Approximation of fractional integrals by means of derivatives

We obtain a new decomposition of the Riemann-Liouville operators of fractional integration as a series involving derivatives (of integer order). The new formulas are valid for functions of class C-n, n is an element of N, and allow us to develop suitable numerical approximations with known estimations for the error. The usefulness of the obtained results, in solving fractional integral equations and fractional problems of the calculus of variations, is illustrated

Expansion formulas in terms of integer-order derivatives for the Hadamard fractional integral and derivative

We obtain series expansion formulas for the Hadamard fractional integral and fractional derivative of a smooth function. When considering finite sums only, an upper bound for the error is given. Numerical simulations show the efficiency of the approximation method

Fractional order optimal control problems with free terminal time

We consider fractional order optimal control problems in which the dynamic control system involves integer and fractional order derivatives and the terminal time is free. Necessary conditions for a state/control/terminal- time triplet to be optimal are obtained. Situations with constraints present at the end time are also considered. Under appropriate assumptions, it is shown that the obtained necessary optimality conditions become sufficient. Numer- ical methods to solve the problems are presented, and some computational simulations are discussed in detail

No differences in value-based decision-making due to use of oral contraceptives

Fluctuating ovarian hormones have been shown to affect decision-making processes in women. While emerging evidence suggests effects of endogenous ovarian hormones such as estradiol and progesterone on value-based decision-making in women, the impact of exogenous synthetic hormones, as in most oral contraceptives, is not clear. In a between-subjects design, we assessed measures of value-based decision-making in three groups of women aged 18 to 29 years, during (1) active oral contraceptive intake (N = 22), (2) the early follicular phase of the natural menstrual cycle (N = 20), and (3) the periovulatory phase of the natural menstrual cycle (N = 20). Estradiol, progesterone, testosterone, and sex-hormone binding globulin levels were assessed in all groups via blood samples. We used a test battery which measured different facets of value-based decision-making: delay discounting, risk-aversion, risk-seeking, and loss aversion. While hormonal levels did show the expected patterns for the three groups, there were no differences in value-based decision-making parameters. Consequently, Bayes factors showed conclusive evidence in support of the null hypothesis. We conclude that women on oral contraceptives show no differences in value-based decision-making compared to the early follicular and periovulatory natural menstrual cycle phases. Copyright © 2022 Lewis, Kimmig, Kroemer, Pooseh, Smolka, Sacher and Derntl

impulsivity and the impact of contextual cues on instrumental behavior in alcohol dependence

Alcohol-related cues acquire incentive salience through Pavlovian conditioning and then can markedly affect instrumental behavior of alcohol-dependent patients to promote relapse. However, it is unclear whether similar effects occur with alcohol-unrelated cues. We tested 116 early-abstinent alcohol- dependent patients and 91 healthy controls who completed a delay discounting task to assess choice impulsivity, and a Pavlovian-to-instrumental transfer (PIT) paradigm employing both alcohol-unrelated and alcohol-related stimuli. To modify instrumental choice behavior, we tiled the background of the computer screen either with conditioned stimuli (CS) previously generated by pairing abstract pictures with pictures indicating monetary gains or losses, or with pictures displaying alcohol or water beverages. CS paired to money gains and losses affected instrumental choices differently. This PIT effect was significantly more pronounced in patients compared to controls, and the group difference was mainly driven by highly impulsive patients. The PIT effect was particularly strong in trials in which the instrumental stimulus required inhibition of instrumental response behavior and the background CS was associated to monetary gains. Under that condition, patients performed inappropriate approach behavior, contrary to their previously formed behavioral intention. Surprisingly, the effect of alcohol and water pictures as background stimuli resembled that of aversive and appetitive CS, respectively. These findings suggest that positively valenced background CS can provoke dysfunctional instrumental approach behavior in impulsive alcohol- dependent patients. Consequently, in real life they might be easily seduced by environmental cues to engage in actions thwarting their long-term goals. Such behaviors may include, but are not limited to, approaching alcohol

Dynamic Modelling of Mental Resilience in Young Adults: Protocol for a Longitudinal Observational Study (DynaM-OBS)

Background Stress-related mental disorders are highly prevalent and pose a substantial burden on individuals and society. Improving strategies for the prevention and treatment of mental disorders requires a better understanding of their risk and resilience factors. This multicenter study aims to contribute to this endeavor by investigating psychological resilience in healthy but susceptible young adults over 9 months. Resilience is conceptualized in this study as the maintenance of mental health or quick recovery from mental health perturbations upon exposure to stressors, assessed longitudinally via frequent monitoring of stressors and mental health. Objective This study aims to investigate the factors predicting mental resilience and adaptive processes and mechanisms contributing to mental resilience and to provide a methodological and evidence-based framework for later intervention studies. Methods In a multicenter setting, across 5 research sites, a sample with a total target size of 250 young male and female adults was assessed longitudinally over 9 months. Participants were included if they reported at least 3 past stressful life events and an elevated level of (internalizing) mental health problems but were not presently affected by any mental disorder other than mild depression. At baseline, sociodemographic, psychological, neuropsychological, structural, and functional brain imaging; salivary cortisol and α-amylase levels; and cardiovascular data were acquired. In a 6-month longitudinal phase 1, stressor exposure, mental health problems, and perceived positive appraisal were monitored biweekly in a web-based environment, while ecological momentary assessments and ecological physiological assessments took place once per month for 1 week, using mobile phones and wristbands. In a subsequent 3-month longitudinal phase 2, web-based monitoring was reduced to once a month, and psychological resilience and risk factors were assessed again at the end of the 9-month period. In addition, samples for genetic, epigenetic, and microbiome analyses were collected at baseline and at months 3 and 6. As an approximation of resilience, an individual stressor reactivity score will be calculated. Using regularized regression methods, network modeling, ordinary differential equations, landmarking methods, and neural net–based methods for imputation and dimension reduction, we will identify the predictors and mechanisms of stressor reactivity and thus be able to identify resilience factors and mechanisms that facilitate adaptation to stressors. Results Participant inclusion began in October 2020, and data acquisition was completed in June 2022. A total of 249 participants were assessed at baseline, 209 finished longitudinal phase 1, and 153 finished longitudinal phase 2. Conclusions The Dynamic Modelling of Resilience–Observational Study provides a methodological framework and data set to identify predictors and mechanisms of mental resilience, which are intended to serve as an empirical foundation for future intervention studies. International Registered Report Identifier (IRRID) DERR1-10.2196/3981

A numerical method to solve higher-order fractional differential equations

In this paper, we present a new numerical method to solve fractional differential equations. Given a fractional derivative of arbitrary real order, we present an approximation formula for the fractional operator that involves integer-order derivatives only. With this, we can rewrite FDEs in terms of a classical one and then apply any known technique. With some examples, we show the accuracy of the method

Numerical approximations of fractional derivatives with applications

Two approximations, derived from continuous expansions of Riemann-Liouville fractional derivatives into series involving integer order derivatives, are studied. Using those series, one can formally transform any problem that contains fractional derivatives into a classical problem in which only derivatives of integer order are present. Corresponding approximations provide useful numerical tools to compute fractional derivatives of functions. Application of such approximations to fractional differential equations and fractional problems of the calculus of variations are discussed. Illustrative examples show the advantages and disadvantages of each approximation. © 2012 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society