Serious fears across cultures about the collapse of the world order

This chapter approaches resilience from an evolutionary psychology (socio-biological) perspective. It argues that the internal constitution and mental toughness of the individual will provide a core protection for life’s inevitable tests in the innumerable micro and macro environments humans find themselves. The many descriptors of the construct of resilience used in various studies are explored. Finally, the difference psychologists can make in the therapy of clients whose resilience is being tested, is examined by means of case examples.

Alternatives to youth imprisonment : evaluating the Victorian youth attendance order

On 22 June 1988 the then Minister for Community Services Victoria, Race Matthews, officially launched the Youth Attendance Order (YAO), a high tariff alternative for young offenders aged between 15 and 18 years who were facing a term of detention. Throughout the order's gestation, much debate occurred about the impact it would have on rates of juvenile incarceration as well as about the potential ‘net widening’ effect it could have on less serious offenders. In May 1994 the National Centre For Socio-Legal Studies at La Trobe University submitted its report evaluating the Victorian Youth Attendance Order. This article presents some of the major findings of that report and examines the future options for this high tariff order in juvenile justice

The effect of exercise intensity on fat oxidation and non-exercise activity thermogenesis in overweight/obese men

It is frequently reported that the actual weight loss achieved through exercise interventions is less than theoretically expected. Amongst other compensatory adjustments that accompany exercise training (e.g., increases in resting metabolic rate and energy intake), a possible cause of the less than expected weight loss is a failure to produce a marked increase in total daily energy expenditure due to a compensatory reduction in non-exercise activity thermogenesis (NEAT). Therefore, there is a need to understand how behaviour is modified in response to exercise interventions. The proposed benefits of exercise training are numerous, including changes to fat oxidation. Given that a diminished capacity to oxidise fat could be a factor in the aetiology of obesity, an exercise training intensity that optimises fat oxidation in overweight/obese individuals would improve impaired fat oxidation, and potentially reduce health risks that are associated with obesity. To improve our understanding of the effectiveness of exercise for weight management, it is important to ensure exercise intensity is appropriately prescribed, and to identify and monitor potential compensatory behavioural changes consequent to exercise training. In line with the gaps in the literature, three studies were performed. The aim of Study 1 was to determine the effect of acute bouts of moderate- and high-intensity walking exercise on NEAT in overweight and obese men. Sixteen participants performed a single bout of either moderate-intensity walking exercise (MIE) or high-intensity walking exercise (HIE) on two separate occasions. The MIE consisted of walking for 60-min on a motorised treadmill at 6 km.h-1. The 60-min HIE session consisted of walking in 5-min intervals at 6 km.h-1 and 10% grade followed by 5-min at 0% grade. NEAT was assessed by accelerometer three days before, on the day of, and three days after the exercise sessions. There was no significant difference in NEAT vector magnitude (counts.min-1) between the pre-exercise period (days 1-3) and the exercise day (day 4) for either protocol. In addition, there was no change in NEAT during the three days following the MIE session, however NEAT increased by 16% on day 7 (post-exercise) compared with the exercise day (P = 0.32). During the post-exercise period following the HIE session, NEAT was increased by 25% on day 7 compared with the exercise day (P = 0.08), and by 30-33% compared with the pre-exercise period (day 1, day 2 and day 3); P = 0.03, 0.03, 0.02, respectively. To conclude, a single bout of either MIE or HIE did not alter NEAT on the exercise day or on the first two days following the exercise session. However, extending the monitoring of NEAT allowed the detection of a 48 hour delay in increased NEAT after performing HIE. A longer-term intervention is needed to determine the effect of accumulated exercise sessions over a week on NEAT. In Study 2, there were two primary aims. The first aim was to test the reliability of a discontinuous incremental exercise protocol (DISCON-FATmax) to identify the workload at which fat oxidation is maximised (FATmax). Ten overweight and obese sedentary male men (mean BMI of 29.5 ¡Ó 4.5 kg/m2 and mean age of 28.0 ¡Ó 5.3 y) participated in this study and performed two identical DISCON-FATmax tests one week apart. Each test consisted of alternate 4-min exercise and 2-min rest intervals on a cycle ergometer. The starting work load of 28 W was increased every 4-min using 14 W increments followed by 2-min rest intervals. When the respiratory exchange ratio was consistently >1.0, the workload was increased by 14 W every 2-min until volitional exhaustion. Fat oxidation was measured by indirect calorimetry. The mean FATmax, ƒtV O2peak, %ƒtV O2peak and %Wmax at which FATmax occurred during the two tests were 0.23 ¡Ó 0.09 and 0.18 ¡Ó 0.08 (g.min-1); 29.7 ¡Ó 7.8 and 28.3 ¡Ó 7.5 (ml.kg-1.min-1); 42.3 ¡Ó 7.2 and 42.6 ¡Ó 10.2 (%ƒtV O2max) and 36.4 ¡Ó 8.5 and 35.4 ¡Ó 10.9 (%), respectively. A paired-samples T-test revealed a significant difference in FATmax (g.min-1) between the tests (t = 2.65, P = 0.03). The mean difference in FATmax was 0.05 (g.min-1) with the 95% confidence interval ranging from 0.01 to 0.18. Paired-samples T-test, however, revealed no significant difference in the workloads (i.e. W) between the tests, t (9) = 0.70, P = 0.4. The intra-class correlation coefficient for FATmax (g.min-1) between the tests was 0.84 (95% confidence interval: 0.36-0.96, P < 0.01). However, Bland-Altman analysis revealed a large disagreement in FATmax (g.min-1) related to W between the two tests; 11 ¡Ó 14 (W) (4.1 ¡Ó 5.3 ƒtV O2peak (%)).These data demonstrate two important phenomena associated with exercise-induced substrate oxidation; firstly, that maximal fat oxidation derived from a discontinuous FATmax protocol differed statistically between repeated tests, and secondly, there was large variability in the workload corresponding with FATmax. The second aim of Study 2 was to test the validity of a DISCON-FATmax protocol by comparing maximal fat oxidation (g.min-1) determined by DISCON-FATmax with fat oxidation (g.min-1) during a continuous exercise protocol using a constant load (CONEX). Ten overweight and obese sedentary males (BMI = 29.5 ¡Ó 4.5 kg/m2; age = 28.0 ¡Ó 4.5 y) with a ƒtV O2max of 29.1 ¡Ó 7.5 ml.kg-1.min-1 performed a DISCON-FATmax test consisting of alternate 4-min exercise and 2-min rest intervals on a cycle ergometer. The 1-h CONEX protocol used the workload from the DISCON-FATmax to determine FATmax. The mean FATmax, ƒtV O2max, %ƒtV O2max and workload at which FATmax occurred during the DISCON-FATmax were 0.23 ¡Ó 0.09 (g.min-1); 29.1 ¡Ó 7.5 (ml.kg-1.min-1); 43.8 ¡Ó 7.3 (%ƒtV O2max) and 58.8 ¡Ó 19.6 (W), respectively. The mean fat oxidation during the 1-h CONEX protocol was 0.19 ¡Ó 0.07 (g.min-1). A paired-samples T-test revealed no significant difference in fat oxidation (g.min-1) between DISCON-FATmax and CONEX, t (9) = 1.85, P = 0.097 (two-tailed). There was also no significant correlation in fat oxidation between the DISCON-FATmax and CONEX (R=0.51, P = 0.14). Bland- Altman analysis revealed a large disagreement in fat oxidation between the DISCONFATmax and CONEX; the upper limit of agreement was 0.13 (g.min-1) and the lower limit of agreement was ¡V0.03 (g.min-1). These data suggest that the CONEX and DISCONFATmax protocols did not elicit different rates of fat oxidation (g.min-1). However, the individual variability in fat oxidation was large, particularly in the DISCON-FATmax test. Further research is needed to ascertain the validity of graded exercise tests for predicting fat oxidation during constant load exercise sessions. The aim of Study 3 was to compare the impact of two different intensities of four weeks of exercise training on fat oxidation, NEAT, and appetite in overweight and obese men. Using a cross-over design 11 participants (BMI = 29 ¡Ó 4 kg/m2; age = 27 ¡Ó 4 y) participated in a training study and were randomly assigned initially to: [1] a lowintensity (45%ƒtV O2max) exercise (LIT) or [2] a high-intensity interval (alternate 30 s at 90%ƒtV O2max followed by 30 s rest) exercise (HIIT) 40-min duration, three times a week. Participants completed four weeks of supervised training and between cross-over had a two week washout period. At baseline and the end of each exercise intervention,ƒtV O2max, fat oxidation, and NEAT were measured. Fat oxidation was determined during a standard 30-min continuous exercise bout at 45%ƒtV O2max. During the steady state exercise expired gases were measured intermittently for 5-min periods and HR was monitored continuously. In each training period, NEAT was measured for seven consecutive days using an accelerometer (RT3) the week before, at week 3 and the week after training. Subjective appetite sensations and food preferences were measured immediately before and after the first exercise session every week for four weeks during both LIT and HIIT. The mean fat oxidation rate during the standard continuous exercise bout at baseline for both LIT and HIIT was 0.14 ¡Ó 0.08 (g.min-1). After four weeks of exercise training, the mean fat oxidation was 0.178 ¡Ó 0.04 and 0.183 ¡Ó 0.04 g.min-1 for LIT and HIIT, respectively. The mean NEAT (counts.min-1) was 45 ¡Ó 18 at baseline, 55 ¡Ó 22 and 44 ¡Ó 16 during training, and 51 ¡Ó 14 and 50 ¡Ó 21 after training for LIT and HIIT, respectively. There was no significant difference in fat oxidation between LIT and HIIT. Moreover, although not statistically significant, there was some evidence to suggest that LIT and HIIT tend to increase fat oxidation during exercise at 45% ƒtV O2max (P = 0.14 and 0.08, respectively). The order of training treatment did not significantly influence changes in fat oxidation, NEAT, and appetite. NEAT (counts.min-1) was not significantly different in the week following training for either LIT or HIIT. Although not statistically significant (P = 0.08), NEAT was 20% lower during week 3 of exercise training in HIIT compared with LIT. Examination of appetite sensations revealed differences in the intensity of hunger, with higher ratings after LIT compared with HIIT. No differences were found in preferences for high-fat sweet foods between LIT and HIIT. In conclusion, the results of this thesis suggest that while fat oxidation during steady state exercise was not affected by the level of exercise intensity, there is strong evidence to suggest that intense exercise could have a debilitative effect on NEAT.

China and the "Responsibility to Protect" : the implications of the Libyan intervention

The emerging principle of a “responsibility to protect” (R2P) presents a direct challenge to China’s traditional emphasis on the twin principles of non-intervention in the domestic affairs of other states and non-use of military force. This paper considers the evolution of China’s relationship with R2P over the past ten years. In particular, it examines how China engaged with R2P during the recent Libyan crisis, and considers what impact this conflict may have first, on Chinese attitudes to R2P, and second, on the future development and implementation of the doctrine itself. This paper argues that China’s decision to allow the passage of Security Council resolution 1973, authorising force in Libya, was shaped by an unusual set of political and factual circumstances, and should not be viewed as evidence of a dramatic shift in Chinese attitudes towards R2P. More broadly, controversy over the scope of NATO’s military action in Libya has raised questions about R2P’s legitimacy, which have contributed to a lack of timely international action in Syria. In the short term at least, this post-Libya backlash against R2P is likely to constrain the Security Council’s ability to respond decisively to other civilian protection situations.

Design of fixed order robust power oscillation damper for TCSC

This paper illustrates robust fixed order power oscillation damper design for mitigating power systems oscillations. From implementation and tuning point of view, such low and fixed structure is common practice for most practical applications, including power systems. However, conventional techniques of optimal and robust control theory cannot handle the constraint of fixed-order as it is, in general, impossible to ensure a target closed-loop transfer function by a controller of any given order. This paper deals with the problem of synthesizing or designing a feedback controller of dynamic order for a linear time-invariant plant for a fixed plant, as well as for an uncertain family of plants containing parameter uncertainty, so that stability, robust stability and robust performance are attained. The desired closed-loop specifications considered here are given in terms of a target performance vector representing a desired closed-loop design. The performance of the designed controller is validated through non-linear simulations for a range of contingencies.

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.

Quality of work life and turnover intention in primary healthcare organisations : a cross-sectional study of registered nurses in Saudi Arabia

Purpose: The purpose of this study was to improve the retention of primary healthcare (PHC) nurses through exploring and assessing their quality of work life (QWL) and turnover intention. Design and methods: A cross-sectional survey design was used in this study. Data were collected using a questionnaire comprising four sections (Brooks’ survey of Quality of Nursing Work Life [QNWL], Anticipated Turnover Intention, open-ended questions and demographic characteristics). A convenience sample was recruited from 143 PHC centres in Jazan, Saudi Arabia. A response rate of 87% (n = 508/585) was achieved. The SPSS v17 for Windows and NVivo 8 were used for analysis purposes. Procedures and tests used in this study to analyse the quantitative data were descriptive statistics, t-test, ANOVA, General Linear Model (GLM) univariate analysis, standard multiple regression, and hierarchical multiple regression. Qualitative data obtained from responses to the open-ended questions were analysed using the NVivo 8. Findings: Quantitative findings suggested that PHC nurses were dissatisfied with their work life. Respondents’ scores ranged between 45 and 218 (mean = 139.45), which is lower than the average total score on Brooks’ Survey (147). Major influencing factors were classified under four dimensions. First, work life/home life factors: unsuitable working hours, lack of facilities for nurses, inability to balance work with family needs and inadequacy of vacations’ policy. Second, work design factors: high workload, insufficient workforce numbers, lack of autonomy and undertaking many non-nursing tasks. Third, work context factors: management practices, lack of development opportunities, and inappropriate working environment in terms of the level of security, patient care supplies and unavailability of recreation room. Finally, work world factors: negative public image of nursing, and inadequate payment. More positively, nurses were notably satisfied with their co-workers. Conversely, 40.4% (n = 205) of the respondents indicated that they intended to leave their current employment. The relationships between QWL and demographic variables of gender, age, marital status, dependent children, dependent adults, nationality, ethnicity, nursing tenure, organisational tenure, positional tenure, and payment per month were significant (p < .05). The eta squared test for these demographics indicates a small to medium effect size of the variation in QWL scores. Using the GLM univariate analysis, education level was also significantly related to the QWL (p < .05). The relationships between turnover intention and demographic variables including gender, age, marital status, dependent children, education level, nursing tenure, organisational tenure, positional tenure, and payment per month were significant (p < .05). The eta squared test for these demographics indicates a small to moderate effect size of the variation in the turnover intention scores. Using the GLM univariate analysis, the dependent adults’ variable was also significantly related to turnover intention (p < .05). Turnover intention was significantly related to QWL. Using standard multiple regression, 26% of the variance in turnover intention was explained by the QWL F (4,491), 43.71, p < .001, with R² = .263. Further analysis using hierarchical multiple regression found that the total variance explained by the model as a whole (demographics and QWL) was 32.1%, F (17.433) = 12.04, p < .001. QWL explained an additional 19% of the variance in turnover intention, after controlling for demographic variables, R squared change =.19, F change (4, 433) = 30.190, p < .001. The work context variable makes the strongest unique contribution (-.387) to explain the turnover intention, followed by the work design dimension (-.112). The qualitative findings reaffirmed the quantitative findings in terms of QWL and turnover intention. However, the home life/work life and work world dimensions were of great important to both QWL and turnover intention. The qualitative findings revealed a number of new factors that were not included in the survey questionnaire. These included being away from family, lack of family support, social and cultural aspects, accommodation facilities, transportation, building and infrastructure of PHC, nature of work, job instability, privacy at work, patients and community, and distance between home and workplace. Conclusion: Creating and maintaining a healthy work life for PHC nurses is very important to improve their work satisfaction, reduce turnover, enhance productivity and improve nursing care outcomes. Improving these factors could lead to a higher QWL and increase retention rates and therefore reinforcing the stabilisation of the nursing workforce. Significance of the research: Many countries are examining strategies to attract and retain the health care workforce, particularly nurses. This study identified factors that influence the QWL of PHC nurses as well as their turnover intention. It also determined the significant relationship between QWL and turnover intention. In addition, the present study tested Brooks’ survey of QNWL on PHC nurses for the first time. The qualitative findings of this study revealed a number of new variables regarding QWL and turnover intention of PHC nurses. These variables could be used to improve current survey instruments or to develop new research surveys. The study findings could be also used to develop and appropriately implement plans to improve QWL. This may help to enhance the home and work environments of PHC nurses, improve individual and organisational performance, and increase nurses’ commitment. This study contributes to the existing body of research knowledge by presenting new data and findings from a different country and healthcare system. It is the first of its kind in Saudi Arabia, especially in the field of PHC. It has examined the relationship between QWL and turnover intention of PHC nurses for the first time using nursing instruments. The study also offers a fresh explanation (new framework) of the relationship between QWL and turnover intention among PHC nurses, which could be used or tested by researchers in other settings. Implications for further research: Review of the extant literature reveals little in-depth research on the PHC workforce, especially in terms of QWL and organisational turnover in developing countries. Further research is required to develop a QWL tool for PHC nurses, taking into consideration the findings of the current study along with the local culture. Moreover, the revised theoretical framework of the current study could be tested in further research in other regions, countries or healthcare systems in order to identify its ability to predict the level of PHC nurses’ QWL and their intention to leave. There is a need to conduct longitudinal research on PHC organisations to gain an in-depth understanding of the determents of and changes in QWL and turnover intention of PHC nurses at various points of time. An intervention study is required to improve QWL and retention among PHC nurses using the findings of the current study. This would help to assess the impact of such strategies on reducing turnover of PHC nurses. Focusing on the location of the current study, it would be valuable to conduct another study in five years’ time to examine the percentage of actual turnover among PHC nurses compared with the reported turnover intention in the current study. Further in-depth research would also be useful to assess the impact of the local culture on the perception of expatriate nurses towards their QWL and their turnover intention. A comparative study is required between PHC centres and hospitals as well as the public and private health sector agencies in terms of QWL and turnover intention of nursing personnel. Findings may differ from sector to sector according to variations in health systems, working environments and the case mix of patients.

Students as advisors in an intervention program for at-risk students : The QUT experience

A managed team of discipline-experienced and trained later year students are used as Student Success Advisors (SSAs) in the Student Success Program, an intervention program that manages student engagement by identifying and supporting first year students at-risk of disengaging from learning. This report focuses on the recruitment and training of SSAs and the day-to-day challenges they and their managers face. The Nuts & Bolts session provides participants with opportunities to discuss the applicability to their institutional contexts of the recruitment and training processes and the “solutions” to the challenges used at QUT.

A small molecule that walks non-directionally along a track without external intervention

Charge of the light brigade: A molecule is able to walk back and forth upon a five-foothold pentaethylenimine track without external intervention. The 1D random walk is highly processive (mean step number 530) and exchange takes place between adjacent amine groups in a stepwise fashion. The walker performs a simple task whilst walking: quenching of the fluorescence of an anthracene group sited at one end of the track. Copyright © 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Experiences of a walking intervention in a University community : Qualitative perspectives

Background: This study explored the experiences of university employees that participated in a walking intervention that encouraged individuals to walk more throughout their workday. The 10-week program was comprised of 5 phases (i.e. baseline, anticipating barriers, short planned walks, longer planned walks and maintenance) and utilized a pedometer diary and an online website for logging steps. The pedometer diary included “action plans” for addressing barriers and planning walking and the online dashboard provided graphical outputs that allowed participants to visualize whether they were reaching or exceeding their step targets. Methods: A subsample of 12 academic and administrative employees from the study completed open ended questionnaires at the end of the study. The questions focused on capturing the major themes of benefits/mediators and problems/moderators of the program and were assessed using phenomenological approaches. Results: Participants found a raised consciousness of physical inactivity throughout the work day. They also found it useful to have a graphical display of physical activity patterns, but found time constraints and lack of managerial support to be the primary barriers/moderators of the program. Those most likely to withdraw from the program experienced technical difficulties with objective monitors and the online website. Conclusions: Findings highlight the value in being involved in a group forum and provide insights into the challenges of supporting such programs within the workplace.

Evaluation of an intervention to promote protective infant feeding practices to prevent childhood obesity : outcomes of the NOURISH RCT at 14 months of age and 6 months post the first of two intervention modules

OBJECTIVE: To evaluate a universal obesity prevention intervention, which commenced at infant age 4-6 months, using outcome data assessed 6-months after completion of the first of two intervention modules and 9 months from baseline. DESIGN: Randomised controlled trial of a community-based early feeding intervention. SUBJECTS AND METHODS: 698 first-time mothers (mean age 30±5 years) with healthy term infants (51% male) aged 4.3±1.0 months at baseline. Mothers and infants were randomly allocated to self-directed access to usual care or to attend two group education modules, each delivered over three months, that provided anticipatory guidance on early feeding practices. Outcome data reported here were assessed at infant age 13.7±1.3 months. Anthropometrics were expressed as z-scores (WHO reference). Rapid weight gain was defined as change in weight-for-age z-score (WAZ) > +0.67. Maternal feeding practices were assessed via self-administered questionnaire. RESULTS: There were no differences according to group allocation on key maternal and infant characteristics. At follow up (n=598 [86%]) the intervention group infants had lower BMIZ (0.42±0.85 vs 0.23±0.93, p=0.009) and infants in the control group were more likely to show rapid weight gain from baseline to follow up (OR=1.5 CI95%1.1-2.1, p=0.014). Mothers in the control group were more likely to report using non- responsive feeding practices that fail to respond to infant satiety cues such as encouraging eating by using food as a reward (15% vs 4%, p=0.001) or using games ( 67% vs 29%, p<0.001). CONCLUSIONS: These results provide early evidence that anticipatory guidance targeting the ‘when, what and how’ of solid feeding can be effective in changing maternal feeding practices and, at least in the short term, reducing anthropometric indicators of childhood obesity risk. Analyses of outcomes at later ages are required to determine if these promising effects can be sustained.

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.

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

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

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.