We carry out the extended symmetry analysis of an ultraparabolic Fokker–Planck equation with three independent variables, which is also called the Kolmogorov equation and is singled out within the class of such Fokker–Planck equations by its remarkable symmetry properties. In particular, its essential Lie invariance algebra is eight-dimensional, which is the maximum dimension within the above class. We compute the complete point symmetry pseudogroup of the Fokker–Planck equation using the direct method, analyse its structure and single out its essential subgroup. After listing inequivalent one- and two-dimensional subalgebras of the essential and maximal Lie invariance algebras of this equation, we exhaustively classify its Lie reductions, carry out its peculiar generalised reductions and relate the latter reductions to generating solutions with iterative action of Lie-symmetry operators. As a result, we construct wide families of exact solutions of the Fokker–Planck equation, in particular, those parameterised by an arbitrary finite number of arbitrary solutions of the (1+1)-dimensional linear heat equation. We also establish the point similarity of the Fokker–Planck equation to the (1+2)-dimensional Kramers equations whose essential Lie invariance algebras are eight-dimensional, which allows us to find wide families of exact solutions of these Kramers equations in an easy way.

In this paper, we present numerical simulations that demonstrate the effect of the particular choice of the equation of state (EoS) relating the surfactant concentration to the surface tension in surfactant-driven thin liquid films. Previous choices of the model EoS have been an ad-hoc decreasing function. Here, we instead propose an empirically motivated EoS; this provides a route to resolve some discrepancies and raises new issues to be pursued in future experiments. In addition, we test the influence of the choice of initial conditions and values for the non-dimensional groups. We demonstrate that the choice of EoS improves the agreement in surfactant distribution morphology between simulations and experiments, and influences the dynamics of the simulations. Because an empirically motivated EoS has regions with distinct gradients, future mathematical models may be improved by considering more than one timescale. We observe that the non-dimensional number controlling the relative importance of gravitational versus capillary forces has a larger influence on the dynamics than the other non-dimensional groups, but is nonetheless not a likely cause of discrepancy between simulations and experiments. Finally, we observe that the experimental approach using a ring to contain the surfactant could affect the surfactant and fluid dynamics if it disrupts the intended initial surfactant distribution. However, the fluid meniscus itself does not significantly affect the dynamics.

A mathematical model is developed for mud cake growth and fluid invasion around a horizontal wellbore during drilling. The non-axisymmetry of the invasion front is addressed for the case of water based drilling mud and isotropic rock formation. It is shown that the invasion profile may lose convexity due to gravitation even though the filtrate density is the same as that of the pore fluid in the rock.

This paper gives a brief overview of some configurations in which high-frequency wave propagation modelled by Helmholtz equation gives rise to solutions that vary rapidly across thin layers. The configurations are grouped according to their mathematical structure and tractability and one of them concerns a famous open problem of mathematical physics.

The following first problem is posed: is a correct ‘entropy solution’ of the Cauchy problem for the fifth-order degenerate non-linear dispersion equations (NDEs), same as for the classic Euler one ut + uux = 0, These two quasi-linear degenerate partial differential equations (PDEs) are chosen as typical representatives; so other (2m + 1)th-order NDEs of non-divergent form admit such shocks waves. As a related second problem, the opposite initial shock S+(x) = −S−(x) = sign x is shown to be a non-entropy solution creating a rarefaction wave, which becomes C∞ for any t > 0. Formation of shocks leads to non-uniqueness of any ‘entropy solutions’. Similar phenomena are studied for a fifth-order in time NDE uttttt = (uux)xxxx in normal form.

On the other hand, related NDEs, such as are shown to admit smooth compactons, as oscillatory travelling wave solutions with compact support. The well-known non-negative compactons, which appeared in various applications (first examples by Dey, 1998, Phys. Rev. E, vol. 57, pp. 4733–4738, and Rosenau and Levy, 1999, Phys. Lett. A, vol. 252, pp. 297–306), are non-existent in general and are not robust relative to small perturbations of parameters of the PDE.

A general method is presented to facilitate the solution of a class of polydisperse spray problems in which a cloud of droplets can be described using a sectional or group model. The procedure involves replacing the original coupled droplet sectional variables conservation equations by a set of uncoupled sectional equations for auxiliary variables. The form of these latter equations is identical to that of the single spray equation for a quasi-monodisperse spray, solutions of which are more readily attainable even for multidimensional spray problems. Thus, these ready-made solutions can be exploited directly for the auxiliary variables, from which solutions can then be constructed in a straightforward manner for the desired original sectional variables. Three illustrative examples for spray diffusion flames with different features of complexity highlight the potential applicability of the proposed method, and indicate the sensitivity of flame characteristics to initial spray conditions and in-spray related phenomena.

The linearized shape stability of melting and solidifying fronts with surface tension is discussed in this paper by using asymptotic analysis. We show that the melting problem is always linearly stable regardless of the presence of surface tension, and that the solidification problem is linearly unstable without surface tension, but with surface tension it is linearly stable for those modes whose wave numbers lie outside a certain finite interval determined by the parameters of the problem. We also show that if the perturbed initial data is zero in the vicinity of the front, but otherwise quite general, it does not affect the stability. The present results complement those in Chadam & Ortoleva [4] which are only valid asymptotically for large time or equivalently for slow-moving interfaces. The theoretical results are verified numerically.

Estimates for sample paths of fast–slow stochastic ordinary differential equations have become a key mathematical tool relevant for theory and applications. In particular, there have been breakthroughs by Berglund and Gentz to prove sharp exponential error estimates. In this paper, we take the first steps in order to generalise this theory to fast–slow stochastic partial differential equations. In a simplified setting with a natural decomposition into low- and high-frequency modes, we demonstrate that for a short-time period the probability for the corresponding sample path to leave a neighbourhood around the stable slow manifold of the system is exponentially small as well.

A semi-infinite body, modelling the leading edge of a cutting tool or submerged hydrofoil, lies beneath a free surface in a uniform stream of infinitely-deep inviscid incompressible fluid flowing steadily under gravity. The body has horizontal upper and lower surfaces. The oncoming flow is partly diverted over, and partly under the body. The flow in that portion that travels over the body can be supercritical or subcritical. When it is supercritical it approaches a stream of some (to be determined) speed in a channel of some (to be determined) depth. When it is subcritical, there is also a train of waves on that stream, whose amplitude is also to be determined. Semi-analytic high-speed and low-speed solutions are obtained, and a numerical solution for finite speeds. There is a ‘forbidden’ intermediate speed range, within which steady flow may not be possible.

We consider two problems related to the b–l and b–ε models of propagation of turbulent bursts. We show that these equations admit some particular exact solutions which reduce to a finite-dimensional dynamical system. This makes it possible to describe a singular effect of finite-time extinction, and in particular, nonsymmetric solutions which do not exhibit symmetrization in the asymptotic behaviour. We show that in the multi-dimensional equation related to the b–l model, the nonsymmetric extinction behaviour is governed by the first-order equation. For the b–ε model with α=β=1 and γ<1, using such particular solutions, we establish that the ω-limit set of all the rescaled extinction orbits is essentially infinite-dimensional.

In this paper I consider a system of equations describing two stratified fluids flowing in closed, slightly inclined ducts. In the framework of the shallow water approximation with turbulent friction acting on the wall and at the interface, I investigate a special class of periodic travelling waves containing stable moving jumps, namely, roll waves and periodic slug flows. The modulation equations of roll waves and slugs are derived and a nonlinear stability criterion is obtained. As for plug flow, only a gas–liquid system is considered. The stability criterion is expressed in terms of an integro-differential relation. For self-similar cross-section, this criterion is simplified to a relation for a function of one variable only.

Silicon is produced in submerged arc furnaces which are heated by electric currents passing through the furnace. It is important to understand the distribution of heating within the furnace in order to accurately model the silicon production process, yet many existing studies neglect aspects of this current flow. In the present paper, we formulate a model that couples the electrical current to thermal, material flow and chemical processes in the furnace. We then exploit disparate timescales to homogenise the model over the timescale of the alternating current, deriving averaged equations for the slow evolution of the system. Our numerical simulations predict a minimum applied current that is required in order to obtain steady-state solutions of the homogenised model and show that for high enough applied currents, two spatially heterogeneous steady-state solutions exist, with distinct crater sizes. We show that the system evolves to the steady state with a larger crater radius and explain this behaviour in terms of the overall power balance typically found within a furnace. We find that the industrial practice of stoking furnaces increases the overall rate of material consumption in the furnace, thereby improving the efficiency of silicon production.

We introduce novel multi-agent interaction models of entropic spatially inhomogeneous evolutionary undisclosed games and their quasi-static limits. These evolutions vastly generalise first- and second-order dynamics. Besides the well-posedness of these novel forms of multi-agent interactions, we are concerned with the learnability of individual payoff functions from observation data. We formulate the payoff learning as a variational problem, minimising the discrepancy between the observations and the predictions by the payoff function. The inferred payoff function can then be used to simulate further evolutions, which are fully data-driven. We prove convergence of minimising solutions obtained from a finite number of observations to a mean-field limit, and the minimal value provides a quantitative error bound on the data-driven evolutions. The abstract framework is fully constructive and numerically implementable. We illustrate this on computational examples where a ground truth payoff function is known and on examples where this is not the case, including a model for pedestrian movement.

Cells and organisms follow aligned structures in their environment, a process that can generate persistent migration paths. Kinetic transport equations are a popular modelling tool for describing biological movements at the mesoscopic level, yet their formulations usually assume a constant turning rate. Here we relax this simplification, extending to include a turning rate that varies according to the anisotropy of a heterogeneous environment. We extend known methods of parabolic and hyperbolic scaling and apply the results to cell movement on micropatterned domains. We show that inclusion of orientation dependence in the turning rate can lead to persistence of motion in an otherwise fully symmetric environment and generate enhanced diffusion in structured domains.

When individuals move together in large groups, as seen in schools of fish, they adapt their speed and direction to that of their neighbours. We present and analyse a model for the speed adaptation process in the case in which all individuals move in the same or in two opposite directions. The model consists of a hyperbolic conservation law for the density of individuals coupled to a parabolic or elliptic equation for speed. A detailed linear analysis reveals several mechanisms for the appearance of instabilities of the homogeneous steady state, which trigger the formation of schools, herds, flocks, etc. Long-term existence of weak solutions is shown using the vanishing viscosity approach.

While a classic result by Merton (1973, Bell J. Econ. Manage. Sci., 141–183) is that one should never exercise an American call option just before expiration if the underlying stock pays no dividends, the conclusion of a very recent empirical study conducted by Jensen and Pedersen (2016, J. Financ. Econ.121(2), 278–299) suggests that one should ‘never say never’. This paper complements Jensen and Pedersen's empirical study by presenting a theoretical study on how to price American call options under a hard-to-borrow stock model proposed by Avellaneda and Lipkin (2009, Risk22(6), 92–97). Our study confirms that it is the lending fee that results in the early exercise of American call options and we shall also demonstrate to what extent lending fees have affected the early exercise decision.

We explore the effect of disjoining pressure on a thin film equation in the presence of a non-uniform body force, motivated by a model describing the reverse draining of a magnetic film. To this end, we use a combination of numerical investigations and analytical considerations. The disjoining pressure has a regularizing influence on the evolution of the system and appears to select a single steady-state solution for fixed height boundary conditions; this is in contrast with the existence of a continuum of locally attracting solutions that exist in the absence of disjoining pressure for the same boundary conditions. We numerically implement matched asymptotic expansions to construct equilibrium solutions and also investigate how they behave as the disjoining pressure is sent to zero. Finally, we consider the effect of the competition between forcing and disjoining pressure on the coarsening dynamics of the thin film for fixed contact angle boundary conditions.

The existence of a symmetric mode in an elastic solid wedge for all admissible values of the Poisson ratio and arbitrary interior angles close to π has been proven.

The one-dimensional porous medium equation ht= (hmhx)x (m>0) admits waiting-time solutions, whose front remains motionless during a finite time interval tw before starting to move. We consider a family of initial value problems, and investigate the asymptotics, close to the front and near start-up, which we expect to be self-similar. We obtain numerical solutions for viscous gravity currents (m=3) and power-law initial conditions (h∝xp, h is proportional to the thickness of the fluid, x is the distance to the front). We find that: (a) if p<2/3 the front starts moving immediately, (b) if p=2/3 the front remains motionless during a finite time, (c) if p>2/3 one obtains waiting-time solutions in which a moving corner layer (a small interval Δx in which hx varies strongly) appears behind the front; the front starts moving when it is overrun by the corner layer. The corner layer strengthens (Δx reduces and the variation of hx increases) as it approaches the front. Our initial conditions produce waiting-time solutions whose front starts moving with nonzero velocity. We determine tw(p) and study the motion of the corner layer and the front, as well as other properties of the solutions. We compare the results with the theoretical upper and lower bounds of tw. We investigate the asymptotics of the numerical solutions for p>2/3, close to the corner layer and the front, and near start-up. To represent this asymptotics various kinds of similarity solutions are available, that can be classified according to the self-similarity exponent δ. We find that only two types (called L and A) are relevant. The L solutions correspond to 1<δ<13/10, and have an infinite series of corner layers that accumulate at the front. The part of these solutions behind the first corner layer of the series represents the asymptotics of the numerical solutions in a domain that excludes the region between the corner layer and the front, for a time interval excluding the neighbourhood of start-up. The A solutions have δ⩽1, and represent the evolution of the strong corner layer that is arriving at the front. The numerical evidence shows that the constant front velocity solution (type A with δ=1) describes the asymptotics close to, and including start-up, so that the motion of the corner layer joins smoothly with that of the front.

We develop a mathematical model for a three-phase free boundary problem in one dimension that involves interactions between gas, water and ice. The dynamics are driven by melting of the ice layer, while the pressurized gas also dissolves within the meltwater. The model incorporates the Stefan condition at the water–ice interface along with Henry's law for dissolution of gas at the gas–water interface. We employ a quasi-steady approximation for the phase temperatures and then derive a series solution for the interface positions. A non-standard feature of the model is an integral free boundary condition that arises from mass conservation owing to changes in gas density at the gas–water interface, which makes the problem non-self-adjoint. We derive a two-scale asymptotic series solution for the dissolved gas concentration, which because of the non-self-adjointness gives rise to a Fourier series expansion in eigenfunctions that do not satisfy the usual orthogonality conditions. Numerical simulations of the original governing equations are used to validate series approximations.