一类分数阶微积分的中值定理及应用

Riemann-Liouville分数阶微积分算子是一类带有一个函数的分数阶微积分算子的特殊情形，以Riemann-Liouville分数阶微积分算子的积分中值定理和微分中值定理为基础，我们得到了一类带有一个函数的分数阶微积分算子的积分中值定理和微分中值定理，并给出其在计算方面的一些应用。     

Robust adaptive passivity-based PIλD control

In this article, we develop proportional, fractional-integral, and derivative () controllers for the regulation and tracking problems of nonlinear systems. The analytic results are obtained by extending the passivity-based approach to include fractional operators. Robustness under parametric uncertainty is dealt with by a combination with an adaptive scheme. It is also shown their robustness under additive noise and their robustness under uncertainty in the derivation order. The advantages in the controlled system performance and in the control energy consumption in comparison to classic PI and proportional integral derivative controllers are illustrated for the quadratic boost converter and a benchmark system in the literature.     

Eight-node conforming straight-side quadrilateral element with high-order completeness (QH8-C1)

A new eight-node conforming quadrilateral element with high-order completeness, denoted as QH8-C1, is proposed in this article. First, expressions for the interpolation displacement function satisfying the requirements for high-order completeness in the global coordinate system are constructed. Second, the displacement function expression in global coordinates is transformed into isoparametric coordinates, and the relationships between the two series of coefficients for the two kinds of displacement function expressions are found. Third, the displacement function expression is modified to satisfy the requirements of nodal freedom and interelement boundary continuity. The key to the new element construction is the derivation of the linear relationship expressions among 12 coefficients of element displacement interpolation polynomials in the global and isoparametric coordinate systems. As a result, the relationship between quadratic completeness and interelement continuity is explicitly given, and a proof of the completeness and the continuity was conducted to theoretically guarantee the validity of the derivation results. Furthermore, in order to verify the correctness of the theoretical work, nine numerical examples were performed. The computation results from these examples demonstrate that QH8-C1 exhibited excellent performance, including high simulation accuracy, fast convergence, insensitivity to mesh distortion, and monotonic convergence.     

Models for circular data from time series spectra

Circular data are those for which the natural support is the unit circle and its toroidal extensions. Numerous constructions have been proposed which can be used to generate models for such data. We propose a new, very general, one based on the normalization of the spectra of complex-valued stationary processes. As illustrations of the new construction's application, we study models for univariate circular data obtained from the spectra of autoregressive moving average models and relate them to existing models in the literature. We also propose and investigate multivariate circular models obtained from the high-order spectra of stationary stochastic processes generated using linear filtering with an autoregressive moving average response function. A new family of distributions for a Markov process on the circle is also introduced. Results for asymptotically optimal inference for dependent observations on the circle are presented which provide a new paradigm for inference with circular models. The application of one of the new families of spectra-generated models is illustrated in an analysis of wind direction data.     

Novel organic dyes containing N-bridged oligothiophene coplanar cores for dye-sensitized solar cells

Three novel organic dyes adopting fully-fused coplanar heteroarene as the donor moieties end-capped with two cyanoacrylic acids as acceptors and anchoring groups have been synthesized, characterized, and used as the sensitizers for dye-sensitized solar cells (DSSCs). The photophysical and electrochemical properties of the novel dyes and the characteristics of the DSSCs based on the novel organic dyes were investigated. The incorporation of the coplanar cores with electron-donating N-bridges are beneficial for the better intramolecular charge transfer (ICT), giving these new dyes good light-harvesting capability. The LUMO energy levels of these coplanar heteroacene-based dyes are sufficiently high for the efficient electron injection to TiO₂ upon photo-excitation, while the suitable HOMOs allow the regeneration of oxidized dyes with the electrolyte redox (I⁻/I₃⁻). The structural features of the coplanar cores (penta vs. hexa heteroarene) as well as the alkyl substitutions play crucial roles in governing the physical properties and device performance. Among these three novel organic sensitizers, the EHTt dye composed of a fully fused hexa-arene core and less bulky N-alkyl groups caused the DSSC to show the best photovoltaic performance with an open-circuit voltage (V_OC) of 0.58 V, a short-circuit photocurrent density (J_SC) of 13.72 mA/cm², and a fill factor (FF) of 0.69, yielding an overall power conversion efficiency (PCE) of 5.52% under AM 1.5G solar irradiation.     

The Investigation of the Fractional-View Dynamics of Helmholtz Equations Within Caputo Operator

It is eminent that partial differential equations are extensively meaningful in physics, mathematics and engineering. Natural phenomena are formulated with partial differential equations and are solved analytically or numerically to interrogate the system’s dynamical behavior. In the present research, mathematical modeling is extended and the modeling solutions Helmholtz equations are discussed in the fractional view of derivatives. First, the Helmholtz equations are presented in Caputo’s fractional derivative. Then Natural transformation, along with the decomposition method, is used to attain the series form solutions of the suggested problems. For justification of the proposed technique, it is applied to several numerical examples. The graphical representation of the solutions shows that the suggested technique is an accurate and effective technique with a high convergence rate than other methods. The less calculation and higher rate of convergence have confirmed the present technique’s reliability and applicability to solve partial differential equations and their systems in a fractional framework.     

Fractional fixed-structure H∞ controller design using Augmented Lagrangian Particle Swarm Optimization with Fractional Order Velocity

The optimization of a fixed-structure controller is in general computationally intractable, posing a challenge to control system engineers. This paper introduces a novel technique for designing such control techniques by formulating it as a nonlinear non-convex constrained problem. A novel PSO algorithm, namely the Augmented Lagrangian Particle Swarm Optimization with Fractional Order Velocity (ALPSOFV), is proposed for improving the convergence rate. The structure of the controller is selectable and, therefore, the Fractional Order Proportional Integral Derivative (FOPID) algorithm is chosen in the scope of the study. The results for three test examples show the good performance of the ALPSOFV.     

Determination of supercapacitor parameters based on fractional differential equations

In this paper, we estimate the parameter values of a fractional-order model of supercapacitors involving fractional derivatives of Liouville-Caputo, Caputo-Fabrizio, and Atangana-Baleanu and fractional conformable derivative in the Liouville-Caputo sense. We present the exact solution of the considered model using the properties of the Laplace transform operator together with the convolution theorem. They developed numerical simulations using each one of the fractional derivatives; the results were compared graphically with experimental data obtained from different supercapacitors using standard laboratory equipment. The nonlocal parameters involved in the equivalent electrical circuit for the supercapacitor model are recalculated for each fractional derivative using a particle swarm optimization algorithm for generating optimal solutions.     

Stumped nature hyperjerk system with fractional order and exponential nonlinearity: Analog simulation,bifurcation analysis and cryptographic applications

This article presents the construction and consumption of hyperjerk system having chaotic nature with the commensurate ordered fractional derivative on rapidly digitalized emerging technologies. The system analyzed analytically using Lipschitz condition and numerically with the help of predictor corrector approaches. Originality of constructed system is confirmed using log error plots which gives satisfactory small values on finite time scale. The dynamical performances of the complex system are termed in multiple phase planes by the mean of their significant nature. Stability and bifurcation analysis around the fractional derivative is also studies for the various parameter of system to check the visibility of chaotic solution. The system is also designed in analog circuit simulator with the aid of operational amplifier, anti-parallel semiconductor diodes for validation of the system. With the help of random number generators (RNGs), binary array generated from the hyperjerk system and plugged in to the NIST 800–22 test suite to measure the randomness. The array having high randomness is used as strong cypher key for the cryptographic execution straightforwardly in both direction encryption and decryption.