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.

Multilevel converter structure and control

In recent years, multilevel converters are becoming more popular and attractive than traditional converters in high voltage and high power applications. Multilevel converters are particularly suitable for harmonic reduction in high power applications where semiconductor devices are not able to operate at high switching frequencies or in high voltage applications where multilevel converters reduce the need to connect devices in series to achieve high switch voltage ratings. This thesis investigated two aspects of multilevel converters: structure and control. The first part of this thesis focuses on inductance between a DC supply and inverter components in order to minimise loop inductance, which causes overvoltages and stored energy losses during switching. Three dimensional finite element simulations and experimental tests have been carried out for all sections to verify theoretical developments. The major contributions of this section of the thesis are as follows: The use of a large area thin conductor sheet with a rectangular cross section separated by dielectric sheets (planar busbar) instead of circular cross section wires, contributes to a reduction of the stray inductance. A number of approximate equations exist for calculating the inductance of a rectangular conductor but an assumption was made that the current density was uniform throughout the conductors. This assumption is not valid for an inverter with a point injection of current. A mathematical analysis of a planar bus bar has been performed at low and high frequencies and the inductance and the resistance values between the two points of the planar busbar have been determined. A new physical structure for a voltage source inverter with symmetrical planar bus bar structure called Reduced Layer Planar Bus bar, is proposed in this thesis based on the current point injection theory. This new type of planar busbar minimises the variation in stray inductance for different switching states. The reduced layer planar busbar is a new innovation in planar busbars for high power inverters with minimum separation between busbars, optimum stray inductance and improved thermal performances. This type of the planar busbar is suitable for high power inverters, where the voltage source is supported by several capacitors in parallel in order to provide a low ripple DC voltage during operation. A two layer planar busbar with different materials has been analysed theoretically in order to determine the resistance of bus bars during switching. Increasing the resistance of the planar busbar can gain a damping ratio between stray inductance and capacitance and affects the performance of current loop during switching. The aim of this section is to increase the resistance of the planar bus bar at high frequencies (during switching) and without significantly increasing the planar busbar resistance at low frequency (50 Hz) using the skin effect. This contribution shows a novel structure of busbar suitable for high power applications where high resistance is required at switching times. In multilevel converters there are different loop inductances between busbars and power switches associated with different switching states. The aim of this research is to consider all combinations of the switching states for each multilevel converter topology and identify the loop inductance for each switching state. Results show that the physical layout of the busbars is very important for minimisation of the loop inductance at each switch state. Novel symmetrical busbar structures are proposed for multilevel converters with diode-clamp and flying-capacitor topologies which minimise the worst case in stray inductance for different switching states. Overshoot voltages and thermal problems are considered for each topology to optimise the planar busbar structure. In the second part of the thesis, closed loop current techniques have been investigated for single and three phase multilevel converters. The aims of this section are to investigate and propose suitable current controllers such as hysteresis and predictive techniques for multilevel converters with low harmonic distortion and switching losses. This section of the thesis can be classified into three parts as follows: An optimum space vector modulation technique for a three-phase voltage source inverter based on a minimum-loss strategy is proposed. One of the degrees of freedom for optimisation of the space vector modulation is the selection of the zero vectors in the switching sequence. This new method improves switching transitions per cycle for a given level of distortion as the zero vector does not alternate between each sector. The harmonic spectrum and weighted total harmonic distortion for these strategies are compared and results show up to 7% weighted total harmonic distortion improvement over the previous minimum-loss strategy. The concept of SVM technique is a very convenient representation of a set of three-phase voltages or currents used for current control techniques. A new hysteresis current control technique for a single-phase multilevel converter with flying-capacitor topology is developed. This technique is based on magnitude and time errors to optimise the level change of converter output voltage. This method also considers how to improve unbalanced voltages of capacitors using voltage vectors in order to minimise switching losses. Logic controls require handling a large number of switches and a Programmable Logic Device (PLD) is a natural implementation for state transition description. The simulation and experimental results describe and verify the current control technique for the converter. A novel predictive current control technique is proposed for a three-phase multilevel converter, which controls the capacitors' voltage and load current with minimum current ripple and switching losses. The advantage of this contribution is that the technique can be applied to more voltage levels without significantly changing the control circuit. The three-phase five-level inverter with a pure inductive load has been implemented to track three-phase reference currents using analogue circuits and a programmable logic device.

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.

Bearing damage analysis by calculation of capacitive coupling between inner and outer races of a ball bearing

Bearing damage in modern inverter-fed AC drive systems is more common than in motors working with 50 or 60 Hz power supply. Fast switching transients and common mode voltage generated by a PWM inverter cause unwanted shaft voltage and resultant bearing currents. Parasitic capacitive coupling creates a path to discharge current in rotors and bearings. In order to analyze bearing current discharges and their effect on bearing damage under different conditions, calculation of the capacitive coupling between the outer and inner races is needed. During motor operation, the distances between the balls and races may change the capacitance values. Due to changing of the thickness and spatial distribution of the lubricating grease, this capacitance does not have a constant value and is known to change with speed and load. Thus, the resultant electric field between the races and balls varies with motor speed. The lubricating grease in the ball bearing cannot withstand high voltages and a short circuit through the lubricated grease can occur. At low speeds, because of gravity, balls and shaft voltage may shift down and the system (ball positions and shaft) will be asymmetric. In this study, two different asymmetric cases (asymmetric ball position, asymmetric shaft position) are analyzed and the results are compared with the symmetric case. The objective of this paper is to calculate the capacitive coupling and electric fields between the outer and inner races and the balls at different motor speeds in symmetrical and asymmetrical shaft and balls positions. The analysis is carried out using finite element simulations to determine the conditions which will increase the probability of high rates of bearing failure due to current discharges through the balls and races.

Analytical and numerical solutions for the time and space-symmetric fractional diffusion equation

We consider a time and space-symmetric fractional diffusion equation (TSS-FDE) under homogeneous Dirichlet conditions and homogeneous Neumann conditions. The TSS-FDE is obtained from the standard diffusion equation by replacing the first-order time derivative by a Caputo fractional derivative, and the second order space derivative by a symmetric fractional derivative. First, a method of separating variables expresses the analytical solution of the TSS-FDE in terms of the Mittag--Leffler function. Second, we propose two numerical methods to approximate the Caputo time fractional derivative: the finite difference method; and the Laplace transform method. The symmetric space fractional derivative is approximated using the matrix transform method. Finally, numerical results demonstrate the effectiveness of the numerical methods and to confirm the theoretical claims.

Analytical and numerical solutions for the time and space-symmetric fractional diffusion equation

We consider a time and space-symmetric fractional diffusion equation (TSS-FDE) under homogeneous Dirichlet conditions and homogeneous Neumann conditions. The TSS-FDE is obtained from the standard diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative and the second order space derivative by the symmetric fractional derivative. Firstly, a method of separating variables is used to express the analytical solution of the tss-fde in terms of the Mittag–Leffler function. Secondly, we propose two numerical methods to approximate the Caputo time fractional derivative, namely, the finite difference method and the Laplace transform method. The symmetric space fractional derivative is approximated using the matrix transform method. Finally, numerical results are presented to demonstrate the effectiveness of the numerical methods and to confirm the theoretical claims.

Wavelet based approaches and high resolution methods for complex process models

Many industrial processes and systems can be modelled mathematically by a set of Partial Differential Equations (PDEs). Finding a solution to such a PDF model is essential for system design, simulation, and process control purpose. However, major difficulties appear when solving PDEs with singularity. Traditional numerical methods, such as finite difference, finite element, and polynomial based orthogonal collocation, not only have limitations to fully capture the process dynamics but also demand enormous computation power due to the large number of elements or mesh points for accommodation of sharp variations. To tackle this challenging problem, wavelet based approaches and high resolution methods have been recently developed with successful applications to a fixedbed adsorption column model. Our investigation has shown that recent advances in wavelet based approaches and high resolution methods have the potential to be adopted for solving more complicated dynamic system models. This chapter will highlight the successful applications of these new methods in solving complex models of simulated-moving-bed (SMB) chromatographic processes. A SMB process is a distributed parameter system and can be mathematically described by a set of partial/ordinary differential equations and algebraic equations. These equations are highly coupled; experience wave propagations with steep front, and require significant numerical effort to solve. To demonstrate the numerical computing power of the wavelet based approaches and high resolution methods, a single column chromatographic process modelled by a Transport-Dispersive-Equilibrium linear model is investigated first. Numerical solutions from the upwind-1 finite difference, wavelet-collocation, and high resolution methods are evaluated by quantitative comparisons with the analytical solution for a range of Peclet numbers. After that, the advantages of the wavelet based approaches and high resolution methods are further demonstrated through applications to a dynamic SMB model for an enantiomers separation process. This research has revealed that for a PDE system with a low Peclet number, all existing numerical methods work well, but the upwind finite difference method consumes the most time for the same degree of accuracy of the numerical solution. The high resolution method provides an accurate numerical solution for a PDE system with a medium Peclet number. The wavelet collocation method is capable of catching up steep changes in the solution, and thus can be used for solving PDE models with high singularity. For the complex SMB system models under consideration, both the wavelet based approaches and high resolution methods are good candidates in terms of computation demand and prediction accuracy on the steep front. The high resolution methods have shown better stability in achieving steady state in the specific case studied in this Chapter.

Thermal studies of D.C. traction motors

This paper describes a thorough thermal study on a fleet of DC traction motors which were found to suffer from overheating after 3 years of full operation. Overheating of these traction motors is attributed partly because of the higher than expected number of starts and stops between train terminals. Another probable cause of overheating is the design of the traction motor and/or its control strategy. According to the motor manufacturer, a current shunt is permanently connected across the motor field winding. Hence, some of the armature current is bypassed into the current shunt. The motor then runs above its rated speed in the field weakening mode. In this study, a finite difference model has been developed to simulate the temperature profile at different parts inside the traction motor. In order to validate the simulation result, an empty vehicle loaded with drums of water was also used to simulate the full pay-load of a light rail vehicle experimentally. The authors report that the simulation results agree reasonably well with experimental data, and it is likely that the armature of the traction motor will run cooler if its field shunt is disconnected at low speeds

Dynamics of pattern writing and erasure in photorefractive lithium niobate for data storage applications

The present rate of technological advance continues to place significant demands on data storage devices. The sheer amount of digital data being generated each year along with consumer expectations, fuels these demands. At present, most digital data is stored magnetically, in the form of hard disk drives or on magnetic tape. The increase in areal density (AD) of magnetic hard disk drives over the past 50 years has been of the order of 100 million times, and current devices are storing data at ADs of the order of hundreds of gigabits per square inch. However, it has been known for some time that the progress in this form of data storage is approaching fundamental limits. The main limitation relates to the lower size limit that an individual bit can have for stable storage. Various techniques for overcoming these fundamental limits are currently the focus of considerable research effort. Most attempt to improve current data storage methods, or modify these slightly for higher density storage. Alternatively, three dimensional optical data storage is a promising field for the information storage needs of the future, offering very high density, high speed memory. There are two ways in which data may be recorded in a three dimensional optical medium; either bit-by-bit (similar in principle to an optical disc medium such as CD or DVD) or by using pages of bit data. Bit-by-bit techniques for three dimensional storage offer high density but are inherently slow due to the serial nature of data access. Page-based techniques, where a two-dimensional page of data bits is written in one write operation, can offer significantly higher data rates, due to their parallel nature. Holographic Data Storage (HDS) is one such page-oriented optical memory technique. This field of research has been active for several decades, but with few commercial products presently available. Another page-oriented optical memory technique involves recording pages of data as phase masks in a photorefractive medium. A photorefractive material is one by which the refractive index can be modified by light of the appropriate wavelength and intensity, and this property can be used to store information in these materials. In phase mask storage, two dimensional pages of data are recorded into a photorefractive crystal, as refractive index changes in the medium. A low-intensity readout beam propagating through the medium will have its intensity profile modified by these refractive index changes and a CCD camera can be used to monitor the readout beam, and thus read the stored data. The main aim of this research was to investigate data storage using phase masks in the photorefractive crystal, lithium niobate (LiNbO3). Firstly the experimental methods for storing the two dimensional pages of data (a set of vertical stripes of varying lengths) in the medium are presented. The laser beam used for writing, whose intensity profile is modified by an amplitudemask which contains a pattern of the information to be stored, illuminates the lithium niobate crystal and the photorefractive effect causes the patterns to be stored as refractive index changes in the medium. These patterns are read out non-destructively using a low intensity probe beam and a CCD camera. A common complication of information storage in photorefractive crystals is the issue of destructive readout. This is a problem particularly for holographic data storage, where the readout beam should be at the same wavelength as the beam used for writing. Since the charge carriers in the medium are still sensitive to the read light field, the readout beam erases the stored information. A method to avoid this is by using thermal fixing. Here the photorefractive medium is heated to temperatures above 150�C; this process forms an ionic grating in the medium. This ionic grating is insensitive to the readout beam and therefore the information is not erased during readout. A non-contact method for determining temperature change in a lithium niobate crystal is presented in this thesis. The temperature-dependent birefringent properties of the medium cause intensity oscillations to be observed for a beam propagating through the medium during a change in temperature. It is shown that each oscillation corresponds to a particular temperature change, and by counting the number of oscillations observed, the temperature change of the medium can be deduced. The presented technique for measuring temperature change could easily be applied to a situation where thermal fixing of data in a photorefractive medium is required. Furthermore, by using an expanded beam and monitoring the intensity oscillations over a wide region, it is shown that the temperature in various locations of the crystal can be monitored simultaneously. This technique could be used to deduce temperature gradients in the medium. It is shown that the three dimensional nature of the recording medium causes interesting degradation effects to occur when the patterns are written for a longer-than-optimal time. This degradation results in the splitting of the vertical stripes in the data pattern, and for long writing exposure times this process can result in the complete deterioration of the information in the medium. It is shown in that simply by using incoherent illumination, the original pattern can be recovered from the degraded state. The reason for the recovery is that the refractive index changes causing the degradation are of a smaller magnitude since they are induced by the write field components scattered from the written structures. During incoherent erasure, the lower magnitude refractive index changes are neutralised first, allowing the original pattern to be recovered. The degradation process is shown to be reversed during the recovery process, and a simple relationship is found relating the time at which particular features appear during degradation and recovery. A further outcome of this work is that the minimum stripe width of 30 ìm is required for accurate storage and recovery of the information in the medium, any size smaller than this results in incomplete recovery. The degradation and recovery process could be applied to an application in image scrambling or cryptography for optical information storage. A two dimensional numerical model based on the finite-difference beam propagation method (FD-BPM) is presented and used to gain insight into the pattern storage process. The model shows that the degradation of the patterns is due to the complicated path taken by the write beam as it propagates through the crystal, and in particular the scattering of this beam from the induced refractive index structures in the medium. The model indicates that the highest quality pattern storage would be achieved with a thin 0.5 mm medium; however this type of medium would also remove the degradation property of the patterns and the subsequent recovery process. To overcome the simplistic treatment of the refractive index change in the FD-BPM model, a fully three dimensional photorefractive model developed by Devaux is presented. This model shows significant insight into the pattern storage, particularly for the degradation and recovery process, and confirms the theory that the recovery of the degraded patterns is possible since the refractive index changes responsible for the degradation are of a smaller magnitude. Finally, detailed analysis of the pattern formation and degradation dynamics for periodic patterns of various periodicities is presented. It is shown that stripe widths in the write beam of greater than 150 ìm result in the formation of different types of refractive index changes, compared with the stripes of smaller widths. As a result, it is shown that the pattern storage method discussed in this thesis has an upper feature size limit of 150 ìm, for accurate and reliable pattern storage.

Remediation strategies of shaft and common mode voltages in adjustable speed drive systems

AC motors are largely used in a wide range of modern systems, from household appliances to automated industry applications such as: ventilations systems, fans, pumps, conveyors and machine tool drives. Inverters are widely used in industrial and commercial applications due to the growing need for speed control in ASD systems. Fast switching transients and the common mode voltage, in interaction with parasitic capacitive couplings, may cause many unwanted problems in the ASD applications. These include shaft voltage and leakage currents. One of the inherent characteristics of Pulse Width Modulation (PWM) techniques is the generation of the common mode voltage, which is defined as the voltage between the electrical neutral of the inverter output and the ground. Shaft voltage can cause bearing currents when it exceeds the amount of breakdown voltage level of the thin lubricant film between the inner and outer rings of the bearing. This phenomenon is the main reason for early bearing failures. A rapid development in power switches technology has lead to a drastic decrement of switching rise and fall times. Because there is considerable capacitance between the stator windings and the frame, there can be a significant capacitive current (ground current escaping to earth through stray capacitors inside a motor) if the common mode voltage has high frequency components. This current leads to noises and Electromagnetic Interferences (EMI) issues in motor drive systems. These problems have been dealt with using a variety of methods which have been reported in the literature. However, cost and maintenance issues have prevented these methods from being widely accepted. Extra cost or rating of the inverter switches is usually the price to pay for such approaches. Thus, the determination of cost-effective techniques for shaft and common mode voltage reduction in ASD systems, with the focus on the first step of the design process, is the targeted scope of this thesis. An introduction to this research – including a description of the research problem, the literature review and an account of the research progress linking the research papers – is presented in Chapter 1. Electrical power generation from renewable energy sources, such as wind energy systems, has become a crucial issue because of environmental problems and a predicted future shortage of traditional energy sources. Thus, Chapter 2 focuses on the shaft voltage analysis of stator-fed induction generators (IG) and Doubly Fed Induction Generators DFIGs in wind turbine applications. This shaft voltage analysis includes: topologies, high frequency modelling, calculation and mitigation techniques. A back-to-back AC-DC-AC converter is investigated in terms of shaft voltage generation in a DFIG. Different topologies of LC filter placement are analysed in an effort to eliminate the shaft voltage. Different capacitive couplings exist in the motor/generator structure and any change in design parameters affects the capacitive couplings. Thus, an appropriate design for AC motors should lead to the smallest possible shaft voltage. Calculation of the shaft voltage based on different capacitive couplings, and an investigation of the effects of different design parameters are discussed in Chapter 3. This is achieved through 2-D and 3-D finite element simulation and experimental analysis. End-winding parameters of the motor are also effective factors in the calculation of the shaft voltage and have not been taken into account in previous reported studies. Calculation of the end-winding capacitances is rather complex because of the diversity of end winding shapes and the complexity of their geometry. A comprehensive analysis of these capacitances has been carried out with 3-D finite element simulations and experimental studies to determine their effective design parameters. These are documented in Chapter 4. Results of this analysis show that, by choosing appropriate design parameters, it is possible to decrease the shaft voltage and resultant bearing current in the primary stage of generator/motor design without using any additional active and passive filter-based techniques. The common mode voltage is defined by a switching pattern and, by using the appropriate pattern; the common mode voltage level can be controlled. Therefore, any PWM pattern which eliminates or minimizes the common mode voltage will be an effective shaft voltage reduction technique. Thus, common mode voltage reduction of a three-phase AC motor supplied with a single-phase diode rectifier is the focus of Chapter 5. The proposed strategy is mainly based on proper utilization of the zero vectors. Multilevel inverters are also used in ASD systems which have more voltage levels and switching states, and can provide more possibilities to reduce common mode voltage. A description of common mode voltage of multilevel inverters is investigated in Chapter 6. Chapter 7 investigates the elimination techniques of the shaft voltage in a DFIG based on the methods presented in the literature by the use of simulation results. However, it could be shown that every solution to reduce the shaft voltage in DFIG systems has its own characteristics, and these have to be taken into account in determining the most effective strategy. Calculation of the capacitive coupling and electric fields between the outer and inner races and the balls at different motor speeds in symmetrical and asymmetrical shaft and balls positions is discussed in Chapter 8. The analysis is carried out using finite element simulations to determine the conditions which will increase the probability of high rates of bearing failure due to current discharges through the balls and races.

An advanced implicit meshless approach for fractional partial differential equation in computational mechanics

Recently, the numerical modelling and simulation for fractional partial differential equations (FPDE), which have been found with widely applications in modern engineering and sciences, are attracting increased attentions. The current dominant numerical method for modelling of FPDE is the explicit Finite Difference Method (FDM), which is based on a pre-defined grid leading to inherited issues or shortcomings. This paper aims to develop an implicit meshless approach based on the radial basis functions (RBF) for numerical simulation of time fractional diffusion equations. The discrete system of equations is obtained by using the RBF meshless shape functions and the strong-forms. The stability and convergence of this meshless approach are then discussed and theoretically proven. Several numerical examples with different problem domains are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. The results obtained by the meshless formations are also compared with those obtained by FDM in terms of their accuracy and efficiency. It is concluded that the present meshless formulation is very effective for the modelling and simulation for FPDE.

Drug diffusion from polymeric delivery devices : a problem with two moving boundaries

An existing model for solvent penetration and drug release from a spherically-shaped polymeric drug delivery device is revisited. The model has two moving boundaries, one that describes the interface between the glassy and rubbery states of polymer, and another that defines the interface between the polymer ball and the pool of solvent. The model is extended so that the nonlinear diffusion coefficient of drug explicitly depends on the concentration of solvent, and the resulting equations are solved numerically using a front-fixing transformation together with a finite difference spatial discretisation and the method of lines. We present evidence that our scheme is much more accurate than a previous scheme. Asymptotic results in the small-time limit are presented, which show how the use of a kinetic law as a boundary condition on the innermost moving boundary dictates qualitative behaviour, the scalings being very different to the similar moving boundary problem that arises from modelling the melting of an ice ball. The implication is that the model considered here exhibits what is referred to as ``non-Fickian'' or Case II diffusion which, together with the initially constant rate of drug release, has certain appeal from a pharmaceutical perspective.