In a recent paper, a new three-parameter class of Abel type equations, so-called AIR, all of whose members can be mapped into Riccati equations, is shown. Most of the Abel equations with solution presented in the literature belong to the AIR class. Three canonical forms were shown to generate this class, according to the roots of a cubic. In this paper, a connection between those canonical forms and the differential equations for the hypergeometric functions $_2{\rm F}_1$, $_1{\rm F}_1$ and $_0{\rm F}_1$ is unveiled. This connection provides a closed form $_p{\rm F}_q$ solution for all Abel equations of the AIR class.

Non-local degenerate parabolic systems arise in three-phase capillary flows in porous media under a pressure control at the inflow- and outflow-boundaries. A mathematical study of such systems is performed for a class of capillarity pressure functions corresponding to triangular capillarity-diffusion tensors. To this end a theory of non-degenerate parabolic approximations is developed: the unique global solvability of initial boundary-value problems is proved.

Free surface potential flows past surfboards and sluice gates are considered. The problem is solved numerically by boundary integral equation methods. In addition weakly nonlinear solutions are presented. It is shown among the six possible types of steady flows, only three exist. The physical relevance of these solutions is discussed in terms of the radiation condition (which requires that there is no energy coming from infinity). In particular, it is shown there are no steady subcritical flows which satisfy the radiation condition. Similarly there are no solutions for the flow under a sluice gate.

The parabolic equation \[u_t + u_{xxxx} + u_{xx} = - (|u_x|^\alpha)_{xx}, \qquad \alpha>1\], is studied under the boundary conditions $u_x|_{\partial\Omega}=u_{xxx}|_{\partial\Omega}=0$ in a bounded real interval $\Omega$. Solutions from two different regularity classes are considered: It is shown that unique mild solutions exist locally in time for any $\alpha>1$ and initial data $u_0 \in W^{1,q}(\Omega)$ ($q>\alpha$), and that they are global if $\alpha \le \frac{5}{3}$. Furthermore, from a semidiscrete approximation scheme global weak solutions are constructed for $\alpha < \frac{10}{3}$, and for suitable transforms of such solutions the existence of a bounded absorbing set in $L^1(\Omega)$ is proved for $\alpha \in [2,\frac{10}{3})$. The article closes with some numerical examples which do not only document the roughening and coarsening phenomena expected for thin film growth, but also illustrate our results about absorbing sets.

Thin-film models for the flow of a low reduced-Reynolds-number poroviscous droplet over a planar substrate are developed. One of the formulations is used to develop a minimal model for active animal cell motion in which the microscopic mechanisms of polymerisation and depolymerisation near the outer cell periphery are modelled by specifying the rate of mass transfer between the phases at the contact-line in terms of the velocity of the latter. An asymptotic analysis in the limit corresponding to strong cell-substrate adhesion is shown to lead to a novel class of multi-valued contact-line laws, a qualitative analysis of which leads in two dimensions to some intriguing behaviour, including (i) periodic contraction and expansion (pulsation), (ii) steady propagation at a contant speed, (iii) an unsteady combination of pulsation and propagation, and (iv) a bistable regime in which both non-motile and motile solutions are admissible, each of them being stable to sufficiently small perturbations, but with sufficiently large perturbations being able to ‘prod’ a stationary cell into motion or halt a moving one; these qualitative predictions are where possible compared with experiment. The contact-line behaviour is likely to be highly sensitive to environmental signals; the formulation may, therefore, provide a useful ‘minimal’ modelling framework for investigation of chemotactic effects at the cell scale. The corresponding extensional flow formulations are also noted.

For superconductors of type II the phenomenon of vortex pinning plays an important role in technological applications. Several models have been proposed for this effect (Kim et al., 1963; Bean, 1964; Bossavit, 1994). In Du et al. (1999) and Prigozhin (1996), some of these models are analyzed. In this work we want to contribute to the analysis for the two-dimensional, rate-independent model proposed in Chapman (2000), which has the special feature that vortex movement and creation is an activated process occurring only when a threshold value of the magnetic field is reached. For analytical studies of related rate-dependent models we refer to Chapman et al. (1996), Schätzle & Styles (1999) and Elliott & Styles (2000).

We study the effect of stochastic volatility on option prices. In the fast mean-reversion model for stochastic volatility of [5], we show that there is a full asymptotic expansion for the option price, centered at the Black-Scholes price. We show how to callibrate the first two terms in the expansion with the implied volatility surface. We show, however, that this price does not converge in a strong sense to Black-Scholes as the mean-reversion rate increases.

The wave equation $(u_{t} + u u_{x})_{x} - u = 0$ is a model for shallow water waves with Coriolis force, sound waves in a bubbly liquid and more generally “is the canonical asymptotic equation for genuinely nonlinear waves that are nondispersive as their wavelength tends to zero” in the words of Hunter [13]. This Ostrovsky–Hunter equation has steadily-translating, spatially periodic solutions which exist only when $c \leq c_{limit}$. The limiting wave (‘parabolic wave’) is exactly given by a piecewise quadratic polynomial in $x$ with a discontinuous slope at the crest. We show that near the limit, the travelling waves (‘paraboloidal waves’) can be approximated by matched asymptotic expansions: the inner solution rounds off the point while the outer solution, valid over most of the spatial domain, is to lowest order just the parabolic wave. In the opposite limit of small amplitude, we derive a Fourier-and-powers-of-amplitude expansion (‘Stokes' series’). We show that this is remarkably accurate even very close to the limiting wave and converges to the limiting wave for unit amplitude. We demonstrate also that the Fourier pseudospectral method gives first order convergence even for the slope-discontinuous parabolic wave.

The aim of this paper is to provide a survey on the recent development in level set methods in inverse problems and optimal design. We give introductions on the general features of such problems involving geometries and on the general framework of the level set method. In subsequent parts we discuss shape sensitivity analysis and its relation to level set methods, various approaches on constructing optimization algorithms based on the level set approach, and special tools needed for the application of level set based optimization methods to ill-posed problems. Furthermore, we provide a review on numerical methods important in this context, and give an overview of applications treated with level set methods. Finally, we provide a discussion of the most challenging and interesting open problems in this field, that might be of interest for scientists who plan to start future research in this field.

We undertake the numerical analysis of a reaction-diffusion system of ‘$\lambda$–$\omega$’ type [26]. Results are presented for a fully-practical piecewise linear finite element method by mimicking results in the continuous case [11]. We establish a priori estimates and error bounds for a semi-discrete and a fully discrete finite element approximation. The theoretical results are illustrated and verified via the numerical solution of periodic plane waves in one space dimension. Experiments in two space dimensions led to‘target patterns’ and spiral wave break-up.

This paper deals with a dynamical inverse problem for a composite beam formed by two connected beams. The vibrations of the composite beam are governed by a differential system where a coupling takes place between longitudinal and bending motions. In this paper, we neglect bending motions and we only deal with the longitudinal motions. These motions are governed by a two-by-two second order system coupled in the lower order terms by the shearing stiffness coefficient, which models the connection between the two beams and which contains direct information on the integrity of the system. We prove that the shearing stiffness coefficient can be reconstructed from the frequency response function of the system evaluated at one end of the beam.

We construct an asymptotic representation for the solution $u(x,t)$ of a singularly-perturbed linear fifth-order evolution equation which accounts for the relevant exponentially-small terms in all regions of the complex $x$ plane. The particular equation that we study is chosen in part to highlight the complexities that arise in high-order examples, resulting in particular from the non-existence of a suitable (steady-state) heteroclinic connection. Key points of this calculation are the identification, location and evolution of the active (in the sense that non-zero, though exponentially-small, terms are switched on across them) Stokes lines, and of the higher-order Stokes lines across which these can be activated or inactivated. In doing so, we need in particular to analyse two ‘levels’ of higher-order Stokes lines and to present the associated mechanisms by which they can themselves be activated or inactivated. By piecing together the information concerning which Stokes lines (both ordinary and higher-order) are active, we are able to deduce systematically which of the competing exponentials that can potentially arise within the asymptotic solution are actually present in each region of the complex plane.

We formulate a number of open problems for time-harmonic inverse electromagnetic scattering theory focusing on uniqueness theorems, the determination of the support of a scattering object and the determination of material parameters

The aim of this paper is to give a detailed introduction to the generalized hyperbolic option pricing model. We discuss European and discrete Asian options, especially arithmetic average options, and compare the results with the classical Black–Scholes model.