A multiscale collocation method for fractional differential problems

We introduce a multiscale collocation method to numerically solve differential problems involving both ordinary and fractional derivatives of high order. The proposed method uses multiresolution analyses (MRA) as approximating spaces and takes advantage of a finite difference formula that allows us to express both ordinary and fractional derivatives of the approximating function in a closed form. Thus, the method is easy to implement, accurate and efficient. The convergence and the stability of the multiscale collocation method are proved and some numerical results are shown.We introduce a multiscale collocation method to numerically solve differential problems involving both ordinary and fractional derivatives of high order. The proposed method uses multiresolution analyses (MRA) as approximating spaces and takes advantage of a finite difference formula that allows us to express both ordinary and fractional derivatives of the approximating function in a closed form. Thus, the method is easy to implement, accurate and efficient. The convergence and the stability of the multiscale collocation method are proved and some numerical results are shown

A fractional spline collocation method for the fractional order logistic equation

We construct a collocation method based on the fractional B-splines to solve a nonlinear differential problem that involves fractional derivative, i.e. the fractional order logistic equation. The use of the fractional B-splines allows us to express the fractional derivative of the approximating function in an analytic form. Thus, the fractional collocation method is easy to implement, accurate and efficient. Several numerical tests illustrate the efficiency of the proposed collocation method.We construct a collocation method based on the fractional B-splines to solve a nonlinear differential problem that involves fractional derivative, i.e. the fractional order logistic equation. The use of the fractional B-splines allows us to express the fractional derivative of the approximating function in an analytic form. Thus, the fractional collocation method is easy to implement, accurate and efficient. Several numerical tests illustrate the efficiency of the proposed collocation method

A collocation method based on discrete spline quasi-interpolatory operators for the solution of time fractional differential equations

In many applications, real phenomena are modeled by differential problems having a time fractional derivative that depends on the history of the unknown function. For the numerical solution of time fractional differential equations, we propose a new method that combines spline quasi-interpolatory operators and collocation methods. We show that the method is convergent and reproduces polynomials of suitable degree. The numerical tests demonstrate the validity and applicability of the proposed method when used to solve linear time fractional differential equations

In-situ acoustic-based analysis system for physical and chemical properties of the lower Martian atmosphere

The Environmental Acoustic Reconnaissance and Sounding experiment (EARS), is composed of two parts: the Environmental Acoustic Reconnaissance (EAR) instrument and the Environmental Acoustic Sounding Experiment (EASE). They are distinct, but have the common objective of characterizing the acoustic environment of Mars. The principal goal of the EAR instrument is "listening" to Mars. This could be a most significant experiment if one thinks of everyday life experience where hearing is possibly the most important sense after sight. Not only will this contribute to opening up this important area of planetary exploration, which has been essentially ignored up until now, but will also bring the general public closer in contact with our most proximate planet. EASE is directed at characterizing acoustic propagation parameters, specifically sound velocity and absorption, and will provide information regarding important physical and chemical parameters of the lower Martian atmosphere; in particular, water vapor content, specific heat capacity, heat conductivity and shear viscosity, which will provide specific constraints in determining its composition. This would enable one to gain a deeper understanding of Mars and its analogues on Earth. Furthermore, the knowledge of the physical and chemical parameters of the Martian atmosphere, which influence its circulation, will improve the comprehension of its climate now and in the past, so as to gain insight on the possibility of the past presence of life on Mars. These aspect are considered strategic in the contest of its exploration, as is clearly indicated in NASA's four main objectives on "Long Term Mars Exploration Program" (http://marsweb.jpl.nasa.gov/mer/science).Comment: 16 pages including figure

Shear stress fluctuations in the granular liquid and solid phases

We report on experimentally observed shear stress fluctuations in both granular solid and fluid states, showing that they are non-Gaussian at low shear rates, reflecting the predominance of correlated structures (force chains) in the solidlike phase, which also exhibit finite rigidity to shear. Peaks in the rigidity and the stress distribution's skewness indicate that a change to the force-bearing mechanism occurs at the transition to fluid behaviour, which, it is shown, can be predicted from the behaviour of the stress at lower shear rates. In the fluid state stress is Gaussian distributed, suggesting that the central limit theorem holds. The fibre bundle model with random load sharing effectively reproduces the stress distribution at the yield point and also exhibits the exponential stress distribution anticipated from extant work on stress propagation in granular materials.Comment: 11 pages, 3 figures, latex. Replacement adds journal reference and addresses referee comment

Statistical properties of acoustic emission signals from metal cutting processes

Acoustic Emission (AE) data from single point turning machining are analysed in this paper in order to gain a greater insight of the signal statistical properties for Tool Condition Monitoring (TCM) applications. A statistical analysis of the time series data amplitude and root mean square (RMS) value at various tool wear levels are performed, �nding that ageing features can be revealed in all cases from the observed experimental histograms. In particular, AE data amplitudes are shown to be distributed with a power-law behaviour above a cross-over value. An analytic model for the RMS values probability density function (pdf) is obtained resorting to the Jaynes' maximum entropy principle (MEp); novel technique of constraining the modelling function under few fractional moments, instead of a greater amount of ordinary moments, leads to well-tailored functions for experimental histograms.Comment: 16 pages, 7 figure

Cardinal Filters

Si analizza un particolare filtro basato su funzioni raffinabili e lo si utilizza per l'analisi di segnali

New families of wavelets on the interval.

We will compare two different procedures for constructing wavelet bases and the corresponding dual bases on a bounded interval. Both procedures start from a class of multiresolution analysis which has shown in several applications better performances than B-spline multiresolution analysis

Numerical solution of the fractional oscillation equation by a refinable collocation method

Fractional Calculus is widely used to model real-world phenomena. In fact, the fractional derivative allows one to easily introduce into the model memory effects in time or nonlocality in space. To solve fractional differential problems efficient numerical methods are required. In this paper we solve the fractional oscillation equation by a collocation method based on refinable bases on the semi-infinite interval. We carry out some numerical tests showing the good performance of the method