Fractional Integration Research Articles

Effects of Fractional Order on Performance of FOSMC for Speed Control of PMSM

Fractional order sliding mode control has been applied for speed control of PMSM. However, in many previous works, the effects of the controller's parameters have not been studied. This paper investigates the effects of fractional order on performance of FOSMC speed control of PMSM. In this work, fractional order, α and β of FOSMS-PID were varied, and their performances were compared. The simulation and experimental results show that variation of order of fractional order integration, α and order of fractional order differentiation, β can affect the performance of the FOSMC-PID controller. Selection of α and β values determines balancing strategies between integral and differentiation portion of the controller. Proper value selection and combination of these variables can further contribute to obtain optimum speed tracking, disturbance rejection and chattering reduction abilities.

Solution Method for Systems of Nonlinear Fractional Differential Equations Using Third Kind Chebyshev Wavelets

Chebyshev Wavelets of the third kind are proposed in this study to solve nonlinear systems of FDEs. The main goal of the method is to convert the nonlinear FDE into a nonlinear system of algebraic equations that can be easily solved using matrix methods. In order to achieve this, we first generate the operational matrices for the fractional integration using third kind Chebyshev Wavelets and block-pulse functions (BPF) for function approximation. Since the obtained operational matrices are sparse, the obtained numerical method is fast and computationally efficient. The original nonlinear FDE is transformed into a system of algebraic equations in a vector-matrix form using the obtained operational matrices. The collocation points are then used to solve the system of algebraic equations. Numerical results for various examples and comparisons are presented.

Evidence of Inflation Using Harmonized Consumer Price Indices in Some Euro Countries: France, Germany, Italy, and Spain, along with the Euro Zone

This paper deals with the analysis of the persistence in the Harmonized Indices of Consumer Prices in France, Germany, Italy, and Spain. The degree of persistence is measured through fractional integration or I (d) techniques, using monthly data from January 2010 to February 2023. We first conducted the analysis with data ending in December 2019, that is, with data prior to the COVID-19 pandemic. Then, we extended the sample, first up to December 2021 and finally to February 2023. Our results show that the findings of our series are highly persistent, with values of the differencing parameter about one or higher than one in the majority of cases. In fact, mean reversion is only observed in the case of Germany with pre-pandemic data. Generally, we observed an increase in the degree of persistence of the series as a consequence of both the COVID-19 pandemic and the Russia–Ukraine war, with the only exception being Spain, where we observe a reduction in the order of integration when including 2022–2023 data.

Convergence analysis of a novel fractional product integration method for solving the second kind weakly singular Volterra integral equations with non-smooth solutions based on Jacobi polynomials

In this paper, we introduce a new fractional basis function based on Lagrange polynomials. We define the new interpolation formula for approximation of the solutions of the second kind weakly singular Volterra integral equations. The product integration method is used for the numerical solution of these equations based on Jacobi polynomials. It is known that the weakly singular Volterra integral equations typically have solutions whose derivatives are unbounded at the left end-point of the interval of integration. We use the suitable transformations to overcome this non-smooth behaviour. An upper error bound of the proposed method is determined and the convergence analysis is discussed. Finally, some numerical examples with non-smooth solutions are prepared to test the efficiency and accuracy of the method.

The home price–income relationship for US states

ABSTRACT There are conflicting theories on whether house prices and income should share a long-run relationship. Empirical work on the topic has yielded mixed results. Most previous studies have investigated whether the house price/income ratio is stationary (short memory) or non-stationary (has a unit root) but have not allowed for the intermediate possibility of long memory or fractional integration. We estimate fractional integration for the house price/income ratio for US states. We find most states exhibit long memory in their ratios. The states with the most long memory tend to be in the high-priced east coast and California. Southern and great plains states, in contrast, tend to exhibit the least persistence in the house price/income metric. In some housing markets – some east coast states, California, Arizona, Florida, and Nevada, home costs can become less and less affordable for local residents, with no tendency to reverse this unaffordability within a reasonable time horizon for potential buyers. In addition to the univariate estimates, multivariate fractional cointegration tests are implemented, and the results support the findings of non-affordability hypothesis.

An inventory model for partial backlogging items with memory effect

Inventory control is considered one of the most widely documented topics in the reality. Fractional derivatives and integration is the part of fractional calculus. Fractional calculus is the generalized part of ordinary calculus. The memory of physical phenomena is a highly concerning topic but it is neglected with describing in terms of integer-order differential equation. To discuss the memory of the inventory model, fractional derivative tools are considered. A fractional derivative at any point gives the previous marginal output and current point output for any current point input. In this model, a shortage is considered, and during the shortage period, demand is partially backlogged. Depending on the low partial backlogging rate and high partial backlogging rate, the result of the memory effect varies on the total average cost and optimal ordering interval. Moreover, in order to show the relationship between fractional models and ordinary classical model, two types of memory indices have been considered: (i) differential memory index and (ii) integral memory index. For a certain memory effect, minimized total average cost is the same for low partial backlogging rate and high partial backlogging rate.

Convergence properties concerning Lebesgue‐p norm of iterative learning control for a class of fractional differential systems

AbstractThis paper investigates the monotonic convergence and speed comparison of first‐ and second‐order proportional‐α‐order‐integral‐derivative‐type ( type) iterative learning control (ILC) schemes for a linear time‐invariant (LTI) system, which is governed by the fractional differential equation with order . By introducing the Lebesgue‐p ( ) norm and utilizing the property of the Mittag‐Leffler function and the boundedness feature of the fractional integration operator, the sufficient condition for the monotonic convergence of the first‐order updating law is strictly analyzed. Therewith, the sufficient condition of the second‐order learning law is established using the same means as the first one. The obtained results objectively reveal the impact of the inherent attributes of system dynamics and the constructive mode of the ILC rule on convergence. Based on the sufficient condition of first/second‐order updating law, the convergence speed of first‐ and second‐order schemes is determined quantitatively. The quantitative result demonstrates that the convergence speed of second‐order law can be faster than the first one when the learning gains and weighting coefficients are properly selected. Finally, the effectiveness of the proposed methods is illustrated by the numerical simulations.

Generalized AB-Fractional Operator Inclusions of Hermite–Hadamard’s Type via Fractional Integration

The aim of this research is to explore fractional integral inequalities that involve interval-valued preinvex functions. Initially, a new set of fractional operators is introduced that uses the extended generalized Mittag-Leffler function Eμ,α,lγ,δ,k,c(τ;p) as a kernel in the interval domain. Additionally, a new form of Atangana–Baleanu operator is defined using the same kernel, which unifies multiple existing integral operators. By varying the parameters in Eμ,α,lγ,δ,k,c(τ;p), several new fractional operators are obtained. This study then utilizes the generalized AB integral operators and the preinvex interval-valued property of functions to establish new Hermite–Hadamard, Pachapatte, and Hermite–Hadamard–Fejer inequalities. The results are supported by numerical examples, graphical illustrations, and special cases.

Assessing the nexus between human capital, green energy, and load capacity factor: Policymaking for achieving sustainable development goals

Climate control is one of the important targets of Sustainable Development Goals (SDGs), which along with other SDGs plays an imperative role in environmental sustainability strategies. Recently, studies on the driving forces of climate change and environmental quality focus on using environmental indicators like CO2 and ecological footprint (EF). Evidently, CO2 emissions have been criticized on the ground that they cannot indicate environmental deterioration to a large extent. EF also focuses only on the consumption of resources and overlooks biocapacity (supply side). The adoption of green energy, which is the fundamental agenda of SDG 7, is the most practical solution toward energy transition and controlling fossil fuel consumption. Also, human-induced environmental challenges require considering factors like education and awareness to cultivate a green society. Against this background, this study employs the load capacity factor (LCF), which considers both supply and demand sides to gauge the environmental quality, and inspects the human capital (HUC), green energy (GEN), and environmental quality nexus using the data from the Association of Southeast Asian Nations (ASEAN) from 1986 to 2018 in the context of the load capacity curve (LCC) hypothesis. Results of the cross-sectionally augmented ARDL, robust to heterogeneity, fractional integration, endogeneity, and cross-sectional dependence (CSD), established that the load capacity curve holds in the context of ASEAN. Thus, achieving a higher income level will support ASEAN to expand LCF and increase environmental quality. HUC is found to expand LCF and promote environmental quality. Similarly, green energy is found to stimulate LCF. However, population density and economic globalization expand environmental deterioration. The paper is concluded by providing detailed policies to enhance environmental quality.

Superlinear solutions of sublinear fractional differential equations and regular variation

We consider a sublinear fractional equation of the order in the interval (1, 2). We give conditions guaranteeing that this equation possesses asymptotically superlinear solutions. We show that all of these solutions are regularly varying and establish precise asymptotic formulae for them. Further we prove non-improvability of the conditions. In addition to the asymptotically superlinear solutions we discuss also other classes of solutions, some of them having no ODE analogy. In the very special case, when the coefficient is asymptotically equivalent to a power function and the order of the equation is 2, we get known results in their full generality. We reveal substantial differences between the integer order and non-integer order case. Among other tools, we utilize the fractional Karamata integration theorem and the fractional generalized L’Hospital rule which are proved in the paper. Several examples illustrating our results but serving also in alternative proofs are given too. We provide also numerical simulations.

Hermite Wavelet Method for Nonlinear Fractional Differential Equations

Nonlinear fractional differential equations (FDEs) constitute the basis for many dynamical systems in various areas of engineering and applied science. Obtaining the numerical solutions to those nonlinear FDEs has quickly gained importance for the purposes of accurate modelling and fast prototyping among many others in recent years. In this study, we use Hermite wavelets to solve nonlinear FDEs. To this end, utilizing Hermite wavelets and block-pulse functions (BPF) for function approximation, we first derive the operational matrices for the fractional integration. The novel contribution provided by this method involves combining the orthogonal Hermite wavelets with their corresponding operational matrices of integrations to obtain sparser conversion matrices. Sparser conversion matrices require less computational load, and also converge rapidly. Using the generated approximate matrices, the original nonlinear FDE is converted into an algebraic equation in vector-matrix form. The obtained algebraic equation is then solved using the collocation points. The proposed method is used to find a number of nonlinear FDE solutions. Numerical results for several resolutions and comparisons are provided to demonstrate the value of the method. The convergence analysis is also provided for the proposed method.

Energy prices in Europe. Evidence of persistence across markets

This paper deals with the behavior of energy price changes and how their shocks exert an impact on suppliers and consumers in different markets. For this purpose, a fractional integration model is used to evaluate the persistence and mean reversion in prices across the major European markets (Germany, France, Italy, UK, Spain). We compare the results with other major players as the US and Japan, to understand, first, if the European behavior is different, and second, if geopolitical shocks that are affecting this market are expected to be permanent. Empirical results show evidence of mean reversion properties in European prices, though some minor differences arise from market to market that apparently, are not associated with the energy generation strategies followed by each country. Thus, it will likely be expected following the current energy shocks the series will recover due to natural market forces, without the need for additional policies.

Evaluation of European Union carbon and energy networks: Evidence from market efficiency

This article investigates dynamic networks of the informational efficiency between European Union carbon and energy markets in a quantile-based framework. The market (in)efficiency degree is gauged by capturing the potentially fractional integration order of the market price series using the Feasible Exact Local Whittle estimator on a rolling basis. Rather than relying on a mean-based method, we employ the causality-in-quantiles approach, through which the cross-market linkage of the inefficiency degree is uncovered in a full-distributional characterization. The dynamic linkage is identified through the perspectives of data in return and volatility, respectively. We find that both carbon and energy markets are inefficient, while the inefficiency degree of the former is slightly higher. The inefficiency degree of the two types of markets is featured by asymmetry and non-linearity over the data distribution, while its cross-market linkage is found to be bi-directional. The spillover degree of the dynamic linkage remains the highest at around the median quantile of the “information giver,” while such the degree weakens at the extreme quantiles. Our findings possess insightful implications for effective operations of the emission trading toward a low carbon transition.

Coronavirus, Vaccination and the Reaction of Consumer Sentiment in The United States: Time Trends and Persistence Analysis

At the beginning of the COVID-19 pandemic, the entire world was waiting for a medical solution (for example, vaccines) in order to return to normality. Sanitary restrictions changed our consumption behaviors and feelings. Therefore, this paper analyzes the stochastic properties of consumer sentiment during the COVID-19 episode and the appearance of vaccines against the virus in December 2020 in the United States of America. This study adds a new dimension to the literature because it is the first research paper that uses advanced methodologies based on fractional integration and fractional cointegration analysis to understand the statistical properties of these time series and their behavior in the long term. The results using fractional integration methodologies exhibit a high degree of persistence, finding behavior of mean reversion during the pandemic episode. Therefore, the shock duration in consumer sentiment will be transitory, recovering to its previous trend in the short run. Focusing on the cointegrating part, we arrive at two main conclusions. First, an increase in total vaccination produces a positive reaction or impact on the behavior of consumers. On the other hand, an increase in new COVID-19 cases negatively affects the behavior of the consumer.

The impact of containment measures and monetary and fiscal responses on US financial markets during the COVID-19 pandemic

This paper analyses the effects of containment measures and monetary and fiscal responses on US financial markets during the Covid-19 pandemic. More specifically, it applies fractional integration methods to analyse their impact on the daily S&P500, the US Treasury Bond Index (USTB), the S&P Green Bond Index (GREEN) and the Dow Jones (DJ) Islamic World Market Index (ISLAM) over the period 1/01/2020–10/03/2021. The results suggest that all four indices are highly persistent and exhibit orders of integration close to 1. A small degree of mean reversion is observed only for the S&P500 under the assumption of white noise errors and USTB with autocorrelated errors; therefore, market efficiency appears to hold in most cases. The mortality rate, surprisingly, seems to have affected stock and bond prices positively with autocorrelated errors. As for the policy responses, both the containment and fiscal measures had a rather limited impact, whilst there were significant announcement effects which lifted markets, especially in the case of monetary announcements. There is also evidence of a significant, positive response to changes in the effective Federal funds rate, which suggests that the financial industry, mainly benefiting from interest rises, plays a dominant role.

US biofuel market persistence and mean reversion properties

This paper investigates the market persistence and mean reversion properties for corn, bioethanol and gasoline prices in the US biofuel industry, evaluating long memory effects with fractional integration techniques from January 1982 to May 2022 with USDA data. Empirical results show evidence of no mean reversion properties for the prices in the three series though some support of it is found when the differences of bioethanol and gasoline are taken with respect to corn. Thus, external shocks in the original series are expected to remain persistent and would require additional policy measures to recover the original trend. Furthermore, the impact of Covid on the time series has been analyzed by comparing the scenarios pre and post pandemic, finding evidence of no major changes in the integration orders in all the series under analysis.

Measuring Persistence in the US Equity Gender Diversity Index

An analysis of the SPDR SSGA Gender Diversity Index ETF using fractional integration or I(d) techniques and daily data from 8 March 2016 to 8 January 2021, reveals that the series is highly persistent with an order of integration smaller than, though very close to 1. However, when estimating d recursively across subsamples, two peaks can be observed. The first peak appears in the sample with 679 observations (ending at 26 December 2018) and the second one occurs in the sample with 974 observations and ending at 28 February 2020, which shows the most significant change in d, moving from values within the I(1) interval to values significantly above 1. The findings indicate that the Covid-19 pandemic has had a significant impact on the persistence of the SPDR SSGA Gender Diversity Index ETF, increasing its magnitude and thus the level of persistence.

Fractional-Order Mittag–Leffler Functions for Solving Multi-dimensional Fractional Pantograph Delay Differential Equations

This manuscript is devoted to presenting an approximation method based upon a new set of fractional functions named fractional-order Mittag–Leffler functions (FM-LFs). This scheme is implemented to approximate the solution of a multi-dimensional fractional pantograph differential equation. To this approach, FM-LFs are introduced. Then, we employ FM-LFs to construct the pseudo-operational matrix of fractional integration and pantograph operational matrix (P-OM). We reduce the considered problems to systems of algebraic equations with the help of the mentioned matrices, and the collocation technique, respectively. Error analysis is proposed. Moreover, several numerical experiments have been considered to confirm the efficiency and applicability of the suggested scheme.

Gaussian RBFNN method for solving FPK and BK equations in stochastic dynamical system with FOPID controller

Solving the Fokker–Planck–Kolmogorov (FPK) equation and the Backward-Kolmogorov (BK) equation is a crucial task to obtain the transient response of stochastic dynamical systems. Fractional order PID (FOPID) is a new efficient controller to change the system response to be the expected one. Therefore, in this paper, the Gaussian Radial Basis Functions Neural Network (RBFNN) is proposed to solve FPK and BK equations, to obtain the transient probability density function and the reliability function for a generalized Van der Pol system under a FOPID controller. The values of the different fractional orders are analyzed to discuss the performance of the FOPID controller. A data collection strategy is adopted to deal with the associated boundary conditions by way of a one-time Monte-Carlo simulation and uniform distribution in our Gaussian RBFNN method. The advantage of this method is that the solution process of FPK and BK equations is converted into solving algebraic equations. Numerical results with regard to the transient system response prove that the Gaussian RBFNN is efficient and accurate in getting the solutions of FPK and BK equations. The order of the fractional integration and the fractional derivative are critical parameters to control the system response. Moreover, we conclude that the fractional order parameters in a FOPID controller can indeed enhance the system’s response to a certain extent and lead to bifurcation.

Growth vs value investing: Persistence and time trend before and after COVID-19

Value investing and growth investing allow economic experts to adopt different investment strategies depending on their chosen specialty; the two investment types have been conditioned by the pandemic, changing the trend of investments and their results. This research aims to analyze the behavior and trends of the different investment strategies before and after the health crisis. We use methodologies based on fractional integration and cointegration to analyze the persistence and trend of the series and their relationship in the long run. We find that the shock is long-lived and causes a change in trend; however, we find no evidence of mean reversion. In addition, we use multivariate wavelet analysis to analyze the correlation between both time series, concluding that a growth-based investment strategy is more successful than a value-based investment strategy. We use neural networks to corroborate our results.