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

Use and abuse of statistics in tobacco industry-funded research on standardised packaging

In this commentary we consider the validity of tobacco industry-funded research on the effects of standardised packaging in Australia. As the first country to introduce standardised packs, Australia is closely watched, and Philip Morris International has recently funded two studies into the impact of the measure on smoking prevalence. Both of these papers are flawed in conception as well as design but have nonetheless been widely publicised as cautionary tales against standardised pack legislation. Specifically, we focus on the low statistical significance of the analytical methods used and the assumption that standardised packaging should have an immediate large impact on smoking prevalence

Good (and Not So Good) practices in computational methods for fractional calculus

The solution of fractional-order differential problems requires in the majority of cases the use of some computational approach. In general, the numerical treatment of fractional differential equations is much more difficult than in the integer-order case, and very often non-specialist researchers are unaware of the specific difficulties. As a consequence, numerical methods are often applied in an incorrect way or unreliable methods are devised and proposed in the literature. In this paper we try to identify some common pitfalls in the use of numerical methods in fractional calculus, to explain their nature and to list some good practices that should be followed in order to obtain correct results

Fractional compartmental models and multi-term Mittag–Leffler response functions

Systems of fractional differential equations (SFDE) have been increasingly used to represent physical and control system, and have been recently proposed for use in pharmacokinetics (PK) by (J Pharmacokinet Pharmacodyn 36:165–178, 2009) and (J Phamacokinet Pharmacodyn, 2010). We contribute to the development of a theory for the use of SFDE in PK by, first, further clarifying the nature of systems of FDE, and in particular point out the distinction and properties of commensurate versus non-commensurate ones. The second purpose is to show that for both types of systems, relatively simple response functions can be derived which satisfy the requirements to represent single-input/single-output PK experiments. The response functions are composed of sums of single- (for commensurate) or two-parameters (for non-commensurate) Mittag–Leffler functions, and establish a direct correspondence with the familiar sums of exponentials used in PK

Why fractional derivatives with nonsingular kernels should not be used

In recent years, many papers discuss the theory and applications of new fractional-order derivatives that are constructed by replacing the singular kernel of the Caputo or Riemann-Liouville derivative by a non-singular (i.e., bounded) kernel. It will be shown here, through rigorous mathematical reasoning, that these non-singular kernel derivatives suffer from several drawbacks which should forbid their use. They fail to satisfy the fundamental theorem of fractional calculus since they do not admit the existence of a corresponding convolution integral of which the derivative is the left-inverse; and the value of the derivative at the initial time t = 0 is always zero, which imposes an unnatural restriction on the differential equations and models where these derivatives can be used. For the particular cases of the so-called Caputo-Fabrizio and Atangana-Baleanu derivatives, it is shown that when this restriction holds the derivative can be simply expressed in terms of integer derivatives and standard Caputo fractional derivatives, thus demonstrating that these derivatives contain nothing new

Absolute dimensions of the unevolved B-type eclipsing binary GG Orionis

We present photometric observations in B and V as well as spectroscopic observations of the detached, eccentric 6.6-day double-lined eclipsing binary GG Ori, a member of the Orion OB1 association. Absolute dimensions of the components, which are virtually identical, are determined to high accuracy (better than 1% in the masses and better than 2% in the radii) for the purpose of testing various aspects of theoretical modeling. We obtain M(A) = 2.342 +/- 0.016 solar masses and R(A) = 1.852 +/- 0.025 solar radii for the primary, and M(B) = 2.338 +/- 0.017 solar masses and R(B) = 1.830 +/- 0.025 solar radii for the secondary. The effective temperature of both stars is 9950 +/- 200 K, corresponding to a spectral type of B9.5. GG Ori is very close to the ZAMS, and comparison with current stellar evolution models gives ages of 65-82 Myr or 7.7 Myr depending on whether the system is considered to be burning hydrogen on the main sequence or still in the final stages of pre-main sequence contraction. We have detected apsidal motion in the binary at a rate of dw/dt = 0.00061 +/- 0.00025 degrees per cycle, corresponding to an apsidal period of U = 10700 +/- 4500 yr. A substantial fraction of this (approximately 70%) is due to the contribution from General Relativity.Comment: To appear in The Astronomical Journal, December 200

Absolute properties of the binary system BB Pegasi

We present a ground based photometry of the low-temperature contact binary BB Peg. We collected all times of mid-eclipses available in literature and combined them with those obtained in this study. Analyses of the data indicate a period increase of 3.0(1) x 10^{-8} days/yr. This period increase of BB Peg can be interpreted in terms of the mass transfer 2.4 x 10^{-8} Ms yr^{-1} from the less massive to the more massive component. The physical parameters have been determined as Mc = 1.42 Ms, Mh = 0.53 Ms, Rc = 1.29 Rs, Rh = 0.83 Rs, Lc = 1.86 Ls, and Lh = 0.94 Ls through simultaneous solution of light and of the radial velocity curves. The orbital parameters of the third body, that orbits the contact system in an eccentric orbit, were obtained from the period variation analysis. The system is compared to the similar binaries in the Hertzsprung-Russell and Mass-Radius diagram.Comment: 17 pages, 3 figures, accepted for Astronomical Journa

Four-colour photometry of eclipsing binaries. XLI uvby light curves for AD Bootis, HW Canis Majoris, SW Canis Majoris, V636 Centauri, VZ Hydrae, and WZ Ophiuchi

CONTEXT: Accurate mass, radius, and abundance determinations from binaries provide important information on stellar evolution, fundamental to central fields in modern astrophysics and cosmology. AIMS: Within the long-term Copenhagen Binary Project, we aim to obtain high-quality light curves and standard photometry for double-lined detached eclipsing binaries with late A, F, and G type main-sequence components, needed for the determination of accurate absolute dimensions and abundances, and for detailed comparisons with results from recent stellar evolutionary models. METHODS: Between March 1985 and July 2007, we carried out photometric observations of AD Boo, HW CMA, SW CMa, V636 Cen, VZ Hya, and WZ Oph at the Str"omgren Automatic Telescope at ESO, La Silla. RESULTS: We obtained complete uvby light curves, ephemerides, and standard uvby\beta indices for all six systems.For V636 Cen and HW CMa, we present the first modern light curves, whereas for AD Boo, SW CMa, VZ Hya, and WZ Oph, they are both more accurate and more complete than earlier data. Due to a high orbital eccentricity (e = 0.50), combined with a low orbital inclination (i = 84.7), only one eclipse, close to periastron, occurs for HW CMa. For the two other eccentric systems, V636 Cen (e = 0.134) and SW CMa (e = 0.316), apsidal motion has been detected with periods of 5270 +/- 335 and 14900 +/- 3600 years, respectively.Comment: Only change is: Bottom lines (hopefully) not truncated anymore. Accepted for publication in Astonomy & Astrophysic

Electron Trap Dynamics in Polymer Light-Emitting Diodes

Semiconducting polymers are being studied intensively for optoelectronic device applications, including solution-processed light-emitting diodes (PLEDs). Charge traps in polymers limit the charge transport and thus the PLED efficiency. It is firmly established that electron transport is hindered by the presence of the universal electron trap density, whereas hole trap formation governs the long-term degradation of PLEDs. Here, the response of PLEDs to electrical driving and breaks covering the timescale from microseconds to (a few) hours is studied, thus focusing on electron traps. As reference polymer, a phenyl-substituted poly(para-phenylene vinylene) (PPV) copolymer termed super yellow (SY) is used. Three different traps with depths between approximate to 0.4 and 0.7 eV, and a total trap site density of approximate to 2 x 10(17) cm(-3) are identified. Surprisingly, filling of deep traps takes minutes to hours, at odds with the common notion that charge trapping is complete after a few hundred microseconds. The slow trap filling feature for PLEDs is confirmed using poly(2-methoxy-5-(2-ethylhexyloxy)-1,4-phenylene vinylene (MEH-PPV) and poly(3-hexylthiophene) (P3HT) as active materials. This unusual phenomenon is explained with trap deactivation upon detrapping and slow trap reactivation. The results provide useful insight to pinpoint the chemical nature of the universal electron traps in semiconducting polymers