Development of a high-speed sensing and detection system for automatic removal of packages with leaky seals from a high-speed food processing and packaging line

Contamination of packaged foods due to micro-organisms entering through air leaks can cause serious public health issues and cost companies large amounts of money due to product recalls, consumer impact and subsequent loss of market share. The main source of contamination is leaks in packaging which allow air, moisture and microorganisms to enter the package. In the food processing and packaging industry worldwide, there is an increasing demand for cost effective state of the art inspection technologies that are capable of reliably detecting leaky seals and delivering products at six-sigma. The new technology will develop non-destructive testing technology using digital imaging and sensing combined with a differential vacuum technique to assess seal integrity of food packages on a high-speed production line. The cost of leaky packages in Australian food industries is estimated close to AUD $35 Million per year. Contamination of packaged foods due to micro-organisms entering through air leaks can cause serious public health issues and cost companies large sums of money due to product recalls, compensation claims and loss of market share. The main source of contamination is leaks in packaging which allow air, moisture and micro-organisms to enter the package. Flexible plastic packages are widely used, and are the least expensive form of retaining the quality of the product. These packets can be used to seal, and therefore maximise, the shelf life of both dry and moist products. The seals of food packages need to be airtight so that the food content is not contaminated due to contact with microorganisms that enter as a result of air leakage. Airtight seals also extend the shelf life of packaged foods, and manufacturers attempt to prevent food products with leaky seals being sold to consumers. There are many current NDT (non-destructive testing) methods of testing the seal of flexible packages best suited to random sampling, and for laboratory purposes. The three most commonly used methods are vacuum/pressure decay, bubble test, and helium leak detection. Although these methods can detect very fine leaks, they are limited by their high processing time and are not viable in a production line. Two nondestructive in-line packaging inspection machines are currently available and are discussed in the literature review. The detailed design and development of the High-Speed Sensing and Detection System (HSDS) is the fundamental requirement of this project and the future prototype and production unit. Successful laboratory testing was completed and a methodical design procedure was needed for a successful concept. The Mechanical tests confirmed the vacuum hypothesis and seal integrity with good consistent results. Electrically, the testing also provided solid results to enable the researcher to move the project forward with a certain amount of confidence. The laboratory design testing allowed the researcher to confirm theoretical assumptions before moving into the detailed design phase. Discussion on the development of the alternative concepts in both mechanical and electrical disciplines enables the researcher to make an informed decision. Each major mechanical and electrical component is detailed through the research and design process. The design procedure methodically works through the various major functions both from a mechanical and electrical perspective. It opens up alternative ideas for the major components that although are sometimes not practical in this application, show that the researcher has exhausted all engineering and functionality thoughts. Further concepts were then designed and developed for the entire HSDS unit based on previous practice and theory. In the future, it would be envisaged that both the Prototype and Production version of the HSDS would utilise standard industry available components, manufactured and distributed locally. Future research and testing of the prototype unit could result in a successful trial unit being incorporated in a working food processing production environment. Recommendations and future works are discussed, along with options in other food processing and packaging disciplines, and other areas in the non-food processing industry.

Investigation to enhance high speed rail accessibility

High Speed Rail (HSR) is rapidly gaining popularity worldwide as a safe and efficient transport option for long-distance travel. Designed to win market shares from air transport, HSR systems optimise their productivity between increasing speeds and station spacing to offer high quality service and gain ridership. Recent studies have investigated the effects that the deployment of HSR infrastructure has on spatial distribution and the economic development of cities and regions. Findings appear mostly positive at higher geographical scales, where HSR links connect major urban centres several hundred kilometres apart and already well positioned within a national or international context. Also, at the urban level, studies have shown regeneration and concentration effects around HSR station areas with positive returns on city’s image and economy. However, doubts persist on the effects of HSR at an intermediate scale, where the accessibility trade off on station spacing limits access to many small and medium agglomerations. Thereby, their ability to participate in the development opportunities facilitated by HSR infrastructure is significantly reduced. The locational advantages deriving from transport improvements appear contrasting especially in regions that tend to have a polycentric structure, where cities may present greater accessibility disparities between those served by HSR and those left behind. This thesis fits in this context where intermediate and regional cities do not directly enjoy the presence of an HSR station while having an existing or planned proximate HSR corridor. With the aim of understanding whether there might be a solution to this apparent incongruity, the research investigates strategies to integrate HSR accessibility at the regional level. While current literature recommends to commit with ancillary investments to the uplift of station areas and the renewal of feeder systems, I hypothesised the interoperability between the HSR and the conventional networks to explore the possibilities offered by mixed traffic and infrastructure sharing. Thus, I developed a methodology to quantify the exchange of benefits deriving from this synergistic interaction. In this way, it was possible to understand which level of service quality offered by alternative transit strategies best facilitates the distribution of accessibility benefits for areas far from actual HSR stations. Therefore, strategies were selected for their type of service capable of regional extensions and urban penetrations, while incorporating a combination of specific advantages (e.g. speed, sub-urbanity, capacity, frequency and automation) in order to emulate HSR quality with increasingly efficient services. The North-eastern Italian macro region was selected as case study to ground the research offering concurrently a peripheral polycentric metropolitan form, the presence of a planned HSR corridor with some portions of HSR infrastructure implementation, and the project to develop a suburban rail service extended regionally. Results show significant distributive potential, in terms of network effects produced in relation with HSR, in increasing proportions for all the strategies considered: a regional metro rail strategy (abbreviated RMR), a regional high speed rail strategy (abbreviated RHSR), a regional light rail transit (abbreviated LRT) strategy, and a non-stopping continuous railway system (abbreviated CRS) strategy. The provision of additional tools to value HSR infrastructure against its accessibility benefits and their regional distribution through alternative strategies beyond the actual HSR stations, would have great implications, both politically and technically, in moving towards new dimensions of HSR evaluation and development.

Sensitivity of building cooling loads to future weather predictions

The interaction and relationship between the global warming and the thermal performance buildings are dynamic in nature. In order to model and understand this behavior, different approaches, including keeping weather variable unchanged, morphing approach and diurnal modelling method, have been used to project and generate future weather data. Among these approaches, various assumptions on the change of solar radiation, air humidity and/or wind characteristics may be adopted. In this paper, an example to illustrate the generation of future weather data for the different global warming scenarios in Australia is presented. The sensitivity of building cooling loads to the possible changes of assumed values used in the future weather data generation is investigated. It is shown that with ± 10% change of the proposed future values for solar radiation, air humidity or wind characteristics, the corresponding change in the cooling load of the modeled sample office building at different Australian capital cities would not exceed 6%, 4% and 1.5% respectively. It is also found that with ±10% changes on the proposed weather variables for both the 2070-high future scenario and the current weather scenario, the corresponding change in the cooling loads at different locations may be weaker (up to 2% difference in Hobart for ±10% change in global solar radiation), similar (less than 0.6%) difference in Hobart for ±10% change in wind speed), or stronger (up to 1.6% difference in Hobart for ±10% change in relative humidity) in the 2070-high future scenario than in the current weather scenario.

QIPC low speed run-overs of young children research project final report

This final report outlines the research conducted by the Centre for Accident Research and Road Safety – Queensland (CARRS-Q) for the research project (title above). This report provides an outline of the project methodology, literature review, three stages of research results (including the focus group discussions, review of organisational records, documentation and initiatives, and analysis of previous CARRS-Q occupational road safety self-report surveys), and recommendations for intervention strategy and initiatives development and implementation.

How Does Information Processing Speed Relate to the Attentional Blink?

Background When observers are asked to identify two targets in rapid sequence, they often suffer profound performance deficits for the second target, even when the spatial location of the targets is known. This attentional blink (AB) is usually attributed to the time required to process a previous target, implying that a link should exist between individual differences in information processing speed and the AB. Methodology/Principal Findings The present work investigated this question by examining the relationship between a rapid automatized naming task typically used to assess information-processing speed and the magnitude of the AB. The results indicated that faster processing actually resulted in a greater AB, but only when targets were presented amongst high similarity distractors. When target-distractor similarity was minimal, processing speed was unrelated to the AB. Conclusions/Significance Our findings indicate that information-processing speed is unrelated to target processing efficiency per se, but rather to individual differences in observers' ability to suppress distractors. This is consistent with evidence that individuals who are able to avoid distraction are more efficient at deploying temporal attention, but argues against a direct link between general processing speed and efficient information selection.

'The comfort lies in all the things you can do': the Australian drive-in: cinema of distraction

Contrary to the claims of some film historians, the drive-in was not a uniquely American invention. Australian drive-in cinemas were, at least in the 1950s and 1960s, distinguishable from their American counterparts by virtue of the profusion of additional amusements (or distractions) they offered alongside film-viewing. This article traces the history of Australian drive-ins as ‘entertainment centres’ and ‘high temples of modernity’. It argues that the drive-in can usefully be understood as a mid-point between the domestic and public spheres, and a powerful symbol of post-WWII Australia, signifying prosperity, gathering consumer confidence and, in metropolitan areas, marking the path of urban development through its concentration in new, outer suburban areas.

Factors influencing speeding in school zones : the role of mindfulness

Background Excessive speed contributes to the incidence and severity of road crashes. The Theory of Planned Behaviour (TPB) has successfully explained variance in speeding intentions and behaviour. However, studies have shown that more than 40% of the variance in outcome measures of speeding remains unexplained, thus, suggesting additional constructs may help to enhance the TPB’s predictive power. Therefore, this study examined mindfulness; a promising construct which has not yet been tested as an additional TPB predictor. Aims The aims of this study were to explore drivers’ beliefs about speeding in school zones using the extended TPB as a framework and to examine the effect that mindfulness had on driver speeding behaviour in school zones. Methods Australian drivers (N = 17) participated in one of four focus group discussions. The overall sample was comprised of five males and twelve females who were aged between 17 to56 years. All participants were recruited via purposive sampling among 1st year psychology students at a large South East Queensland University. The group discussions took approximately one hour and were guided by a structured interview schedule which sought to elicit drivers’ beliefs, thoughts and opinions on speeding in school zones and the factors which motivate such behaviour. Results Overall, thematic analysis revealed some similar issues emerged across the groups. . In particular and perhaps somewhat unsurprisingly, given public concerns regarding the want to ensure the safety of school children, there was much agreement that speeding in school zones was dangerous and unacceptable. Somewhat paradoxically however, some participants also agreed that they had unintentionally or mindlessly sped in school zones. There were several factors that drivers believed influenced their speeding in school zones including their current mood (e.g., if in a bad mood, anxious, or excited they may be more likely to drive without awareness of, and being attentive to, their driving environment) and the extent to which they were familiar with the environment (i.e., more familiar contexts, more likely to drive mindlessly). Thus, although drivers expressed a belief that speeding in school zones was dangerous and acceptable, the extent to which a driver is mindful does influence whether or not a driver may actually engage in speeding in this context. Discussion and conclusions This study highlights the potential role of mindfulness in helping to explain speeding behaviour in school zones. Mindless drivers may speed unintentionally and while unintentional still be endangering the safety and lives of school children. The findings of this research suggest that unintentional speeding, especially in school zones, may be reduced by countermeasures which heighten the extent to which drivers are mindful of approaching and/or driving through a school zone, such as street markings and engineering measures (e.g.,flashing lights and speed bumps).

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.

Speed enforcement in Australia : a changing policy landscape shaped by public opinion

- Speeding and crash involvement in Australia - Speed management in Australia - Jurisdictional differences - National Road Safety Strategy (2011-2020) - Auditor-General reviews of speed camera programs - The role of public opinion/feedback - Implications for speed management

The experiences of learner drivers, provisional drivers and supervisors with graduated driver licensing in two Australian jurisdictions

Newly licensed drivers on a provisional or intermediate licence have the highest crash risk when compared with any other group of drivers. In comparison, learner drivers have the lowest crash risk. Graduated driver licensing is one countermeasure that has been demonstrated to effectively reduce the crashes of novice drivers. This thesis examined the graduated driver licensing systems in two Australian states in order to better understand the behaviour of learner drivers, provisional drivers and the supervisors of learner drivers. By doing this, the thesis investigated the personal, social and environmental influences on novice driver behaviour as well as providing effective baseline data against which to measure subsequent changes to the licensing systems. In the first study, conducted prior to the changes to the graduated driver licensing system introduced in mid-2007, drivers who had recently obtained their provisional licence in Queensland and New South Wales were interviewed by telephone regarding their experiences while driving on their learner licence. Of the 687 eligible people approached to participate at driver licensing centres, 392 completed the study representing a response rate of 57.1 per cent. At the time the data was collected, New South Wales represented a more extensive graduated driver licensing system when compared with Queensland. The results suggested that requiring learners to complete a mandated number of hours of supervised practice impacts on the amount of hours that learners report completing. While most learners from New South Wales reported meeting the requirement to complete 50 hours of practice, it appears that many stopped practising soon after this goal was achieved. In contrast, learners from Queensland, who were not required to complete a specific number of hours at the time of the survey, tended to fall into three groups. The first group appeared to complete the minimum number of hours required to pass the test (less than 26 hours), the second group completed 26 to 50 hours of supervised practice while the third group completed significantly more practice than the first two groups (over 100 hours of supervised practice). Learner drivers in both states reported generally complying with the road laws and were unlikely to report that they had been caught breaking the road rules. They also indicated that they planned to obey the road laws once they obtained their provisional licence. However, they were less likely to intend to comply with recommended actions to reduce crash risk such as limiting their driving at night. This study also identified that there were relatively low levels of unaccompanied driving (approximately 15 per cent of the sample), very few driving offences committed (five per cent of the sample) and that learner drivers tended to use a mix of private and professional supervisors (although the majority of practice is undertaken with private supervisors). Consistent with the international literature, this study identified that very few learner drivers had experienced a crash (six per cent) while on their learner licence. The second study was also conducted prior to changes to the graduated driver licensing system and involved follow up interviews with the participants of the first study after they had approximately 21 months driving experience on their provisional licence. Of the 392 participants that completed the first study, 233 participants completed the second interview (representing a response rate of 59.4 per cent). As with the first study, at the time the data was collected, New South Wales had a more extensive graduated driver licensing system than Queensland. For instance, novice drivers from New South Wales were required to progress through two provisional licence phases (P1 and P2) while there was only one provisional licence phase in Queensland. Among the participants in this second study, almost all provisional drivers (97.9 per cent) owned or had access to a vehicle for regular driving. They reported that they were unlikely to break road rules, such as driving after a couple of drinks, but were also unlikely to comply with recommended actions, such as limiting their driving at night. When their provisional driving behaviour was compared to the stated intentions from the first study, the results suggested that their intentions were not a strong predictor of their subsequent behaviour. Their perception of risk associated with driving declined from when they first obtained their learner licence to when they had acquired provisional driving experience. Just over 25 per cent of participants in study two reported that they had been caught committing driving offences while on their provisional licence. Nearly one-third of participants had crashed while driving on a provisional licence, although few of these crashes resulted in injuries or hospitalisations. To complement the first two studies, the third study examined the experiences of supervisors of learner drivers, as well as their perceptions of their learner’s experiences. This study was undertaken after the introduction of the new graduated driver licensing systems in Queensland and New South Wales in mid- 2007, providing insights into the impacts of these changes from the perspective of supervisors. The third study involved an internet survey of 552 supervisors of learner drivers. Within the sample, approximately 50 per cent of participants supervised their own child. Other supervisors of the learner drivers included other parents or stepparents, professional driving instructors and siblings. For two-thirds of the sample, this was the first learner driver that they had supervised. Participants had provided an average of 54.82 hours (sd = 67.19) of supervision. Seventy-three per cent of participants indicated that their learners’ logbooks were accurate or very accurate in most cases, although parents were more likely than non-parents to report that their learners’ logbook was accurate (F (1,546) = 7.74, p = .006). There was no difference between parents and non-parents regarding whether they believed the log book system was effective (F (1,546) = .01, p = .913). The majority of the sample reported that their learner driver had had some professional driving lessons. Notwithstanding this, a significant proportion (72.5 per cent) believed that parents should be either very involved or involved in teaching their child to drive, with parents being more likely than non-parents to hold this belief. In the post mid-2007 graduated driver licensing system, Queensland learner drivers are able to record three hours of supervised practice in their log book for every hour that is completed with a professional driving instructor, up to a total of ten hours. Despite this, there was no difference identified between Queensland and New South Wales participants regarding the amount of time that they reported their learners spent with professional driving instructors (X2(1) = 2.56, p = .110). Supervisors from New South Wales were more likely to ensure that their learner driver complied with the road laws. Additionally, with the exception of drug driving laws, New South Wales supervisors believed it was more important to teach safety-related behaviours such as remaining within the speed limit, car control and hazard perception than those from Queensland. This may be indicative of more intensive road safety educational efforts in New South Wales or the longer time that graduated driver licensing has operated in that jurisdiction. However, other factors may have contributed to these findings and further research is required to explore the issue. In addition, supervisors reported that their learner driver was involved in very few crashes (3.4 per cent) and offences (2.7 per cent). This relatively low reported crash rate is similar to that identified in the first study. Most of the graduated driver licensing research to date has been applied in nature and lacked a strong theoretical foundation. These studies used Akers’ social learning theory to explore the self-reported behaviour of novice drivers and their supervisors. This theory was selected as it has previously been found to provide a relatively comprehensive framework for explaining a range of driver behaviours including novice driver behaviour. Sensation seeking was also used in the first two studies to complement the non-social rewards component of Akers’ social learning theory. This program of research identified that both Akers’ social learning theory and sensation seeking were useful in predicting the behaviour of learner and provisional drivers over and above socio-demographic factors. Within the first study, Akers’ social learning theory accounted for an additional 22 per cent of the variance in learner driver compliance with the law, over and above a range of socio-demographic factors such as age, gender and income. The two constructs within Akers’ theory which were significant predictors of learner driver compliance were the behavioural dimension of differential association relating to friends, and anticipated rewards. Sensation seeking predicted an additional six per cent of the variance in learner driver compliance with the law. When considering a learner driver’s intention to comply with the law while driving on a provisional licence, Akers’ social learning theory accounted for an additional 10 per cent of the variance above socio-demographic factors with anticipated rewards being a significant predictor. Sensation seeking predicted an additional four per cent of the variance. The results suggest that the more rewards individuals anticipate for complying with the law, the more likely they are to obey the road rules. Further research is needed to identify which specific rewards are most likely to encourage novice drivers’ compliance with the law. In the second study, Akers’ social learning theory predicted an additional 40 per cent of the variance in self-reported compliance with road rules over and above socio-demographic factors while sensation seeking accounted for an additional five per cent of the variance. A number of Aker’s social learning theory constructs significantly predicted provisional driver compliance with the law, including the behavioural dimension of differential association for friends, the normative dimension of differential association, personal attitudes and anticipated punishments. The consistent prediction of additional variance by sensation seeking over and above the variables within Akers’ social learning theory in both studies one and two suggests that sensation seeking is not fully captured within the non social rewards dimension of Akers’ social learning theory, at least for novice drivers. It appears that novice drivers are strongly influenced by the desire to engage in new and intense experiences. While socio-demographic factors and the perception of risk associated with driving had an important role in predicting the behaviour of the supervisors of learner drivers, Akers’ social learning theory provided further levels of prediction over and above these factors. The Akers’ social learning theory variables predicted an additional 14 per cent of the variance in the extent to which supervisors ensured that their learners complied with the law and an additional eight per cent of the variance in the supervisors’ provision of a range of practice experiences. The normative dimension of differential association, personal attitudes towards the use of professional driving instructors and anticipated rewards were significant predictors for supervisors ensuring that their learner complied with the road laws, while the normative dimension was important for range of practice. This suggests that supervisors who engage with other supervisors who ensure their learner complies with the road laws and provide a range of practice to their own learners are more likely to also engage in these behaviours. Within this program of research, there were several limitations including the method of recruitment of participants within the first study, the lower participation rate in the second study, an inability to calculate a response rate for study three and the use of self-report data for all three studies. Within the first study, participants were only recruited from larger driver licensing centres to ensure that there was a sufficient throughput of drivers to approach. This may have biased the results due to the possible differences in learners that obtain their licences in locations with smaller licensing centres. Only 59.4 per cent of the sample in the first study completed the second study. This may be a limitation if there was a common reason why those not participating were unable to complete the interview leading to a systematic impact on the results. The third study used a combination of a convenience and snowball sampling which meant that it was not possible to calculate a response rate. All three studies used self-report data which, in many cases, is considered a limitation. However, self-report data may be the only method that can be used to obtain some information. This program of research has a number of implications for countermeasures in both the learner licence phase and the provisional licence phase. During the learner phase, licensing authorities need to carefully consider the number of hours that they mandate learner drivers must complete before they obtain their provisional driving licence. If they mandate an insufficient number of hours, there may be inadvertent negative effects as a result of setting too low a limit. This research suggests that logbooks may be a useful tool for learners and their supervisors in recording and structuring their supervised practice. However, it would appear that the usage rates for logbooks will remain low if they remain voluntary. One strategy for achieving larger amounts of supervised practice is for learner drivers and their supervisors to make supervised practice part of their everyday activities. As well as assisting the learner driver to accumulate the required number of hours of supervised practice, it would ensure that they gain experience in the types of environments that they will probably encounter when driving unaccompanied in the future, such as to and from education or work commitments. There is also a need for policy processes to ensure that parents and professional driving instructors communicate effectively regarding the learner driver’s progress. This is required as most learners spend at least some time with a professional instructor despite receiving significant amounts of practice with a private supervisor. However, many supervisors did not discuss their learner’s progress with the driving instructor. During the provisional phase, there is a need to strengthen countermeasures to address the high crash risk of these drivers. Although many of these crashes are minor, most involve at least one other vehicle. Therefore, there are social and economic benefits to reducing these crashes. If the new, post-2007 graduated driver licensing systems do not significantly reduce crash risk, there may be a need to introduce further provisional licence restrictions such as separate night driving and peer passenger restrictions (as opposed to the hybrid version of these two restrictions operating in both Queensland and New South Wales). Provisional drivers appear to be more likely to obey some provisional licence laws, such as lower blood alcohol content limits, than others such as speed limits. Therefore, there may be a need to introduce countermeasures to encourage provisional drivers to comply with specific restrictions. When combined, these studies provided significant information regarding graduated driver licensing programs. This program of research has investigated graduated driver licensing utilising a cross-sectional and longitudinal design in order to develop our understanding of the experiences of novice drivers that progress through the system in order to help reduce crash risk once novice drivers commence driving by themselves.

Numerical simulation of a new two-dimensional variable-order fractional percolation equation in non-homogeneous porous media

Percolation flow problems are discussed in many research fields, such as seepage hydraulics, groundwater hydraulics, groundwater dynamics and fluid dynamics in porous media. Many physical processes appear to exhibit fractional-order behavior that may vary with time, or space, or space and time. The theory of pseudodifferential operators and equations has been used to deal with this situation. In this paper we use a fractional Darcys law with variable order Riemann-Liouville fractional derivatives, this leads to a new variable-order fractional percolation equation. In this paper, a new two-dimensional variable-order fractional percolation equation is considered. A new implicit numerical method and an alternating direct method for the two-dimensional variable-order fractional model is proposed. Consistency, stability and convergence of the implicit finite difference method are established. Finally, some numerical examples are given. The numerical results demonstrate the effectiveness of the methods. This technique can be used to simulate a three-dimensional variable-order fractional percolation equation.

Numerical analysis of the time variable fractional order mobile-immobile advection-dispersion model

Many physical processes exhibit fractional order behavior that varies with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. In this paper, we consider the time variable fractional order mobile-immobile advection-dispersion model. Numerical methods and analyses of stability and convergence for the fractional partial differential equations are quite limited and difficult to derive. This motivates us to develop efficient numerical methods as well as stability and convergence of the implicit numerical methods for the fractional order mobile immobile advection-dispersion model. In the paper, we use the Coimbra variable time fractional derivative which is more efficient from the numerical standpoint and is preferable for modeling dynamical systems. An implicit Euler approximation for the equation is proposed and then the stability of the approximation are investigated. As for the convergence of the numerical scheme we only consider a special case, i.e. the time fractional derivative is independent of time variable t. The case where the time fractional derivative depends both the time variable t and the space variable x will be considered in the future work. Finally, numerical examples are provided to show that the implicit Euler approximation is computationally efficient.

Numerical techniques for the variable order time fractional diffusion equation

In this paper we consider the variable order time fractional diffusion equation. We adopt the Coimbra variable order (VO) time fractional operator, which defines a consistent method for VO differentiation of physical variables. The Coimbra variable order fractional operator also can be viewed as a Caputo-type definition. Although this definition is the most appropriate definition having fundamental characteristics that are desirable for physical modeling, numerical methods for fractional partial differential equations using this definition have not yet appeared in the literature. Here an approximate scheme is first proposed. The stability, convergence and solvability of this numerical scheme are discussed via the technique of Fourier analysis. Numerical examples are provided to show that the numerical method is computationally efficient. Crown Copyright © 2012 Published by Elsevier Inc. All rights reserved.

Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation

Anomalous subdiffusion equations have in recent years received much attention. In this paper, we consider a two-dimensional variable-order anomalous subdiffusion equation. Two numerical methods (the implicit and explicit methods) are developed to solve the equation. Their stability, convergence and solvability are investigated by the Fourier method. Moreover, the effectiveness of our theoretical analysis is demonstrated by some numerical examples. © 2011 American Mathematical Society.