The dynamics and oscillatory instabilities of multi-spike solutions to the one-dimensional Gray-Scott reaction–diffusion system on a finite domain are studied in a particular parameter regime. In this parameter regime, a formal singular perturbation method is used to derive a novel ODE–PDE Stefan problem, which determines the dynamics of a collection of spikes for a multi-spike pattern. This Stefan problem has moving Dirac source terms concentrated at the spike locations. For a certain subrange of the parameters, this Stefan problem is quasi-steady and an explicit set of differential-algebraic equations characterizing the spike dynamics is derived. By analysing a nonlocal eigenvalue problem, it is found that this multi-spike quasi-equilibrium solution can undergo a Hopf bifurcation leading to oscillations in the spike amplitudes on an O(1) time scale. In another subrange of the parameters, the spike motion is not quasi-steady and the full Stefan problem is solved numerically by using an appropriate discretization of the Dirac source terms. These numerical computations, together with a linearization of the Stefan problem, show that the spike layers can undergo a drift instability arising from a Hopf bifurcation. This instability leads to a time-dependent oscillatory behaviour in the spike locations.

Nowadays many financial derivatives, such as American or Bermudan options, are of early exercise type. Often the pricing of early exercise options gives rise to high-dimensional optimal stopping problems, since the dimension corresponds to the number of underlying assets. High-dimensional optimal stopping problems are, however, notoriously difficult to solve due to the well-known curse of dimensionality. In this work, we propose an algorithm for solving such problems, which is based on deep learning and computes, in the context of early exercise option pricing, both approximations of an optimal exercise strategy and the price of the considered option. The proposed algorithm can also be applied to optimal stopping problems that arise in other areas where the underlying stochastic process can be efficiently simulated. We present numerical results for a large number of example problems, which include the pricing of many high-dimensional American and Bermudan options, such as Bermudan max-call options in up to 5000 dimensions. Most of the obtained results are compared to reference values computed by exploiting the specific problem design or, where available, to reference values from the literature. These numerical results suggest that the proposed algorithm is highly effective in the case of many underlyings, in terms of both accuracy and speed.

This article presents a method for the numerical quadrature of highly oscillatory integrals with stationary points. We begin with the derivation of a new asymptotic expansion, which has the property that the accuracy improves as the frequency of oscillations increases. This asymptotic expansion is closely related to the method of stationary phase, but presented in a way that allows the derivation of an alternate approximation method that has similar asymptotic behaviour, but with significantly greater accuracy. This approximation method does not require moments.

We propose a delay differential equation model for a single species with stage-structure in which the maturation delay is modelled as a distribution, to allow for the possibility that individuals may take different amounts of time to mature. General birth and death rate functions are used. We find that the dynamics of the model depends largely on the qualitative form of the birth function, which depends on the total number of adults. If it is monotonic increasing and a non-zero equilibrium exists, then the equilibrium is globally stable for all maturation delay distributions with compact support. For the case of a finite spatial domain with impermeable boundaries, a reaction-diffusion extension of the model is rigorously derived using an approach based on the von Foerster diffusion equation. The resulting reaction-diffusion system is nonlocal. The dynamics of the reaction-diffusion system again depends largely on the qualitative form of the birth function. If the latter is non-monotone with a single hump, then the dynamics depends largely on whether the equilibrium is to the left or right of the hump, with oscillatory dynamics a possibility if it is sufficiently far to the right.

Flows with free boundaries in a Hele-Shaw cell provide a unique opportunity to study non-linear boundary dynamics using rigorous analytic approaches. While of limited direct ‘practical value’, these studies give rise to a plethora of new phenomena and insights which may serve as beacons in the turbulent ocean of moving free boundaries and pattern forming. This paper gives a brief summary of the authors' studies of Hele-Shaw flows with free boundaries and some related problems based upon Richardson's approach. Some promising directions of further research are also discussed.

When a solid plate is removed from a pool of fluid, a film of fluid is attached to the plate. There are two possible outcomes. The edge of the fluid may be raised through a finite distance, with the edge slipping on the plate. Alternatively, a continuous film of a certain thickness may be drawn up. For plates which have a small slope, it is shown that the first alternative holds when the speed of withdrawal is sufficiently small, and that, when a critical speed is exceeded, the height of the edge above the fluid level in the pool increases with time. A related problem concerns the shape of a receding meniscus in a channel. If the static contact angle is small, lubrication theory can be applied to the film of fluid adjacent to the wall of the channel, and the results of this part of the solution can be matched to the central part of the meniscus, which is controlled by capillarity and gravity, but for which lubrication theory does not apply. As in the draw-up problem, for small speeds the shape of the meniscus is time-independent, but above a critical speed, a tail of fluid of increasing length remains in the tube.

We study the dynamics of dissipation and blow-up in a critical-case unstable thin film equation. The governing equation is a nonlinear fourth-order degenerate parabolic PDE derived from a generalized model for lubrication flows of thin viscous fluid layers on solid surfaces. %For a special balance between %destabilizing second-order terms and regularizing fourth-order terms, There is a critical mass for blow-up and a rich set of dynamics including families of similarity solutions for finite-time blow-up and infinite-time spreading. The structure and stability of the steady-states and the compactly-supported similarity solutions is studied.

As a paradigm for non-interpenetrating crack models, the Poisson equation in a nonsmooth domain in R2 is considered. The geometrical domain has a cut (a crack) of variable length. At the crack faces, inequality type boundary conditions are prescribed. The behaviour of the energy functional is analysed with respect to the crack length changes. In particular, the derivative of the energy functional with respect to the crack length is obtained. The associated Griffith formula is derived, and properties of the solution are investigated. It is shown that the Rice–Cherepanov integral defined for the solutions of the unilateral problem defined in the nonsmooth domain is path-independent. Finally, a non-negative measure characterising interaction forces between the crack faces is constructed.

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.

Lie symmetry analysis is applied to study the non-linear rotating shallow-water equations. The 9-dimensional Lie algebra of point symmetries admitted by the model is found. It is shown that the rotating shallow-water equations can be transformed to the classical shallow-water model. The derived symmetries are used to generate new exact solutions of the rotating shallow-water equations. In particular, a new class of time-periodic solutions with quasi-closed particle trajectories is constructed and studied. The symmetry reduction method is also used to obtain some invariant solutions of the model. Examples of these solutions are presented with a brief physical interpretation.

We consider a new concept of weak solutions to the phase-field equations with a small parameter ε characterizing the length of interaction. For the standard situation of a single free interface, this concept (in contrast with the common one) leads to the well-known Stefan–Gibbs–Thomson problem as ε→0. For the case of a large number M(ε) (M(ε)→∞ as ε→0) of free interfaces, which corresponds to the ‘wave-train’ interpretation of a ‘mushy region’, this concept allows us to obtain the limit problem as ε→0.

Uniqueness and continuous dependence on the initial temperature are proved for a onedimensional, quasistatic and frictionless contact problem in linear thermoelasticity. First the problem is reformulated in such a way that it decouples. The resulting problem for the temperature is a nonlinear integro-differential equation. Once the temperature is known the displacement is recovered from an appropriate variational inequality. Uniqueness is proved by considering an integral transform of the temperature. The steady solution is obtained and the asymptotic stability is shown. It turns out that the asymptotic behaviour and the steady state are determined by a relation between the coupling constant a and the initial gap.

We aim at understanding transport in porous materials consisting of regions with both high and low diffusivities. We apply a formal homogenisation procedure to the case where the heterogeneities are not arranged in a strictly periodic manner. The result is a two-scale model formulated in x-dependent Bochner spaces. We prove the weak solvability of the limit two-scale model for a prototypical advection–diffusion system of minimal size. A special feature of our analysis is that most of the basic estimates (positivity, L∞-bounds, uniqueness, energy inequality) are obtained in the x-dependent Bochner spaces.

Electronic communications, as well as other categories of interactions within social networks, exhibit bursts of activity localised in time. We adopt a self-exciting Hawkes process model for this behaviour. First we investigate parameter estimation of such processes and find that, in the parameter regime we encounter, the choice of triggering function is not as important as getting the correct parameters once a choice is made. Then we present a relaxed maximum likelihood method for filling in missing data in records of communications in social networks. Our optimisation algorithm adapts a recent curvilinear search method to handle inequality constraints and a non-vanishing derivative. Finally we demonstrate the method using a data set composed of email records from a social network based at the United States Military Academy. The method performs differently on this data and data from simulations, but the performance degrades only slightly as more information is removed. The ability to fill in large blocks of missing social network data has implications for security, surveillance, and privacy.

In this article, we present a simplified means of pricing Asian options using partial differential equations (PDEs). We first provide a concise derivation of the well-known similarity reduction and exact Laplace transform solution. We then analyse the problem afresh as a power series in the volatility-scaled contract duration, with a view to obtaining an asymptotic solution for the low-volatility limit, a limit which presents difficulties in the context of the general Laplace transform solution. The problem is approached anew from the point of view of asymptotic expansions and the results are compared with direct, high precision, inversion of the Laplace transform and with numerical results obtained by V. Linetsky and J. Vecer. Our asymptotic formulae are little more complicated than the standard Black–Scholes formulae and, working to third order in the volatility-scaled expiry, are accurate to at least four significant figures for standard test problems. In the case of zero risk-neutral drift, we have the solution to fifth order and, for practical purposes, the results are effectively exact. We also provide comparisons with the hybrid analytic and finite-difference method of Zhang.

The coupled motion between shallow-water sloshing in a moving vehicle and the vehicle dynamics is considered, with the vehicle dynamics restricted to horizontal motion. The paper is motivated by Cooker's experiments and theory for water waves in a suspended container. A new derivation of the coupled problem in the Eulerian fluid representation is given. However, it is found that transformation to a Lagrangian representation leads to a formulation which has nice properties for numerical simulation. In the Lagrangian representation, a simple and fast numerical algorithm with excellent energy conservation over long times, based on the Störmer–Verlet method, is implemented. Numerical simulations of the coupled dynamics in both the linear and non-linear case are presented.

In this paper domain decomposition methods for radiative transfer problems including conductive heat transfer are treated. The paper focuses on semi-transparent materials, like glass, and the associated conditions at the interface between the materials. Using asymptotic analysis we derive conditions for the coupling of the radiative transfer equations and a diffusion approximation. Several test casts are treated and a problem appearing in glass manufacturing processes is computed. The results clearly show the advantages of a domain decomposition approach. Accuracy equivalent to the solution of the global radiative transfer solution is achieved, whereas computation time is strongly reduced.

We examine the Black–Scholes partial differential equation for the pricing of a traded option (an American call option on an asset paying a continuous dividend) and make comparisons with other well known free boundary diffusion problems, such as the oxygen consumption problem. The pricing of American options can be viewed as a free boundary problem and is, therefore, inherently nonlinear. We consider the short and long time behaviour of the free boundary, present analytic results for the option value in such limits, and consider the formulation of the problem as a variational inequality, and its numerical solution.

We consider a geometric motion associated with the minimization of a curvature dependent functional, which is related to the Willmore functional. Such a functional arises in connection with the image segmentation problem in computer vision theory. We show by using formal asymptotics that the geometric motion can be approximated by the evolution of the zero level set of the solution of a nonlinear fourth-order equation related to the Cahn–Hilliard and Allen–Cahn equations.

We investigate the motion of two immiscible fluids in a porous medium described by a two-phase flow system. In the capillary pressure relation, we include static and dynamic hysteresis. The model is well established in the context of the Richards equation, which is obtained by assuming a constant pressure for one of the two phases. We derive an existence result for this hysteresis two-phase model for non-degenerate permeability and capillary pressure curves. A discretization scheme is introduced and numerical results for fingering experiments are obtained. The main analytical tool is a compactness result for two variables that are coupled by a hysteresis relation.