We present an adaptation of the Monge–Ampère (MA) lattice basis reduction scheme to the MA equation with second boundary value condition, provided the target is a convex set. This yields a fast adaptive method to numerically solve the optimal transport (OT) problem between two absolutely continuous measures, the second of which has convex support. The proposed numerical method actually captures a specific Brenier solution which is minimal in some sense. We prove the convergence of the method as the grid step size vanishes and show with numerical experiments that it is able to reproduce subtle properties of the OT problem.

The modeling of the motion of a contact line, the triple point at which solid, liquid and air meet, is a major outstanding problem in the fluid mechanics of thin films [2, 9]. In this paper, we compare two well-known models in the specific context of Marangoni driven films. The precursor model replaces the contact line by a sharp transition between the bulk fluid and a thin layer of fluid, effectively pre-wetting the solid; the Navier slip model replaces the usual no-slip boundary condition by a singular slip condition that is effective only very near the contact line. We restrict attention to traveling wave solutions of the thin film PDE for a film driven up an inclined planar solid surface by a thermally induced surface tension gradient. This involves analyzing third order ODE that depend on several parameters. The two models considered here have subtle differences in their description, requiring a careful treatment when comparing traveling waves and effective contact angles. Numerical results exhibit broad agreement between the two models, but the closest comparison can be done only for a rather restricted range of parameters. The driven film context gives contact angle results quite different from the case of a film moving under the action of gravity alone. The numerical technique for exploring phase portraits for the third order ODE is also used to tabulate the kinetic relation and nucleation condition, information that can be used with the underlying hyperbolic conservation law to explain the rich combination of wave structures observed in simulations of the PDE and in experiments [3, 15].

The subcritical flow of a stream over a bottom obstruction or depression is considered with particular interest in obtaining solutions with no downstream waves. In the linearised problem this can always be achieved by superposition of multiple obstructions, but it is not clear whether this is possible in a full nonlinear problem. Solutions computed here indicate that there is an effective nonlinear superposition principle at work as no special shape modifications were required to obtain wave-cancelling solutions. Waveless solutions corresponding to one or more trapped waves are computed at a range of different Froude numbers and are shown to provide a rather elaborate mosaic of solution curves in parameter space when both negative and positive obstruction heights are included.

We study an electrical conduction problem in biological tissues in the radiofrequency range, which is governed by an elliptic equation with memory. We prove the time exponential asymptotic stability of the solution. We provide in this way both a theoretical justification to the complex elliptic problem currently used in electrical impedance tomography and additional information on the structure of the complex coefficients appearing in the elliptic equation. Our approach relies on the fact that the elliptic equation with memory is the homogenisation limit of a sequence of problems for which we prove suitable uniform estimates.

We study the existence of grain boundaries in the Swift–Hohenberg equation. The analysis relies on a spatial dynamics formulation of the existence problem and a centre-manifold reduction. In this setting, the grain boundaries are found as heteroclinic orbits of a reduced system of ordinary differential equations in normal form. We show persistence of the leading-order approximation using transversality induced by wavenumber selection.

In this article, we use Mellin transforms to derive alternative results for option pricing and implied volatility estimation when the underlying asset price is governed by jump-diffusion dynamics. The current well known results are restrictive since the jump is assumed to follow a predetermined distribution (e.g., lognormal or double exponential). However, the results we present are general since we do not specify a particular jump-diffusion model within the derivations. In particular, we construct and derive an exact solution to the option pricing problem in a general jump-diffusion framework via Mellin transforms. This approach of Mellin transforms is further extended to derive a Dupire-like partial integro-differential equation, which ultimately yields an implied volatility estimator for assets subjected to instantaneous jumps in the price. Numerical simulations are provided to show the accuracy of the estimator.

We investigate the effect of mass transfer on the evolution of a thin, two-dimensional, partially wetting drop. While the effects of viscous dissipation, capillarity, slip and uniform mass transfer are taken into account, other effects, such as gravity, surface tension gradients, vapour transport and heat transport, are neglected in favour of mathematical tractability. Our focus is on a matched-asymptotic analysis in the small-slip limit, which reveals that the leading-order outer formulation and contact-line law depend delicately on both the sign and the size of the mass transfer flux. This leads, in particular, to novel generalisations of Tanner's law. We analyse the resulting evolution of the drop on the timescale of mass transfer and validate the leading-order predictions by comparison with preliminary numerical simulations. Finally, we outline the generalisation of the leading-order formulations to prescribed non-uniform rates of mass transfer and to three dimensions.

As a result of field fringing, the capacitance of a parallel-plate capacitor differs from that predicted by the textbook formula. Using singular perturbations and conformal mapping techniques, we calculate the leading-order correction to the capacitance in the limit of large aspect ratio. We additionally obtain a comparable approximation for the electrostatic attraction between the plates.

A concentrated capacity problem is posed for the heat equation in a multidimensional domain. In the concentrated capacity (i.e. in a portion of the boundary of the domain) a change of phase takes place, and a Stefan-like problem is posed. This scheme has been introduced in the literature as the formal limiting case of a certain class of diffusion problems.

Our main result is a theorem of continuous dependence of the solution on the data. It is also used to prove the existence of the solution (in a weak sense), assuming only integrability of the data. The solution is found as the limit of the solutions of the approximating problems.

This article provides a borrower's optimal strategies to terminate a mortgage with a fixed interest rate by paying the outstanding balance all at once. The problem is modelled as a free boundary problem for the appropriate analogue of the Black-Scholes pricing equation under the assumption of the Vasicek model for the short-term rate of investment. Here the free boundary provides the optimal time at which the mortgage contract is to be terminated. A number of integral identities are derived and then used to design efficient numerical codes for computing the free boundary. For numerical simulation, parameters for the Vasicek model are estimated via the method of maximum likelihood estimation using 40 years of data from US government bonds. The asymptotic behaviour of the free boundary for the infinite horizon is fully analysed. Interpolating this infinite horizon behaviour and a known near-expiry behaviour, two simple analytical approximation formulas for the optimal exercise boundary are proposed. Numerical evidence shows that the enhanced version of the approximation formula is amazingly accurate; in general, its relative error is less than 1%, for all time before expiry.

We go back to the results of Poincaré [Poincare, H (1891) Sur lintegration des equations differentielles du premier ordre et du premier degre I and II, Rendiconti del circolo matematico di Palermo5, 161–191] on the multipliers of a periodic orbit for proving the C1 non-integrability of differential systems. We apply these results to Lorenz, Rossler and Michelson systems, among others.

The equations for three-wave interaction describe the resonant, quadratic, nonlinear interaction of three waves. They are obtained as amplitude equations in an asymptotic reduction of the basic equations of nonlinear optics, fluid mechanics, and plasma physics. These equations are completely integrable and have been the subject of intensive research in the last years. It is the purpose of this paper to prove exact estimates between the approximations obtained via this system and solutions of the original physical system. Although the three-wave interaction model is believed to describe a number of different physical models we restrict attention to its application as a model of the resonant interaction of water waves subject to weak surface tension.

The main result of this paper is a complete description of local invertible transformations

relating one equation of the form uxy = F(x, y, u, ux, uy) to another equation of the form υxy = G(x, y, υ, υx, υy).

This paper is concerned with modelling the dynamics of social outbursts of activity, such as protests or riots. In this sequel to our work in Berestycki et al. (Networks and Heterogeneous Media, vol. 10, no. 3, 1–34), written in collaboration with J-P. Nadal, we model the effect of restriction of information and explore its impact on the existence of upheaval waves. The system involves the coupling of an explicit variable representing the intensity of rioting activity and an underlying (implicit) field of social tension. We prove the existence of global solutions to the Cauchy problem in ${\mathbb R}^d$ as well as the existence of traveling wave solutions in certain parameter regimes. We furthermore explore the effects of heterogeneities in the environment with the help of numerical simulations, which lead to pulsating waves in certain cases. We analyse the effects of periodic domains as well as the barrier problem with the help of numerical simulations. The barrier problem refers to the potential blockage of a wavefront due to a spatial heterogeneity in the system which leads to an area of low excitability (referred to as the barrier). We conclude with a variety of open problems.

A three-phase ensemble-averaged model is developed for the flow of water and air through a deformable porous matrix. The model predicts a separation of the flow into saturated and unsaturated regions. The model is closed by proposing an experimentally-motivated heuristic elastic law which allows large-strain nonlinear behaviour to be treated in a relatively straightforward manner. The equations are applied to flow in the ‘nip’ area of a roll press machine whose function is to squeeze water out of wet paper as part of the manufacturing process. By exploiting the thin-layer limit suggested by the geometry of the nip, the problem is reduced to a nonlinear convection-diffusion equation with one free boundary. A numerical method is proposed for determining the flow and sample simulations are presented.

We study the motion of voids in conductors subject to intense electrical current densities. We use a free-boundary model in which the flow of current around the insulating void is coupled to a law of motion (kinematic condition) for the void boundary. In the first part of the paper, we apply a new complex variable formulation of the model to an infinite domain and use this to (i) consider the stability of circular and flat front travelling waves, (ii) show that, in the unbounded metal domain, the only travelling waves of finite void area are circular, and (iii) consider possible static solutions. In the second part of the paper, we look at a conducting strip (which can be used to model interconnects in electronic devices) and use asymptotic methods to investigate the motion of long wavelength voids on the conductor boundary. In this case we derive a nonlinear parabolic PDE describing the evolution of the free boundary and, using this simpler model, are able to make some predictions about the evolution of the void over long times.

The purpose of this paper is to give a pricing analysis for the American option in a jump-diffusion model by PDE arguments. Existence and uniqueness of the solution to the obstacle problem for the associated model is shown in suitable spaces. We also prove the unique existence of the solution of the corresponding free boundary problem. Furthermore, smoothness and monotonicity of the free boundary which is the optimal exercise boundary of the option are deduced.

We study the problem of breakup of an air bubble in a Hele-Shaw cell. In particular, we propose some sufficient conditions of breakup of the bubble, and ways to find the contraction points of its parts. We also study regulated contraction of a pair of bubbles (in which the rates of air extraction from the bubbles are controlled) and study various asymptotic questions (such as the asymptotics of contraction of a bubble to a degenerate critical point, and asymptotics of contraction of a small bubble in the presence of a big bubble)

We consider the activator-inhibitor Gierer–Meinhardt reaction-diffusion system of biological pattern formation in a closed bounded domain. The existence and stability of a boundary apike-layer solution to the Gierer–Meinhardt model, and it, so-called shadow limit, is analysed. In the limit of small activator diffusivity, together with a large inhibitor diffusivity, an equilibrium boundary spike-layer solution is constructed that concentrates at a non-degenerate critical point P of the boundary. By non-degenerate we mean that every principal curvature of the boundary has a local maximum at P, and hence the mean curvature at the boundary has a local maximum at P. Rigorous results for the stability of such a boundary spike-layer solution are given.

This work is devoted to the identification of parameters in a problem of pollution modeled by a semi-linear parabolic equation. We use the notion of sentinels introduced by J. L. Lions, (Lions, J. L. 1992 Sentinelles pour les systèmes distribués à données incomplètes. Masson, Paris.) re-visited in a more general framework. We prove the existence of such sentinels by solving a problem of null controllability with constraint on the control. The key of our results is an observability inequality of Carleman type adapted to the constraint.