Fractional Integration Research Articles

A numerical study for solving multi-term fractional-order differential equations

In this article, we extended operational matrices using orthonormal Boubaker polynomials of Riemann-Liouville fractional integration and Caputo derivative to find numerical solution of multi-term fractional-order differential equations (FDE). The proposed method is utilized to convert FDE into a system of algebraic equations. The convergence of the method is proved. Examples are given to explain the simplicity, computational time and accuracy of the method.

Strichartz estimates for the Klein–Gordon equation in ℝ3+1

In this paper we prove standard and reversed Strichartz estimates for the Klein--Gordon equation in $\mathbb{R}^{3+1}$. Instead of the Fourier theory, our analysis is based on fundamental solutions of the free equations and fractional integrations. In the final part of this paper, we apply Strichartz estimates in the study of a semilinear Klein--Gordon equation.

CERTAIN GENERALIZED FRACTIONAL CALCULUS FORMULAS AND INTEGRAL TRANSFORMS OF (p, q)-EXTENDED τ -HYPERGEOMETRIC FUNCTION

In this paper, we established certain image formulas of the (p, q)–extended τ hypergeometric function Rτ p ,q (a, b; c; z) by employing Marichev-Sigo-Maeda(M-S-M) fractional integration and differentation Corresponding special cases for the Saigo’s, Riemann-Liouville and Erdelyi-Kober fractional integral and differ- ential operators are also deduced which are earlier obtained by Solanki et al. [23]. Further certain integral transforms of the (p, q)–extended τ hypergeometric function Rτ (a, b; c; z) are established. All the results are represented in terms of the Hadamard product of the (p, q)–extended τ hypergeometric function Rτ (a, b; c; z) and the Fox-Wright function.

THE IMPACT OF COVID-19 ON THE TOURISM SECTOR IN BOSNIA AND HERZEGOVINA

In this study, we examine the degree of persistence in tourism in Bosnia and Herzegovina and the impact that the COVID-19 pandemic has had on it. The series investigated are foreign arrivals and overnight stays for the time period from January 2008 to December 2021. The methodology used is based on fractional integration. The results indicate that the impact of COVID-19 has been strong on the series, removing the significance of the time trend, increasing the level of persistence and reducing significantly the seasonality factor in both arrivals and overnight stay series.

The effects of us covid-19 policy responses on cryptocurrencies, fintech and artificial intelligence stocks: A fractional integration analysis

The effects of us covid-19 policy responses on cryptocurrencies, fintech and artificial intelligence stocks: A fractional integration analysis

Mean reversion and convergence of ecological footprint in the MENA region: evidence from a fractional integration procedure.

This paper deals with the analysis of mean reversion and convergence of the ecological footprint (EF) in the MENA region. Using a long memory model based on fractional integration, we find that the results are very heterogeneous across countries depending on the assumptions made on the error term and the use of original versus logged data. Nevertheless, some conclusions can be obtained. Thus, mean reversion is decisively found in the case of Tunisia, and other countries showing some degree of reversion to the mean include Israel, Syria, Yemen, and Iran. Dealing with the issue of convergence within the MENA countries, similar conclusions hold and only Tunisia reports statistical evidence of convergence for the two types of errors. Additional evidence is found in the case of Syria, Yemen, and Jordan with uncorrelated errors and for Iran with autocorrelation. It is recommended that environmental policies targeted at stabilizing the trends in EF in the MENA region should not be indiscriminately applied in consideration of the heterogeneous nature of the series in the region.

Non-polynomial fractional spline method for solving Fredholm integral equations

A new type of non-polynomial fractional spline function for approximating solutions of Fredholm-integral equations has been presented. For this purpose, we used a new idea of fractional continuity conditions by using the Caputo fractional derivative and the Riemann Liouville fractional integration to generate fractional spline derivatives. Moreover, the convergence analysis is studied with proven theorems. The approach is also well-explained and supported by four computational numerical findings, which show that it is both accurate and simple to apply.

Unveiling the linkages between emerging stock market indices and cryptocurrencies

This paper investigated the relationship between cryptocurrencies and emerging stock market indices using fractional integration and co-integration technique. Particularly, fractional integration is applied to examine stochastic properties of individual assets and fractional cointegration to analyse bivariate connectedness. Our findings unveil the absence of mean reversion in majority cases which indicates high persistence in series. Furthermore, bivariate analysis reveals disconnection between cryptocurrencies prices and stock indices. Surprisingly, a different picture emerges on using conditional volatility instead of prices. Like, conditional volatility-based estimation uncovers evidence of mean reversion in univariate analysis as expected. There is some evidence of cointegration on volatility grounds between cryptocurrencies and emerging stock market indices. Our findings implies that investment decision regarding digital currencies should be taken cautiously. As cryptocurrencies are extremely volatile with high degree of persistence which can make them counterproductive.

Persistence in the passion investment market

Research on the passion investment market is still limited, especially on the issue of persistence. This paper is the first to investigate it using both R/S analysis and fractional integration in the case of 3 fine wine price indices, 10 diamond price indices, 15 art price indices, and 1 stamp price index at the daily, monthly and quarterly frequency. The results can be summarised as follows: wine prices are found to be highly persistent, whilst stamp prices appear to be only weakly persistent, though they can still be characterised as a long-memory process; as for diamond prices, they can be persistent (Diamonds & Gems), anti-persistent (Diamonds Carat indices) or even random (Polished Prices Diamond Index). The dynamic R/S analysis also shows that persistence is time-varying and tends to fluctuate around the average. These findings can be explained by the different degree of liquidity of the assets examined and have implications for both academics investigating market efficiency and practitioners interested in forecasting prices.

Oil extraction and crude oil price behavior in the United States: a fractional integration and cointegration analysis

ABSTRACT This study reviews the relationship between the different types of oil extraction such as horizontal drilling or fracking, or directional drilling, which is a hybrid between vertical and horizontal, on the behavior of West Texas Intermediate crude oil prices. In doing so the study adds a new dimension to the literature on the relationship between oil price and extraction techniques. The analysis is based on statistical properties using the VAR model of Fractional Cointegration, reflecting evidence of cointegration between the series, and indicating a long-term equilibrium relationship. In addition, we apply the wavelet transform to analyze the structural changes in the price of West Texas Intermediate brought about by changes in drilling technology. Our results show that all three forms of extraction and West Texas Intermediate prices reach high levels of correlation, particularly around 2014. We conclude that a decrease in production based on any form of crude oil extraction leads to an increase in the price of crude oil.

Migrations and culture. Essential reflections on wandering human beings.

It is possible that our century and the one just past will be remembered in the future as the centuries of migration. Faced with the extent of migration today, social scientists have posed several questions, and in particular they have examined the causes of migration, the integration of immigrants into host countries, and the development of their cultural identity. The newcomers are almost always poorer than those who settled before them, and have different languages, physical appearance, customs, beliefs, and religious practices. The widespread perception is that of an upheaval of the social order. For some, it is the dawn of a new world, under the banner of métissage (or hybridization) and universal brotherhood; for most, it is the beginning of an invasion. Immigration is always a matter of borders: Who is "us"? Who is "them"? The host society has the power to define, classify, and construct the social category of immigrants. There are many differences within this category and obviously any strategic policy should be able to manage every specificity. Nevertheless, we need starting by focusing the more general ideal type of what is "otherness" and what it can be in the social representations in order to construct new conditions of encounter. In this scenario, urban spaces represent the stages on which the encounter with difference takes place. The space is never neutral and may affect, sometimes significantly, the conditions of that encounter. The physical form of the city is the result of widespread social representations of all phenomena, but it is also able to act on those same social representations by altering the processes that take shape within it. The urban dimension and the redesign of the urban space become increasingly key to shaping and managing social processes aimed at governing the transformation processes in a multi-ethnic sense of the European societies. Urban policy-makers can either wait for spontaneous processes of integration and virtuous composition of differences, or implement actions to manage differences, to prevent potential conflicts and start processes of active inclusion. In order to support the act of wandering within this second pattern of urban policies, moving from simple tolerance to a Habermasian process of dialogical exchange, at least two conditions are necessary: the existence of shared public spaces, and the quality of policies for regulating the use of urban public spaces.

Interpolating Scaling Functions Tau Method for Solving Space–Time Fractional Partial Differential Equations

This paper is devoted to an innovative and efficient technique for solving space–time fractional differential equations (STFPDEs). To this end, we apply the Tau method such that the bases used are interpolating scaling functions (ISFs). The operational metrics for the derivative operator and fractional integration operator are used to introduce the operational matrix for the Caputo fractional derivative. Due to some characteristics of ISFs, such as interpolation, computation costs can be significantly reduced. We investigate the convergence of the technique, and some numerical implementations show that the method is effective for solving such equations.

Research on Satellite Network Traffic Prediction Based on Improved GRU Neural Network.

The current satellite network traffic forecasting methods cannot fully exploit the long correlation between satellite traffic sequences, which leads to large network traffic forecasting errors and low forecasting accuracy. To solve these problems, we propose a satellite network traffic forecasting method with an improved gate recurrent unit (GRU). This method combines the attention mechanism with GRU neural network, fully mines the characteristics of self-similarity and long correlation among traffic data sequences, pays attention to the importance of traffic data and hidden state, learns the time-dependent characteristics of input sequences, and mines the interdependent characteristics of data sequences to improve the prediction accuracy. Particle Swarm Optimization (PSO) algorithm is used to obtain the best network model Hyperparameter and improve the prediction efficiency. Simulation results show that the proposed method has the best fitting effect with real traffic data, and the errors are reduced by 26.9%, 37.2%, and 57.8% compared with the GRU, Support Vector Machine (SVM), and Fractional Autoregressive Integration Moving Average (FARIMA) models, respectively.

The impact of the COVID-19 pandemic on persistence in the European stock markets

This paper analyses the impact of the Covid-19 pandemic on the degree of persistence of European stock markets. Specifically, it uses fractional integration methods to estimate persistence at the daily, weekly and monthly frequencies in the case of ten major European stock market indices; the effects of the pandemic are assessed by comparing the pre-pandemic estimates (over the period 2005–2019) to those from a sample extended until July 2021 which includes the pandemic period. The approach used is more general than the standard one based on the stationarity versus non-stationarity dichotomy and allows for a wider range of dynamic processes. Three different model specifications are considered, and these are estimated under two alternative assumptions for the disturbances (white noise and autocorrelation). The findings indicate that there has not been any significant impact of the Covid-19 pandemic on the degree of persistence of the European stock market indices, though their volatility persistence has decreased.

Some New Integral Inequalities Involving Fractional Operator with Applications to Probability Density Functions and Special Means

Fractional calculus manages the investigation of supposed fractional derivatives and integrations over complex areas and their applications. Fractional calculus is the purported assignment of differentiations and integrations of arbitrary non-integer order. Lately, it has attracted the attention of several mathematicians because of its real-life applications. More importantly, it has turned into a valuable tool for handling elements from perplexing frameworks within different parts of the pure and applied sciences. Integral inequalities, in association with convexity, have a strong relationship with symmetry. The objective of this article is to introduce the notion of operator refined exponential type convexity. Fractional versions of the Hermite–Hadamard type inequality employing generalized R–L fractional operators are established. Additionally, some novel refinements of Hermite–Hadamard type inequalities are also discussed using some established identities. Finally, we present some applications of the probability density function and special means of real numbers.

Unemployment and COVID-19: an analysis of change in persistence

ABSTRACT This paper examines the degree of persistence in monthly unemployment rates for a group of 24 European countries along with the global rate for the Euro area, the European Union, the G7 and the OECD countries. For this purpose, fractionally integrated methods are employed. Using data from January 2010 to November 2020, our results indicate that fractional integration is present in all countries examined, with the orders of integration of the series ranging in the (0, 1) interval. Comparing the data before COVID-19 with those including it, the significant time trend coefficient and the mean reverting property disappear in most cases when COVID-19 data are considered. This implies that governments should consider that, after the pandemic, shocks on the labour market will have permanent effects. Thus, policies should address unemployment accordingly. Our work, however, does not focus on the analysis of nonlinearities, what could provide a more complete understanding of the series behaviour.

3D trajectories and velocities of rainfall drops in a multifractal turbulent wind field

Abstract. Weather radars measure rainfall in altitude, whereas hydro-meteorologists are mainly interested in rainfall at ground level. During their fall, drops are advected by the wind, which affects the location of the measured field. The governing equation of a rain drop's motion relates the acceleration to the forces of gravity and buoyancy along with the drag force. It depends non-linearly on the instantaneous relative velocity between the drop and the local wind, which yields complex behaviour. Here, the drag force is expressed in a standard way with the help of a drag coefficient expressed as a function of the Reynolds number. Corrections accounting for the oblateness of drops greater than 1–2 mm are suggested and validated through a comparison of the retrieved “terminal fall velocity” (i.e. without wind) with commonly used relationships in the literature. An explicit numerical scheme is then implemented to solve this equation for a 3+1D turbulent wind field, and hence analyse the temporal evolution of the velocities and trajectories of rain drops during their fall. It appears that multifractal features of the input wind are simply transferred to the drop velocity with an additional fractional integration whose level depends on the drop size, and a slight time shift. Using an actual high-resolution 3D sonic anemometer and a scale invariant approach to simulate realistic fluctuations of wind in space, trajectories of drops of various sizes falling form 1500 m are studied. For a strong wind event, drops located within a radar gate in altitude during 5 min are spread on the ground over an area of the size of a few kilometres. The spread for drops of a given diameter is found to cover a few radar pixels. Consequences on measurements of hydro-meteorological extremes that are needed to improve the resilience of urban areas are discussed.

Global diabetes burden: analysis of regional differences to improve diabetes care

IntroductionThe current evaluation processes of the burden of diabetes are incomplete and subject to bias. This study aimed to identify regional differences in the diabetes burden on a universal level...

A Comparative Study for Estimate Fractional Parameter of ARFIMA Model

Long memory analysis is one of the most active areas in econometrics and time series where various methods have been introduced to identify and estimate the long memory parameter in partially integrated time series. One of the most common models used to represent time series that have a long memory is the ARFIMA (Auto Regressive Fractional Integration Moving Average Model) which diffs are a fractional number called the fractional parameter. To analyze and determine the ARFIMA model, the fractal parameter must be estimated. There are many methods for fractional parameter estimation. In this research, the estimation methods were divided into indirect methods, where the Hurst parameter is estimated first, and then the fractional integration parameter is estimated from it by a relation between them. As for direct methods, the fractional integration parameter is estimated directly without relying on Hurst's parameter, and most of them are semi parametric methods. In this paper, some of the most common direct methods were used to estimate the fraction modulus namely (Geweke-Porter-Hudak, Smoothed Geweke-Porter-Hudak, Local Whittle, Wavelet and weighted wavelet), using simulation method with different value of (d) and different size of time series. The comparison between the methods was done using the mean squared error (MSE). It turns out that the best methods to estimate the fractional parameter is (Local Whittle).
 The ARFIMA model was generated by a function programmed by the MATLAB statistical program

The persistence of economic sentiment: a trip down memory lane.

Although various indicators of economic sentiment are often assessed in macroeconomic studies, the generating process of economic sentiment itself is still a puzzle. This paper offers pioneer evidence on the persistence of economic sentiment. Applying a battery of fractional integration tests on the European Economic Sentiment Indicator (ESI) of all individual EU member states, we reveal that ESI is dominantly a long-memory process. This finding is robust across several estimators, and it fairly contradicts the conventional wisdom of ESI as a purely transitory macroeconomic shock. Further on, this is true for both core EU economies and new member states, although the later ones exhibit slightly longer memory. Finally, we reveal that the end of the Great Moderation era has increased ESI’s persistence, but the effect is only marginal. As it seems, a series of macroeconomic turbulences recorded after the global financial crisis has not initiated a significant shift in agents’ collective memory and ESI will likely keep its pivotal role in governing business cycles in the future.