We present a coupled system of elliptic equations describing the steady state of the thermoelectrical behaviour of an aluminium electrolytic cell. The thermal model is a free boundary problem which consists of the heat equation with Joule heating as a source. We neglect the Joule heating in the ledge, and allow for temperature-dependent electrical conductivity. We also formulate a numerical approximation using a finite element method. An iterative algorithm and numerical results are presented. The existence of a weak solution is also proved.

We study the interactions between a nematic liquid crystal disclination and the surface of the half-space which bounds it. When strong anchoring conditions are applied on the boundary, the biaxial core of the disclination affects the repulsive force that tends to drive the disclination away from the surface. If we replace the strong boundary conditions with an anchoring potential, the surface-disclination interaction depends on the surface extrapolation length. In particular, the nematic may expel the disclination if the anchoring strength is below a critical value.

We present an $n$-population generalization of the Lighthill–Whitham and Richards traffic flow model. This model is analytically interesting because of several non-standard features. For instance, it leads to non-classical shocks and enjoys an unexpected stability in spite of the presence of umbilic points. Furthermore, while satisfying all the minimal ‘common sense’ requirements, it also allows for a description of phenomena often neglected by other models, such as overtaking.

An eigenvalue problem is considered for a multi-structure consisting of a three-dimensional finite solid connected to an arbitrary smooth elastic thin shell of revolution. Two-sided estimates are obtained for the first six eigenfrequencies of the multi-structure. Explicit asymptotic formulae are given.

We introduce a differential model to study damage accumulation processes in the presence of chemical reactions. The influence of micro-structure leads to a nonlinear parabolic system characterized by the presence of a characteristic length. Here, we first present an analytical description of the qualitative behaviour of solutions which blow-up in finite time. Numerical simulations are given to describe the shape of solutions near the rupture time and the influence of the chemical reagents. As in the non-reactive model, the failure of the material occurs in a region of positive measure, due to the diffusive effects of the micro-structure, although some localization phenomena are observed. Moreover, if we increase the chemical concentration beyond a given threshold, which depends upon the specific conditions of the material, we observe a strong acceleration in the damage process.

We study a transport model for populations whose individuals move according to a velocity jump process and stop moving in areas which provide shelter or food. This model has direct applications in ecology (e.g. homeranges, territoriality, stream ecosystems, travelling waves) or cellular biology (e.g. movement of bacteria or movement of proteins in the cell nucleus). In this paper we consider a general model from a mathematical point of view. This provides general insight into the features of these models, which in turn is useful in the modelling process. We consider a singular perturbation expansion and show that the leading order term of the outer solution satisfies a reaction-advection-diffusion equation. The advective term describes taxis toward homeranges or toward regions of shelter. The reaction terms are given by “effective” birth and death rates. Within this framework, the parameters of the reaction-advection-diffusion model (like mobility, drift, birth or death rates) are directly related to the individual movement behaviour of the species at hand (like velocity, frequency of directional changes, response to spatial in-homogeneities, death, or reproduction). We prove that in a homogeneous environment the diffusion limit approximates the solution of the resting-phase transport model to second order in the perturbation parameter.

We prove the existence of a family of Travelling Wave (TW) solutions for a large class of scalar reaction-diffusion equations with degenerate, nonlinear diffusion coefficients and monostable nonlinear reaction terms. We also investigate stability. Specifically, we show that, as in the linear diffusion case [6], the slowest TW in the family yields the asymptotic rate of the propagation of disturbances from the unstable rest state in these systems. In addition, we give conditions on the reaction term and diffusion coefficient ensuring the existence of interfaces.

We present a multi-parameter non-constant-invariant class of Abel ordinary differential equations with the following remarkable features. This one class is shown to unify, i.e. it contains as particular cases all the integrable classes presented by Abel, Liouville and Appell, as well as all those shown in Kamke's book and various other references. In addition, the class being presented includes other new and fully integrable subclasses, as well as the most general parameterized class of which we know whose members can systematically be mapped into Riccati equations. Finally, many integrable members of this class can be systematically mapped into an integrable member of a different class. We thus find new integrable classes from previously known ones.

The homogenization problem is considered for the equations of two-phase flow in porous media with a periodic or random small-scale structure of inhomogeneities. The capillary relation between saturation and the drop in pressures at microscales accounts for hysteresis and dynamic memory effects. Homogenized equations are derived, and convergence of solutions to the solution of the homogenized problem is proved. Properties of averaged capillary relation are described in the particular case of a two-component porous medium.

We consider the dynamic process of an elastic body in unilateral frictional contact with a rigid foundation. Friction is modelled with the Coulomb law with a coefficient that depends on the slip velocity. To allow for velocity discontinuities we use the elastodynamic (hyperbolic) framework. Nevertheless, this does not lead to a well-posed problem. To remedy this, we perturb the solution of the elastodynamic problem in a thin layer next to the contact boundary. This is a generalisation of an approach previously studied in a one-dimensional case. We establish existence and uniqueness results for the perturbed and regularised problem and provide an interpretation of this perturbation.

For a pattern-forming system with two unbounded spatial directions that is near the onset to instability, we prove the existence of modulated fronts that connect (i) stable hexagons with the unstable trivial pattern, (ii) stable hexagons with unstable roll solutions, (iii) stable hexagons with unstable hexagons, and (iv) stable roll solutions with unstable hexagons. Our approach is based on spatial dynamics, bifurcation theory, and geometric singular perturbation theory.

We present an algorithm for solving first-order ordinary differential equations by systematically determining symmetries of the form $[\xi=F(x),\, \eta=P(x)\,y+Q(x)]$, where $\xi\; \pa/\pa x + \eta\; \pa/\pa y$ is the symmetry generator. To these linear symmetries one can associate an ordinary differential equation class which embraces all first-order equations mappable into separable ones through linear transformations $\{t=f(x),\,u=p(x)\,y+q(x)\}$. This single class includes as members, for instance, 429 of the 552 solvable first-order examples of Kamke's [12] book. Concerning the solution of this class, a restriction on the algorithm being presented exists, only in the case of Riccati equations, for which linear symmetries always exist, but the algorithm will only partially succeed in finding them.

We study the hole-filling problem for the porous medium equation $u_t= \frac{1}{m} \UDelta u^m$ with $m>1$ in two space dimensions. It is well known that it admits a radially symmetric self-similar focusing solution $u=t^{2\beta-1}F(|x|t^{-\beta})$, and we establish that the self-similarity exponent $\beta$ is a monotone function of the parameter $m$. We subsequently use this information to examine in detail the stability of the radial self-similar solution. We show that it is unstable for any $m>1$ against perturbations with 2-fold symmetry. In addition, we prove that as $m$ is varied there are bifurcations from the radial solution to self-similar solutions with $k$-fold symmetry for each $k=3,4,5,\dots.$ These bifurcations are simple and occur at values $m_3>m_4>m_5> \cdots\to1$.

The standard scales that are used to non-dimensionalize the temperature- and time-dependent Ginzburg–Landau (TTDGL) model developed by Schmid [27], eliminate temperature- dependent parameters, and thus do not allow for superconducting phenomena due to variations in temperature. In this study, a set of new scales is presented to non-dimensionalize the TTDGL model so that the resulting dimensionless system depends upon a temperature parameter as well. Moreover, some properties of solutions to TTDGL system as a function of temperature are explored. Numerical experiments illustrating the temperature-dependency of vortex nucleation in type-II superconductors as well as the transition to the Meissner state in type-I superconductors are presented.

We perform an analytical and numerical study of the crossover from the Josephson effect to that of bulk superconducting flow for a one-dimensional superconductor containing a sandwich layer of normal material. A generalized Ginzburg–Landau (GL) model, proposed in Chapman & Gunzburger [1] is used in modelling the whole structure. When the thickness of the normal layer is very small, the introduction of three effective $\delta$-function potentials of specified strength leads to an exact analytical solution of the modified stationary GL equation. The resulting current density-phase offset relation is analyzed numerically. We show that the critical Josephson current density $j_c$ corresponds to a bifurcation of the solutions of the nonlinear boundary value problem for the modified GL-equation. The influence of the second term in the Fourier-decomposition of the supercurrent density-phase relation is also investigated. We derive also a simple analytical formula for the critical Josephson current.