Spline collocation for strongly elliptic equations on the torus

Summary We prove convergence and error estimates in Sobolev spaces for the collocation method with tensor product splines for strongly elliptic pseudodifferential equations on the torus. Examples of applications include elliptic partial differential equations with periodic boundary conditions but also the classical boundary integral operators of potential theory on torus-shaped domains in three or more dimensions. For odd-degree splines, we prove convergence of nodal collocation for any strongly elliptic operator. For even-degree splines and midpoint collocation, we find an additional condition for the convergence which is satisfied for the classical boundary integral operators. Our analysis is a generalization to higher dimensions of the corresponding analysis of Arnold and Wendland [4].     

Simulation model for shapiro time delay and light deflection experiments

The time delay experiment proposed by I.I. Shapiro in 1964 and conducted in the seventies was the most precise experiment of general relativity until that time. Further experimentation has improved the accuracy level of both the time delay and the light deflection experiments. A simulation model is proposed that involves only a simple mass and time transformation factor involving velocity of light. The light deflection and the time delay experiments are numerically simulated using this model that does not use the general relativistic equations. The computed values presented in this paper compare well with recent levels of accuracy of their respective experimental results.     

A second-order accurate numerical method for a fractional wave equation

We study a generalized Crank–Nicolson scheme for the time discretization of a fractional wave equation, in combination with a space discretization by linear finite elements. The scheme uses a non-uniform grid in time to compensate for the singular behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assuming that the spatial domain is convex or smooth, we show that the error is of order k ² + h ², where k and h are the parameters for the time and space meshes, respectively.     

Monitoring photoproduct formation and photobleaching by fluorescence spectroscopy has the potential to improve PDT dosimetry with a verteporfin-like photosensitizer

In current clinical practice, photodynamic therapy (PDT) is carried out with prescribed drug doses and light doses as well as fixed drug-light intervals and illumination fluence rates. This approach can result in undesirable treatment outcomes of either overtreatment or undertreatment because of biological variations between different lesions and patients. In this study, we explore the possibility of improving PDT dosimetry by monitoring drug photobleaching and photoproduct formation. The study involved 60 mice receiving the same drug dose of a novel verteporfin-like photosensitizer, QLT0074, at 0.3 mg/kg body weight, followed by different light doses of 5, 10, 20, 30, 40 or 50 J/cm2 at 686 nm and a fluence rate of 70 mW/cm2. Photobleaching and photoproduct formation were measured simultaneously, using fluorescence spectroscopy. A ratio technique for data processing was introduced to reliably detect the photoproduct formed by PDT on mouse skin in vivo. The study showed that the QLT0074 photoproduct is stable and can be reliably quantified. Three new parameters, photoproduct score (PPS), photobleaching score (PBS) and percentage photobleaching score (PBS%), were introduced and tested together with the conventional dosimetry parameter, light dose, for performance on predicting PDT-induced outcome, skin necrosis. The statistical analysis of experimental results was performed with an ordinal logistic regression model. We demonstrated that both PPS and PBS improved the prediction of skin necrosis dramatically compared to light dose. PPS was identified as the best single parameter for predicting the PDT outcome.