An Extension of the Well-Posedness Concept for Fractional Differential Equations of Caputo's Type

It is well known that, under standard assumptions, initial value problems for fractional ordinary differential equations involving Caputo-type derivatives are well posed in the sense that a unique solution exists and that this solution continuously depends on the given function, the initial value and the order of the derivative. Here we extend this well-posedness concept to the extent that we also allow the location of the starting point of the differential operator to be changed, and we prove that the solution depends on this parameter in a continuous way too if the usual assumptions are satisfied. Similarly, the solution to the corresponding terminal value problems depends on the location of the starting point and of the terminal point in a continuous way too.Comment: 11 page

First photometric study of the eclipsing binary PS Persei

The CCD photometric observations of the eclipsing binary PS Persei (PS Per) were obtained on two consecutive days in 2009. The 2003 version Wilson-Devinney code was used to analyze the first complete light curves in $V$ and $R$ bands. It is found that PS Per is a short-period Algol-type binary with the less massive component accurately filling its inner critical Roche lobe. The mass ratio of $q=0.518$ and the orbital inclination of $i=89.^{\circ}86$ are obtained. On the other hand, based on all available times of primary light minimum including two new ones, the orbital period has been improved.Comment: 7 pages, 4 figure

The Period Variation of and a Spot Model for the Eclipsing Binary AR Bootis

New CCD photometric observations of the eclipsing system AR Boo were obtained from February 2006 to April 2008. The star's photometric properties are derived from detailed studies of the period variability and of all available light curves. We find that over about 56 years the orbital period of the system has varied due to a combination of an upward parabola and a sinusoid rather than in a monotonic fashion. Mass transfer from the less massive primary to the more massive secondary component is likely responsible for at least a significant part of the secular period change. The cyclical variation with a period of 7.57 yrs and a semi-amplitude of 0.0015 d can be produced either by a light-travel-time effect due to an unseen companion with a scaled mass of $M_3 \sin i_3$=0.081 $M_\odot$ or by a magnetic period modulation in the secondary star. Historical light curves of AR Boo, as well as our own, display season-to-season light variability, which are best modeled by including both a cool spot and a hot one on the secondary star. We think that the spots express magnetic dynamo-related activity and offer limited support for preferring the magnetic interpretation of the 7.57-year cycle over the third-body understanding. Our solutions confirm that AR Boo belongs to the W-subtype contact binary class, consisting of a hotter, less massive primary star with a spectral type of G9 and a companion of spectral type K1.Comment: 30 pages, including 6 figures and 9 tables, accepted for publication in A

Volterra integral equations and fractional calculus: Do neighbouring solutions intersect?

This is the author's PDF version of an article published in Journal of integral equations and applications. The definitive version is available at rmmc.asu.edu/jie/jie.html.This journal article considers the question of whether or not the solutions to two Volterra integral equations which have the same kernel but different forcing terms may intersect at some future time

Pitfalls in fast numerical solvers for fractional differential equations

This is a PDF version of an preprint submitted to Elsevier. The definitive version was published in the Journal of computational and applied mathematics and is available at www.elsevier.comThis preprint discusses the properties of high order methods for the solution of fractional differential equations. A number of fractional multistep methods are are discussed.This article was submitted to the RAE2008 for the University of Chester - Applied Mathematics