Robotics toolbox for MATLAB, release 9 [Software]

The ninth release of the Toolbox, represents over fifteen years of development and a substantial level of maturity. This version captures a large number of changes and extensions generated over the last two years which support my new book “Robotics, Vision & Control”. The Toolbox has always provided many functions that are useful for the study and simulation of classical arm-type robotics, for example such things as kinematics, dynamics, and trajectory generation. The Toolbox is based on a very general method of representing the kinematics and dynamics of serial-link manipulators. These parameters are encapsulated in MATLAB ® objects - robot objects can be created by the user for any serial-link manipulator and a number of examples are provided for well know robots such as the Puma 560 and the Stanford arm amongst others. The Toolbox also provides functions for manipulating and converting between datatypes such as vectors, homogeneous transformations and unit-quaternions which are necessary to represent 3-dimensional position and orientation. This ninth release of the Toolbox has been significantly extended to support mobile robots. For ground robots the Toolbox includes standard path planning algorithms (bug, distance transform, D*, PRM), kinodynamic planning (RRT), localization (EKF, particle filter), map building (EKF) and simultaneous localization and mapping (EKF), and a Simulink model a of non-holonomic vehicle. The Toolbox also including a detailed Simulink model for a quadcopter flying robot.

An estimation of crash risks using a log-linear model and the induced exposure technique

Prevention and safety promotion programmes. Traditionally, in-depth investigations of crash risks are conducted using exposure controlled study or case-control methodology. However, these studies need either observational data for control cases or exogenous exposure data like vehicle-kilometres travel, entry flow or product of conflicting flow for a particular traffic location, or a traffic site. These data are not readily available and often require extensive data collection effort on a system-wide basis. Aim: The objective of this research is to propose an alternative methodology to investigate crash risks of a road user group in different circumstances using readily available traffic police crash data. Methods: This study employs a combination of a log-linear model and the quasi-induced exposure technique to estimate crash risks of a road user group. While the log-linear model reveals the significant interactions and thus the prevalence of crashes of a road user group under various sets of traffic, environmental and roadway factors, the quasi-induced exposure technique estimates relative exposure of that road user in the same set of explanatory variables. Therefore, the combination of these two techniques provides relative measures of crash risks under various influences of roadway, environmental and traffic conditions. The proposed methodology has been illustrated using Brisbane motorcycle crash data of five years. Results: Interpretations of results on different combination of interactive factors show that the poor conspicuity of motorcycles is a predominant cause of motorcycle crashes. Inability of other drivers to correctly judge the speed and distance of an oncoming motorcyclist is also evident in right-of-way violation motorcycle crashes at intersections. Discussion and Conclusions: The combination of a log-linear model and the induced exposure technique is a promising methodology and can be applied to better estimate crash risks of other road users. This study also highlights the importance of considering interaction effects to better understand hazardous situations. A further study on the comparison between the proposed methodology and case-control method would be useful.

Efficient preconditioning of the method of lines for solving nonlinear two-sided space-fractional diffusion equations

A standard method for the numerical solution of partial differential equations (PDEs) is the method of lines. In this approach the PDE is discretised in space using �finite di�fferences or similar techniques, and the resulting semidiscrete problem in time is integrated using an initial value problem solver. A significant challenge when applying the method of lines to fractional PDEs is that the non-local nature of the fractional derivatives results in a discretised system where each equation involves contributions from many (possibly every) spatial node(s). This has important consequences for the effi�ciency of the numerical solver. First, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. Second, since the Jacobian matrix of the system is dense (partially or fully), methods that avoid the need to form and factorise this matrix are preferred. In this paper, we consider a nonlinear two-sided space-fractional di�ffusion equation in one spatial dimension. A key contribution of this paper is to demonstrate how an eff�ective preconditioner is crucial for improving the effi�ciency of the method of lines for solving this equation. In particular, we show how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach.

On the significance of omitted variables in intersection crash modeling

Advances in safety research—trying to improve the collective understanding of motor vehicle crash causes and contributing factors—rest upon the pursuit of numerous lines of research inquiry. The research community has focused considerable attention on analytical methods development (negative binomial models, simultaneous equations, etc.), on better experimental designs (before-after studies, comparison sites, etc.), on improving exposure measures, and on model specification improvements (additive terms, non-linear relations, etc.). One might logically seek to know which lines of inquiry might provide the most significant improvements in understanding crash causation and/or prediction. It is the contention of this paper that the exclusion of important variables (causal or surrogate measures of causal variables) cause omitted variable bias in model estimation and is an important and neglected line of inquiry in safety research. In particular, spatially related variables are often difficult to collect and omitted from crash models—but offer significant opportunities to better understand contributing factors and/or causes of crashes. This study examines the role of important variables (other than Average Annual Daily Traffic (AADT)) that are generally omitted from intersection crash prediction models. In addition to the geometric and traffic regulatory information of intersection, the proposed model includes many spatial factors such as local influences of weather, sun glare, proximity to drinking establishments, and proximity to schools—representing a mix of potential environmental and human factors that are theoretically important, but rarely used. Results suggest that these variables in addition to AADT have significant explanatory power, and their exclusion leads to omitted variable bias. Provided is evidence that variable exclusion overstates the effect of minor road AADT by as much as 40% and major road AADT by 14%.

Multi-method, multi-theoretical, multi-level research in the learning sciences

We examine methodologies and methods that apply to multi-level research in the learning sciences. In so doing we describe how multiple theoretical frameworks informs the use of different methods that apply to social levels involving space-time relationships that are not accessible consciously as social life is enacted. Most of the methods involve analyses of video and audio files. Within a framework of interpretive research we present a methodology of event-oriented social science, which employs video ethnography, narrative, conversation analysis, prosody analysis, and facial expression analysis. We illustrate multi-method research in an examination of the role of emotions in teaching and learning. Conversation and prosody analyses augment facial expression analysis and ethnography. We conclude with an exploration of ways in which multi-level studies can be complemented with neural level analyses.

Non-invasive method for detecting changes in soil moisture using wireless sensor networks

There are several popular soil moisture measurement methods today such as time domain reflectometry, electromagnetic (EM) wave, electrical and acoustic methods. Significant studies have been dedicated in developing method of measurements using those concepts, especially to achieve the characteristics of noninvasiveness. EM wave method provides an advantage because it is non-invasive to the soil and does not need to utilise probes to penetrate or bury in the soil. But some EM methods are also too complex, expensive, and not portable for the application of Wireless Sensor Networks; for example satellites or UAV (Unmanned Aerial Vehicle) based sensors. This research proposes a method in detecting changes in soil moisture using soil-reflected electromagnetic (SREM) wave from Wireless Sensor Networks (WSNs). Studies have shown that different levels of soil moisture will affects soil’s dielectric properties, such as relative permittivity and conductivity, and in turns change its reflection coefficients. The SREM wave method uses a transmitter adjacent to a WSNs node with purpose exclusively to transmit wireless signals that will be reflected by the soil. The strength from the reflected signal that is determined by the soil’s reflection coefficients is used to differentiate the level of soil moisture. The novel nature of this method comes from using WSNs communication signals to perform soil moisture estimation without the need of external sensors or invasive equipment. This innovative method is non-invasive, low cost and simple to set up. There are three locations at Brisbane, Australia chosen as the experiment’s location. The soil type in these locations contains 10–20% clay according to the Australian Soil Resource Information System. Six approximate levels of soil moisture (8, 10, 13, 15, 18 and 20%) are measured at each location; with each measurement consisting of 200 data. In total 3600 measurements are completed in this research, which is sufficient to achieve the research objective, assessing and proving the concept of SREM wave method. These results are compared with reference data from similar soil type to prove the concept. A fourth degree polynomial analysis is used to generate an equation to estimate soil moisture from received signal strength as recorded by using the SREM wave method.

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.

Anomalous subdiffusion equations by the meshless collocation method

This paper aims to develop an implicit meshless collocation technique based on the moving least squares approximation for numerical simulation of the anomalous subdiffusion equation(ASDE). The discrete system of equations is obtained by using the MLS meshless shape functions and the meshless collocation formulation. The stability and convergence of this meshless approach related to the time discretization are investigated theoretically and numerically. The numerical examples with regular and irregular nodal distributions are used to the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling of ASDEs.

RCA method for fault diagnosis in digital substations of power systems

The authors present a Cause-Effect fault diagnosis model, which utilises the Root Cause Analysis approach and takes into account the technical features of a digital substation. The Dempster/Shafer evidence theory is used to integrate different types of fault information in the diagnosis model so as to implement a hierarchical, systematic and comprehensive diagnosis based on the logic relationship between the parent and child nodes such as transformer/circuit-breaker/transmission-line, and between the root and child causes. A real fault scenario is investigated in the case study to demonstrate the developed approach in diagnosing malfunction of protective relays and/or circuit breakers, miss or false alarms, and other commonly encountered faults at a modern digital substation.

Investigating the potential value of individual parameters of histological grading systems in a sheep model of cartilage damage : the modified Mankin method

A total histological grade does not necessarily distinguish between different manifestations of cartilage damage or degeneration. An accurate and reliable histological assessment method is required to separate normal and pathological tissue within a joint during treatment of degenerative joint conditions and to sub-classify the latter in meaningful ways. The Modified Mankin method may be adaptable for this purpose. We investigated how much detail may be lost by assigning one composite score/grade to represent different degenerative components of the osteoarthritic condition. We used four ovine injury models (sham surgery, anterior cruciate ligament/medial collateral ligament instability, simulated anatomic anterior cruciate ligament reconstruction and meniscal removal) to induce different degrees and potentially 'types' (mechanisms) of osteoarthritis. Articular cartilage was systematically harvested, prepared for histological examination and graded in a blinded fashion using a Modified Mankin grading method. Results showed that the possible permutations of cartilage damage were significant and far more varied than the current intended use that histological grading systems allow. Of 1352 cartilage specimens graded, 234 different manifestations of potential histological damage were observed across 23 potential individual grades of the Modified Mankin grading method. The results presented here show that current composite histological grading may contain additional information that could potentially discern different stages or mechanisms of cartilage damage and degeneration in a sheep model. This approach may be applicable to other grading systems.

Alternating direction implicit numerical method for the space and time fractional Bloch-Torrey equation in 3-D

Recently, some authors have considered a new diffusion model–space and time fractional Bloch-Torrey equation (ST-FBTE). Magin et al. (2008) have derived analytical solutions with fractional order dynamics in space (i.e., _ = 1, β an arbitrary real number, 1 < β ≤ 2) and time (i.e., 0 < α < 1, and β = 2), respectively. Yu et al. (2011) have derived an analytical solution and an effective implicit numerical method for solving ST-FBTEs, and also discussed the stability and convergence of the implicit numerical method. However, due to the computational overheads necessary to perform the simulations for nuclear magnetic resonance (NMR) in three dimensions, they present a study based on a two-dimensional example to confirm their theoretical analysis. Alternating direction implicit (ADI) schemes have been proposed for the numerical simulations of classic differential equations. The ADI schemes will reduce a multidimensional problem to a series of independent one-dimensional problems and are thus computationally efficient. In this paper, we consider the numerical solution of a ST-FBTE on a finite domain. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. A fractional alternating direction implicit scheme (FADIS) for the ST-FBTE in 3-D is proposed. Stability and convergence properties of the FADIS are discussed. Finally, some numerical results for ST-FBTE are given.

A new implicit numerical method for the space and time fractional Bloch-Torrey equation

In recent years, it has been found that many phenomena in engineering, physics, chemistry and other sciences can be described very successfully by models using mathematical tools from fractional calculus. Recently, noted a new space and time fractional Bloch-Torrey equation (ST-FBTE) has been proposed (see Magin et al. (2008)), and successfully applied to analyse diffusion images of human brain tissues to provide new insights for further investigations of tissue structures. In this paper, we consider the ST-FBTE on a finite domain. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we propose a new effective implicit numerical method (INM) for the STFBTE whereby we discretize the Riesz fractional derivative using a fractional centered difference. Secondly, we prove that the implicit numerical method for the ST-FBTE is unconditionally stable and convergent, and the order of convergence of the implicit numerical method is ( T2 - α + h2 x + h2 y + h2 z ). Finally, some numerical results are presented to support our theoretical analysis.

Analytical solutions of the multi-term modified power law wave equations in a finite domain

Fractional partial differential equations with more than one fractional derivative term in time, such as the Szabo wave equation, or the power law wave equation, describe important physical phenomena. However, studies of these multi-term time-space or time fractional wave equations are still under development. In this paper, multi-term modified power law wave equations in a finite domain are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals (1, 2], [2, 3), [2, 4) or (0, n) (n > 2), respectively. Analytical solutions of the multi-term modified power law wave equations are derived. These new techniques are based on Luchko’s Theorem, a spectral representation of the Laplacian operator, a method of separating variables and fractional derivative techniques. Then these general methods are applied to the special cases of the Szabo wave equation and the power law wave equation. These methods and techniques can also be extended to other kinds of the multi term time-space fractional models including fractional Laplacian.

A finite volume method for solving the two-sided time-space fractional advection-dispersion equation

The field of fractional differential equations provides a means for modelling transport processes within complex media which are governed by anomalous transport. Indeed, the application to anomalous transport has been a significant driving force behind the rapid growth and expansion of the literature in the field of fractional calculus. In this paper, we present a finite volume method to solve the time-space two-sided fractional advection dispersion equation on a one-dimensional domain. Such an equation allows modelling different flow regime impacts from either side. The finite volume formulation provides a natural way to handle fractional advection-dispersion equations written in conservative form. The novel spatial discretisation employs fractionally-shifted Gr¨unwald formulas to discretise the Riemann-Liouville fractional derivatives at control volume faces in terms of function values at the nodes, while the L1-algorithm is used to discretise the Caputo time fractional derivative. Results of numerical experiments are presented to demonstrate the effectiveness of the approach.

An implicit RBF meshless method for a fractal mobile/immobile transport model

Fractional differential equation is used to describe a fractal model of mobile/immobile transport with a power law memory function. This equation is the limiting equation that governs continuous time random walks with heavy tailed random waiting times. In this paper, we firstly propose a finite difference method to discretize the time variable and obtain a semi-discrete scheme. Then we discuss its stability and convergence. Secondly we consider a meshless method based on radial basis functions (RBF) to discretize the space variable. By contrast to conventional FDM and FEM, the meshless method is demonstrated to have distinct advantages: calculations can be performed independent of a mesh, it is more accurate and it can be used to solve complex problems. Finally the convergence order is verified from a numerical example is presented to describe the fractal model of mobile/immobile transport process with different problem domains. The numerical results indicate that the present meshless approach is very effective for modeling and simulating of fractional differential equations, and it has good potential in development of a robust simulation tool for problems in engineering and science that are governed by various types of fractional differential equations.