An analytical version of the discrete-ordinates method is used here to solve the classical temperature-jump problem based on the BGK model in rarefied-gas dynamics. In addition to a complete development of the discrete-ordinates method for the application considered, the computational algorithm is implemented to yield very accurate results for the temperature jump and the complete temperature and density distributions in the gas. The algorithm is easy to use, and the developed code runs typically in less than a second on a 400 MHz Pentium-based PC.

In this paper we investigate the uniqueness of solutions of the Ginzburg–Landau system for superconductivity in the regime where the thickness 2a of the film is small. We analyse the equations of first variation with respect to two shooting parameters and obtain estimates on all relevant quantities at the end of the film where the boundary conditions are prescribed. Using these estimates, we prove that the bifurcation curve of symmetric solutions is given by a decreasing function of the order parameter β when the film is thin enough. In addition, we prove that there is no curve of asymmetric solutions branching from the symmetric curve.

We consider a model for non-static groundwater flow where the saturation-pressure relation is extended by a dynamic term. This approach, together with a convective term due to gravity, results in a pseudo-parabolic Burgers type equation. We give a rigorous study of global travelling-wave solutions, with emphasis on the role played by the dynamic term and the appearance of fronts.

Motivated by models from fracture mechanics and from biology, we study the infinite system of differential equations

formula here

where A and F are positive parameters. For fixed A > 0 we show that there are monotone travelling waves for F in an interval Fcrit < F < A, and we are able to give a rigorous upper bound for Fcrit, in contrast to previous work on similar problems. We raise the problem of characterizing those nonlinearities (apparently the more common) for which Fcrit > 0. We show that, for the sine nonlinearity, this is true if A > 2. (Our method yields better estimates than this, but does not include all A > 0.) We also consider the existence and multiplicity of time independent solutions when |F|< Fcrit.

An initial-boundary value problem for nonlinear parabolic equations modelling surfactant diffusions is investigated. The boundary conditions are of nonlinear adsorptive types, and the initial value has a single point jump. We study the well-posedness of the problem, the convergence of a numerical scheme, and the regularity as well as quantitative behaviour of solutions.