A coupled system of two singularly perturbed linear reaction-diffusion two-point boundary value problems is examined. The leading term of each equation is multiplied by a small positive parameter, but these parameters may have different magnitudes. The solutions to the system have boundary layers that overlap and interact. The structure of these layers is analysed, and this leads to the construction of a piecewise-uniform mesh that is a variant of the usual Shishkin mesh. On this mesh central differencing is proved to be almost first-order accurate, uniformly in both small parameters. Supporting numerical results are presented for a test problem.

Blood lactate markers are used as summary measures of the underlying model of an athlete's blood lactate response to increasing work rate. Exercise physiologists use these endurance markers, typically corresponding to a work rate in the region of high curvature in the lactate curve, to predict and compare endurance ability. A short theoretical background of the commonly used markers is given and algorithms provided for their calculation. To date, no free software exists that allows the sports scientist to calculate these markers. In this paper, software is introduced for precisely this purpose that will calculate a variety of lactate markers for an individual athlete, an athlete at different instants (e.g. across a season), and simultaneously for a squad.

Axisymmetric hydromagnetic equilibria are obtained, for the case of strongly anisotropic pressure, by an expansion in the inverse aspect ratio. The calculation is carried out by expanding the inverse Grad-Shafranov equation to second order. The calculation shows that non-circular distortion of the magnetic surfaces is driven to lower order than has previously been found. Many shaping harmonics are found to be present in this order

A system of M(≥ 2) coupled singularly perturbed linear reaction-diffusion equations is considered on the unit square. Under certain hypotheses on the coupling, a maximum principle is established for the differential operator. The relationship between compatibility conditions at the corners of the square and the smoothness of the solution on the closed domain is fully described. A decomposition of the solution of the system is constructed. A finite-difference method for the solution of the system on a Shishkin mesh is presented, and it is proved that the computed solution is second-order accurate (up to a logarithmic factor). Numerical results are given to support this result and to investigate the effect of weaker compatibility assumptions on the data.

The numerical solution of a linear singularly-perturbed reaction-diffusion two-point boundary value problem is considered. The method used is adaptive movement of a fixed number of mesh points by monitor-function equidistribution. A partly heuristic argument based on truncation error analysis leads to several suitable monitor functions, but also shows that the standard arc-length monitor function is unsuitable for this problem. Numerical results are provided to demonstrate the effectiveness of our preferred monitor function.