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Abstract. In this paper, we consider spectral approximation of fractional differential equations (FDEs). A main ingredient of our approach is to define a new class of generalized Jacobi functions (GJFs), which is intrinsically related to fractional calculus, and can serve as natural basis functions for properly designed spectral methods for FDEs. We establish spectral approximation results for these GJFs in weighted Sobolev spaces involving fractional derivatives. We construct efficient GJF-Petrov-Galerkin methods for a class of prototypical fractional initial value problems (FIVPs) and fractional boundary value problems (FBVPs) of general order, and show that with an appropriate choice of the parameters in GJFs, the resulted linear systems can be sparse and well-conditioned. Moreover, we derive error estimates with convergence rate only depending on the smoothness of data, so truly spectral accuracy can be attained if the data are smooth enough. The idea and results presented in this paper will be useful to deal with more general FDEs associated with Riemann-Liouville or Caputo fractional derivatives.

Duration, frequency, and temporal distribution of attacks have been objectively estimated in 11 SUNCT patients (3 women and 8 men). The mean age at the time of the study was 69 years (range 52 to 81). The duration of a total of 348 attacks was measured from videotape records, polygraphic tracings, or by stopwatch. The duration of attacks ranged from 5 to 250 seconds, with an unweighted mean of 61 seconds. Both frequency and exact timing of attacks were assessed in four patients who filled in a time chart with the exact onset of 585 consecutive attacks. The majority of attacks occurred during daytime, with a bimodal distribution; ie, morning and afternoon/evening peaks, and only a few attacks were noted at night (ie, 1.2% of the attacks). The unweighted mean frequency of attacks was 28 per day (range 6 to 77). Duration and timing of attacks in SUNCT syndrome may be of help in the differential diagnosis versus other disorders with the same localization, especially first division trigeminal neuralgia.

Abstract. The use of a phase field to describe interfacial phenomena has a long and fruitful tradition. There are two key ingredients to the method: the transformation of Lagrangian description of geometric motions to Eulerian description framework, and the employment of the energetic variational procedure to derive the coupled systems. Several groups have used this theoretical framework to approximate Navier-Stokes systems for two-phase flows. Recently, we have adapted the method to simulate interfacial dynamics in blends of microstructured complex fluids. This review has two objectives. The first is to give a more or less self-contained exposition of the method. We will briefly review the literature, present the governing equations and discuss a numerical scheme based on different numerical schemes, such as spectral methods. The second objective is to elucidate the subtleties of the model that need to be handled properly for certain applications. These points, rarely discussed in the literature, are essential for a realistic representation of the physics and a successful numerical implementation. The advantages and limitations of the method will be illustrated by numerical examples. We hope that this review will encourage readers whose applications may potentially benefit from a similar approach to explore it further.