A finite volume method for solving the two-sided time-space fractional advection-dispersion equation

We present a finite volume method to solve the time-space two-sided fractional advection-dispersion equation on a one-dimensional domain. The spatial discretisation employs fractionally-shifted Grünwald formulas to discretise the Riemann-Liouville fractional derivatives at control volume faces in terms of function values at the nodes. We demonstrate how the finite volume formulation provides a natural, convenient and accurate means of discretising this equation in conservative form, compared to using a conventional finite difference approach. Results of numerical experiments are presented to demonstrate the effectiveness of the approach.

Finite volume methods for simulating anomalous transport

In this thesis a new approach for solving a certain class of anomalous diffusion equations was developed. The theory and algorithms arising from this work will pave the way for more efficient and more accurate solutions of these equations, with applications to science, health and industry. The method of finite volumes was applied to discretise the spatial derivatives, and this was shown to outperform existing methods in several key respects. The stability and convergence of the new method were rigorously established.

Shared genetic basis for migraine and ischemic stroke: A genome-wide analysis of common variants

OBJECTIVE To quantify genetic overlap between migraine and ischemic stroke (IS) with respect to common genetic variation. METHODS We applied 4 different approaches to large-scale meta-analyses of genome-wide data on migraine (23,285 cases and 95,425 controls) and IS (12,389 cases and 62,004 controls). First, we queried known genome-wide significant loci for both disorders, looking for potential overlap of signals. We then analyzed the overall shared genetic load using polygenic scores and estimated the genetic correlation between disease subtypes using data derived from these models. We further interrogated genomic regions of shared risk using analysis of covariance patterns between the 2 phenotypes using cross-phenotype spatial mapping. RESULTS We found substantial genetic overlap between migraine and IS using all 4 approaches. Migraine without aura (MO) showed much stronger overlap with IS and its subtypes than migraine with aura (MA). The strongest overlap existed between MO and large artery stroke (LAS; p = 6.4 x 10(-28) for the LAS polygenic score in MO) and between MO and cardioembolic stroke (CE; p = 2.7 x 10(-20) for the CE score in MO). CONCLUSIONS Our findings indicate shared genetic susceptibility to migraine and IS, with a particularly strong overlap between MO and both LAS and CE pointing towards shared mechanisms. Our observations on MA are consistent with a limited role of common genetic variants in this subtype.