Singularities in Inflationary Cosmological Models

Due to the accelerated expansion of the universe, the possibilities for the formation of singularities has changed from the classical Big Bang and Big Crunch singularities to include a number of new scenarios. In recent papers it has been shown that such singularities may appear in inflationary cosmological models with a fractional power scalar field potential. In this paper we enlarge the analysis of singularities in scalar field cosmological models by the use of generalised power expansions of their Hubble scalars and their scalar fields in order to describe all possible models leading to a singularity, finding other possible cases. Unless a negative scalar field potential is considered, all singularities are weak and of type IV.

Series Representations for Uncertain Fractional IVPs in the Fuzzy Conformable Fractional Sense

Fuzzy differential equations provide a crucial tool for modeling numerous phenomena and uncertainties that potentially arise in various applications across physics, applied sciences and engineering. Reliable and effective analytical methods are necessary to obtain the required solutions, as it is very difficult to obtain accurate solutions for certain fuzzy differential equations. In this paper, certain fuzzy approximate solutions are constructed and analyzed by means of a residual power series (RPS) technique involving some class of fuzzy fractional differential equations. The considered methodology for finding the fuzzy solutions relies on converting the target equations into two fractional crisp systems in terms of ρ-cut representations. The residual power series therefore gives solutions for the converted systems by combining fractional residual functions and fractional Taylor expansions to obtain values of the coefficients of the fractional power series. To validate the efficiency and the applicability of our proposed approach we derive solutions of the fuzzy fractional initial value problem by testing two attractive applications. The compatibility of the behavior of the solutions is determined via some graphical and numerical analysis of the proposed results. Moreover, the comparative results point out that the proposed method is more accurate compared to the other existing methods. Finally, the results attained in this article emphasize that the residual power series technique is easy, efficient, and fast for predicting solutions of the uncertain models arising in real physical phenomena.

Asymptotic stabilization of underactuated surface vehicles with actuator saturation

This paper investigates the problem of global asymptotic stabilization of underactuated surface vessels (USVs) with input saturation. A novel input transformation is presented, so that the USV system can be transformed to a cascade structure. For the obtained system, the improved fractional power control laws are proposed to ensure input signals do not exceed actuator constraints and enhance convergence rates. Finally, stabilization and parameter optimization algorithm of USVs are proposed. Simulations are given to demonstrate the effectiveness of the presented method.

Comparative analysis of suitability of fractional derivatives in modelling the practical capacitor

Purpose The purpose of this paper is to compare the suitability of fractional derivatives in the modelling of practical capacitors. Such suitability refers to ability to provide the analytical capacitance function that matches the experimental ones of each fractional derivative. Design/methodology/approach The analytical capacitance functions based on various fractional derivatives of both local and nonlocal types including the author’s have been derived. The derived capacitance functions have been simulated and compared with the experimental ones of aluminium electrolytic and electrical double layer capacitors (EDLCs). Findings This paper has found that any local fractional derivative with fractional power law-based relationship with the conventional one is suitable for modelling the aluminium electrolytic capacitor (AEC) by incorporating with the conventional capacitance definition. On the other hand, the author’s nonlocal fractional derivatives have been found to be more suitable than the others for modelling the EDLC by incorporating with the revisited definition of capacitance. Originality/value The proposed comparative analysis has been originally presented in this work. The criterion for local fractional derivative, to be suitable for modelling the AEC, has been found. The nonlocal fractional operators which are most suitable for modelling the EDLC have been derived where the unsuitable one has been pointed out.

FRACTIONAL POWER SERIES APPROACH FOR THE SOLUTION OF FRACTIONAL-ORDER INTEGRO-DIFFERENTIAL EQUATIONS

Fractional differential and integral equations are focus of the researchers owing to their tremendous applications in different field of science and technology, such as physics, chemistry, mathematical biology, dynamical system and engineering. In this work, a power series approach called Residual Power Series Method (RPSM) is applied for the solution of fractional (non-integer) order integro-differential equations (FIDEs). The Caputo sense is used for calculating fractional derivatives. Comparison of the obtained solution is made with the Trigonometric Transform Method (TTM) and Optimal Homotopy Asymptotic Method (OHAM). There is no restrictive condition on the proposed solution. The presented technique is simple in applicability and easily computable.

Rhythmic Change of Cortical Hemodynamic Signals Associated with Ongoing Nociception in Awake and Anesthetized Individuals: An Exploratory Functional Near Infrared Spectroscopy Study

Background Patients undergoing surgical procedures are vulnerable to repetitive evoked or ongoing nociceptive barrage. Using functional near infrared spectroscopy, the authors aimed to evaluate the cortical hemodynamic signal power changes during ongoing nociception in healthy awake volunteers and in surgical patients under general anesthesia. The authors hypothesized that ongoing nociception to heat or surgical trauma would induce reductions in the power of cortical low-frequency hemodynamic oscillations in a similar manner as previously reported using functional magnetic resonance imaging for ongoing pain. Methods Cortical hemodynamic signals during noxious stimuli from the fontopolar cortex were evaluated in two groups: group 1, a healthy/conscious group (n = 15, all males) where ongoing noxious and innocuous heat stimulus was induced by a contact thermode to the dorsum of left hand; and group 2, a patient/unconscious group (n = 13, 3 males) receiving general anesthesia undergoing knee surgery. The fractional power of low-frequency hemodynamic signals was compared across stimulation conditions in the healthy awake group, and between patients who received standard anesthesia and those who received standard anesthesia with additional regional nerve block. Results A reduction of the total fractional power in both groups—specifically, a decrease in the slow-5 frequency band (0.01 to 0.027 Hz) of oxygenated hemoglobin concentration changes over the frontopolar cortex—was observed during ongoing noxious stimuli in the healthy awake group (paired t test, P = 0.017; effect size, 0.70), and during invasive procedures in the surgery group (paired t test, P = 0.003; effect size, 2.16). The reduction was partially reversed in patients who received a regional nerve block that likely diminished afferent nociceptive activity (two-sample t test, P = 0.002; effect size, 2.34). Conclusions These results suggest common power changes in slow-wave cortical hemodynamic oscillations during ongoing nociceptive processing in conscious and unconscious states. The observed signal may potentially promote future development of a surrogate signal to assess ongoing nociception under general anesthesia. Editor’s Perspective What We Already Know about This Topic What This Article Tells Us That Is New