact that there is minimal characterization of in vivo persistence of any chlamydial species. It is possible that the induction of persistent infections in the reproductive tract might enhance the development of pathology and thereby enhance the risk of infertility, factors that need to be prevented by vaccination, not enhanced. Overall, this study has shown that the inoculating dose does not affect pathology development in the female reproductive tract of infected mice, but does alter the degree and rate of ascending infection. It has also been shown that intranasal immunization during an active genital infection, or after clearance of one, induces persistent infections in the uterine horns and oviducts of mice. This suggests that potential vaccine candidates will need to have these factors closely examined before progressing to clinical trials. This is significant, because if the same situation occurs in humans, a vaccine administered to an asymptomatic, or previously exposed individual may not afford any extra protection and may in fact enhance the risk of development of infection-associated sequelae. This suggests that a vaccine may serve the community better if administered before the commencement of sexual activity.

Timing of births and endometrial cancer risk in Swedish women

While a protective long-term effect of parity on endometrial cancer risk is well established, the impact of timing of births is not fully understood. We examined the relationship between endometrial cancer risk and reproductive characteristics in a population-based cohort of 2,674,465 Swedish women, 20–72 years of age. During follow-up from 1973 through 2004, 7,386 endometrial cancers were observed. Compared to uniparous women, nulliparous women had a significantly elevated endometrial cancer risk (hazard ratio [HR] = 1.32, 95% confidence interval [CI], 1.22–1.42). Endometrial cancer risk decreased with increasing parity; compared to uniparous women, women with ≥4 births had a HR=0.66 (95% CI, 0.59–0.74); p-trend < 0.001. Among multiparous women, we observed no relationship of risk with age at first birth after adjustment for other reproductive factors. While we initially observed a decreased risk with later ages at last birth, this appeared to reflect a stronger relationship with time since last birth, with women with shorter times being at lowest risk. In models for multiparous women that included number of births, age at first and last birth, and time since last birth, age at last birth was not associated with endometrial cancer risk, while shorter time since last birth and increased parity were associated with statistically significantly reduced endometrial cancer risks. The HR was 3.95 (95%CI; 2.17–7.20; p-trend=<0.0001) for women with ≥25 years since a last birth compared to women having given birth within 4 years. Our findings support that clearance of initiated cells during delivery may be important in endometrial carcinogenesis. Keywords: endometrial carcinoma, parity, registry, reproductive factors

Computationally efficient methods for solving time-variable-order time-space fractional reaction-diffusion equation

Fractional differential equations are becoming more widely accepted as a powerful tool in modelling anomalous diffusion, which is exhibited by various materials and processes. Recently, researchers have suggested that rather than using constant order fractional operators, some processes are more accurately modelled using fractional orders that vary with time and/or space. In this paper we develop computationally efficient techniques for solving time-variable-order time-space fractional reaction-diffusion equations (tsfrde) using the finite difference scheme. We adopt the Coimbra variable order time fractional operator and variable order fractional Laplacian operator in space where both orders are functions of time. Because the fractional operator is nonlocal, it is challenging to efficiently deal with its long range dependence when using classical numerical techniques to solve such equations. The novelty of our method is that the numerical solution of the time-variable-order tsfrde is written in terms of a matrix function vector product at each time step. This product is approximated efficiently by the Lanczos method, which is a powerful iterative technique for approximating the action of a matrix function by projecting onto a Krylov subspace. Furthermore an adaptive preconditioner is constructed that dramatically reduces the size of the required Krylov subspaces and hence the overall computational cost. Numerical examples, including the variable-order fractional Fisher equation, are presented to demonstrate the accuracy and efficiency of the approach.

A second-order accurate numerical approximation for the Riesz space fractional advection-dispersion equation

In this paper, we consider a space Riesz fractional advection-dispersion equation. The equation is obtained from the standard advection-diffusion equation by replacing the ¯rst-order and second-order space derivatives by the Riesz fractional derivatives of order β 1 Є (0; 1) and β2 Є(1; 2], respectively. Riesz fractional advection and dispersion terms are approximated by using two fractional centered difference schemes, respectively. A new weighted Riesz fractional ¯nite difference approximation scheme is proposed. When the weighting factor Ѳ = 1/2, a second- order accurate numerical approximation scheme for the Riesz fractional advection-dispersion equation is obtained. Stability, consistency and convergence of the numerical approximation scheme are discussed. A numerical example is given to show that the numerical results are in good agreement with our theoretical analysis.