Fractional Integral

Various methods for estimating the self-similarity parameter and/or the intensityof long-range dependence in a time series are available. Some are morereliable than others. To discover the ones that work best, we apply the differentmethods to simulated sequences of fractional Gaussian noise and fractionalARIMA(0; d; 0). We also provide here a theoretical justification for the methodof residuals of regression.1 IntroductionTime series

We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating these operators in a way accessible to applied scientists, avoiding unproductive generalities and excessive mathematical rigor. By applying this technique we shall derive the analytical solutions of the most simple linear integral and differential equations of fractional order. We show the fundamental role of the Mittag-Leffler function, whose properties are reported in an ad hoc Appendix. The topics discussed here will be: (a) essentials of Riemann-Liouville fractional calculus with basic formulas of Laplace transforms, (b) Abel type integral equations of first and second kind, (c) relaxation and oscillation type differential equations of fractional order.

We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating these operators in a way accessible to applied ...

We numerically investigate chaotic behavior in autonomous nonlinear models of fractional order. Linear transfer function approximations of the fractional integrator block are calculated for a set of fractional orders in (0,1], based on frequency domain arguments, and the resulting equivalent models are studied. Two chaotic models are considered in this study; an electronic chaotic oscillator, and a mechanical chaotic “jerk”

Abstract: Our aim is three-fold: to point out that the fractional integral actions are coming from Stieltjes actions, to find the roots and the geometry of some Euler-Lagrange or Hamilton ODEs or PDEs, to evidentiate some ideas that include the fractal theory of solids. Section 1 discusses the Euler-Lagrange ODEs associated to single-time Stieltjes actions. Teir dual Hamilton ODEs are

International Journal Pure & Applied Mathematical Sciences. ISSN 0972-9828 Vol.3 No.1 (2006), pp. 47-52 © GBS Publishers & Distributors (India) http://www.gbspublisher.com/ijpams .htm ... Mahmoud M. El-Borai, Khairia El-Said El-Nadi, Osama L. Mostafa and Hamdy M. ...

A discretization of differential and integral operators of integer and non-integer orders is suggested. New type of differences, which are represented by infinite series, is proposed. A characteristic feature of the suggested differences is an implementation of the same algebraic properties that have the operator of differentiation (property of algebraic correspondence). Therefore the suggested differences are considered as an exact discretiza-tion of derivatives. These differences have a property of universality, which means that these operators do not depend on the form of differential equations and the parameters of these equations. The suggested differences operators allows us to have difference equations whose solutions are equal to the solutions of corresponding differential equations. The exact discretization of the derivatives of integer orders is given by the suggested differences of the same integer orders. Similarly, the exact discretization of the Riesz derivatives and integrals of integer and non-integer order is given by the proposed fractional differences of the same order.

In transportation analyses, autoregressive integrated moving average (ARIMA) and generalized autoregressive conditional heteroskedasticity (GARCH) models have been widely used mainly because of their well established theoretical foundation and ease of application. However, they lack the ability to capture long memory properties and do not jointly treat the mean and variance (variability) of a time-series. We employ fractionally integrated dual memory models and compare results to classical time-series models in a traffic engineering context. Results indicate that dual memory models offer better representation of the original time-series than classical models; further, forcing the differentiation parameter of ARIMA model to equal 1 leads to over-inflated moving average terms and, consequently, to questionable models with artificial correlation structures.

In this paper, fractional integrating dynamics in the return and the volatility series of stock market indices are investigated. The investigation is conducted using wavelet ordinary least squares, wavelet weighted least squares and the approximate Maximum Likelihood estimator. It is shown that the long memory property in stock returns is approximately associated with emerging markets rather than developed ones while strong evidence of long range dependence is found for all volatility series. The relevance of the wavelet-based estimators, especially, the approximate Maximum Likelihood and the weighted least squares techniques is proved in terms of stability and estimation accuracy.