8,185 research outputs found
Fourier spectral methods for fractional-in-space reaction-diffusion equations
Fractional differential equations are becoming increasingly used as a powerful modelling approach for understanding the many aspects of nonlocality and spatial heterogeneity. However, the numerical approximation of these models is computationally demanding and imposes a number of computational constraints. In this paper, we introduce Fourier spectral methods as an attractive and easy-to-code alternative for the integration of fractional-in-space reactiondiffusion equations. The main advantages of the proposed schemes is that they yield a fully diagonal representation of the fractional operator, with increased accuracy and efficiency when compared to low-order counterparts, and a completely straightforward extension to two and three spatial dimensions. Our approach is show-cased by solving several problems of practical interest, including the fractional Allen–Cahn, FitzHugh–Nagumo and Gray–Scott models,together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator
Fractional order differentiation by integration with Jacobi polynomials
The differentiation by integration method with Jacobi polynomials was
originally introduced by Mboup, Join and Fliess. This paper generalizes this
method from the integer order to the fractional order for estimating the
fractional order derivatives of noisy signals. The proposed fractional order
differentiator is deduced from the Jacobi orthogonal polynomial filter and the
Riemann-Liouville fractional order derivative definition. Exact and simple
formula for this differentiator is given where an integral formula involving
Jacobi polynomials and the noisy signal is used without complex mathematical
deduction. Hence, it can be used both for continuous-time and discrete-time
models. The comparison between our differentiator and the recently introduced
digital fractional order Savitzky-Golay differentiator is given in numerical
simulations so as to show its accuracy and robustness with respect to
corrupting noises
Anomalous volatility scaling in high frequency financial data
Volatility of intra-day stock market indices computed at various time
horizons exhibits a scaling behaviour that differs from what would be expected
from fractional Brownian motion (fBm). We investigate this anomalous scaling by
using empirical mode decomposition (EMD), a method which separates time series
into a set of cyclical components at different time-scales. By applying the EMD
to fBm, we retrieve a scaling law that relates the variance of the components
to a power law of the oscillating period. In contrast, when analysing 22
different stock market indices, we observe deviations from the fBm and Brownian
motion scaling behaviour. We discuss and quantify these deviations, associating
them to the characteristics of financial markets, with larger deviations
corresponding to less developed markets.Comment: 25 pages, 11 figure, 5 table
Fractional integration and cointegration in stock prices and exchange rates
This paper examines the relationships between the CAC40 index, the Dow Jones index and the Euro/USD exchange rate using daily data over the period 1999-2008. We find that these variables are I(1) nonstationary series, but they are fractionally cointegrated: equilibrium errors exhibit slow mean reversion, responding slowly to shocks. Therefore, with regard to the recent empirical cointegration literature, taking into account fractional cointegration techniques appears as a promising way to study the long-run relationships between stock prices and exchange rates.fractional cointegration, long memory, stock prices, exchange rates
- …