Long-wavelength models for van der Waals driven rupture of a free thin viscous sheet and for capillary pinchoff of a viscous fluid thread both give rise to families of first-type similarity solutions. The scaling exponents in these solutions are independent of the dimensionality of problem. However, the structure of the similarity solutions exhibits an intriguing geometric dependence on the dimensionality of the system: van der Waals driven sheet rupture proceeds symmetrically, whereas thread rupture is inherently asymmetric. To study the bifurcation of rupture from symmetric to asymmetric forms, we generalize the governing equations with the dimension serving as a control parameter. The bifurcation is governed by leading-order inviscid dynamics in which viscous effects are asymptotically small but nevertheless provide the selection mechanism.

We propose a novel algorithm which allows to sample paths from an underlying price process in a local volatility model and to achieve a substantial variance reduction when pricing exotic options. The new algorithm relies on the construction of a discrete multinomial tree. The crucial feature of our approach is that – in a similar spirit to the Brownian Bridge – each random path runs backward from a terminal fixed point to the initial spot price. We characterize the tree in two alternative ways: (i) in terms of the optimal grids originating from the Recursive Marginal Quantization algorithm, (ii) following an approach inspired by the finite difference approximation of the diffusion's infinitesimal generator. We assess the reliability of the new methodology comparing the performance of both approaches and benchmarking them with competitor Monte Carlo methods.

We consider an anisotropic diffusion equation of the form ut = ∇∇(D(x)u) in two dimensions, which arises in various applications, including the modelling of wolf movement along seismic lines and the invasive spread of certain brain tumours along white matter neural fibre tracts. We consider a degenerate case, where the diffusion tensor D(x) has a zero-eigenvalue for certain values of x. Based on a regularisation procedure and various pointwise and integral a priori estimates, we establish the global existence of very weak solutions to the degenerate limit problem. Moreover, we show that in the large time limit these solutions approach profiles that exhibit a Dirac-type mass concentration phenomenon on the boundary of the region in which diffusion is degenerate, which is quite surprising for a linear diffusion equation. The results are illustrated by numerical examples.

Surfaces arising in amorphous thin-film-growth are often described by certain classes of stochastic PDEs. In this paper we address the question of existence of a solution and statistical quantities (e.g. mean interface width or correlation functions). Moreover, we discuss the approximations of such statistical quantities by the spectral Galerkin method. This is an important question, as the numerical computation of statistical quantities plays a key role in the verification of the models.

The motion of an initially circular drop of viscous fluid surrounded by inviscid fluid in a Hele-Shaw cell withdrawn from an eccentric point sink is considered. Using a numerical algorithm based on a boundary integral equation, the solution for small, finite surface tension is observed. It is found that the zero-surface-tension formation of a cusp is avoided, and instead a narrow finger of inviscid fluid forms, which then rapidly propagates towards the sink. The scaling of the finger in the sink vicinity is determined.

We model a thin liquid film moving down a slope using the lubrication approximation with a slip condition. The travelling-wave solution is derived for small inclination angle α, using singular perturbation methods, and compared to the numerical solution. For the linear stability analysis we combine numerical methods with the long-wave approximation and find a small but finite critical α* below which the flow remains linearly stable to spanwise perturbations. This is contrasted with the vanishing of the hump of the travelling-wave solution. Finally, the prevailing linear stability of the travelling-wave at small inclination angles is compared with recent related results using a precursor model. Here, though, a strong dependence on the magnitude of the contact angle is found, which we think has not been observed before.

In this work, we analyse a broad class of generalized Poisson–Boltzmann equations and reveal a common mathematical structure. In the limit of a wide electrode, we show that a broad class of generalized Poisson–Boltzmann equations admits a reduction that affords an explicit connection between the functional form of the corresponding free energy and the associated differential capacitance data. We exploit the relation to we show that differential capacitance curves generically undergo an inflection transition with increasing salt concentration, shifting from a local minimum near the point of zero charge for dilute solutions to a local maximum point near the point of zero charge for concentrated solutions. In addition, we develop a robust numerical method for solving generalized Poisson–Boltzmann equations which is easily applicable to the broad class of generalized Poisson–Boltzmann equations with very few code adjustments required for each model

Cooperative coordination in multi-agent systems has been a topic of interest in networked control theory in recent years. In contrast to cooperative agents, Byzantine agents in a network are capable to manipulate their data arbitrarily and send bad messages to neighbors, causing serious network security issues. This paper is concerned with resilient tracking consensus over a time-varying random directed graph, which consists of cooperative agents, Byzantine agents and a single leader. The objective of resilient tracking consensus is the convergence of cooperative agents to the leader in the presence of those deleterious Byzantine agents. We assume that the number and identity of the Byzantine agents are not known to cooperative agents, and the communication edges in the graph are dynamically randomly evolving. Based upon linear system analysis and a martingale convergence theorem, we design a linear discrete-time protocol to ensure tracking consensus almost surely in a purely distributed manner. Some numerical examples are provided to verify our theoretical results.

We present two new models describing the dynamic behavior of an automotive thermostat, involving delay-differential equations with hysteresis. Existence, uniqueness, and regularity of the solutions for both models are obtained by a continuation argument. We establish sufficient conditions for the models to exhibit intrinsic oscillations. We also present an algorithm for numerical approximations of the solutions and give some representative numerical simulations. These reveal a rather interesting dynamical behavior of the solutions.

The classical Monge–Kantorovich (MK) problem as originally posed is concerned with how best to move a pile of soil or rubble to an excavation or fill with the least amount of work relative to some cost function. When the cost is given by the square of the Euclidean distance, one can define a metric on densities called the Wasserstein distance. In this note, we formulate a natural matrix counterpart of the MK problem for positive-definite density matrices. We prove a number of results about this metric including showing that it can be formulated as a convex optimisation problem, strong duality, an analogue of the Poincaré–Wirtinger inequality and a Lax–Hopf–Oleinik–type result.

We propose a multi-level method to increase the accuracy of machine learning algorithms for approximating observables in scientific computing, particularly those that arise in systems modelled by differential equations. The algorithm relies on judiciously combining a large number of computationally cheap training data on coarse resolutions with a few expensive training samples on fine grid resolutions. Theoretical arguments for lowering the generalisation error, based on reducing the variance of the underlying maps, are provided and numerical evidence, indicating significant gains over underlying single-level machine learning algorithms, are presented. Moreover, we also apply the multi-level algorithm in the context of forward uncertainty quantification and observe a considerable speedup over competing algorithms.

We consider polymerization–crystallization waves in a cylindrical reactor, in which monomer is converted to polymer in a planar front. The polymer is subsequently crystallized in a wider zone behind the front. Specifically, we study uniformly propagating polymerization–crystallization waves, and determine profiles of temperature, and concentrations of polymer and crystallized polymer, as well as the propagation velocity. A linear stability analysis of the travelling wave solutions indicates the possibility of Hopf bifurcation, which describes the transition to the experimentally observed spinning mode of propagation, in which a hot spot is observed to propagate along a helical path on the surface of the cylinder. Since conditions at the time of conversion determine the nature of the polymer produced, spiral hollows, which trace out a helical path, appear on the surface of the crystallized polymer product.

This paper deals with Maxwell's equations in a quasi-stationary electromagnetic field subject to the effects of temperature. This model is encountered in the penetration of a magnetic field in substances where the electrical conductivity depends on the temperature. Similar phenomena also occur in some industrial problems such as the thermistor. Taking the effect of Joule heating into the consideration, we obtain a strongly coupled nonlinear system. Global solvability is established for this system.

The Stefan problem with surface tension in the three-dimensional case with spherical symmetry is considered. We first establish the existence and uniqueness of the classical solution with surface tension and kinetic undercooling effects for all time, and then pass to the limit as the kinetic undercooling tends to zero. The limiting solution is the global-in-time weak solution and the local-in-time classical solution for the Stefan problem with surface tension. This solution cannot be the global-in-time classical solution. If S(t) is the radius of a solid ball in a supercooled liquid, then (1) there exists at least one point t* of discontinuity of the function S(t):

or (2) the continuous function S(t) cannot be absolutely continuous, and it maps some zero-measure set of (t*, T*) onto some set of Ω with a strictly positive measure.

We consider the following Gray–Scott model in $B_R (0) = \{x: |x|\lt R\} \subset \R^N, N=2,3$: \[ \left\{ \begin{array}{@{}ll} v_t= \ep^2 \oldDelta v- v+ A v^2 u &\mbox{ in } B_R (0),\\ \tau u_t=\oldDelta u+1-u - v^2 u & \mbox{ in } B_R(0), \\ u, v\gt 0;~~ \frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu} =0 &\hspace*{3pt}\mbox{on} \ \partial B_R(0) \end{array} \right. \] where $\ep>0$ is a small parameter. We assume that $A= \skew{7}\hat{A} \ep^{\frac{1}{2}}$. For each $\skew{7}\hat{A} < +\infty$ and $R<\infty$, we construct ring-like solutions which concentrate on an $(N-1)$-dimensional sphere for the stationary system for all sufficiently small $\ep$. More precisely, it is proved the above problem has a radially symmetric steady state solution $(v_{\ep,R}, u_{\ep, R})$ with the property that $v_{\ep,R} (r) \to 0 $ in $\R^N \backslash \{ r \not = r_0 \}$ for some $ r_0 \in (0, R)$. Then we show that for $N=2$ such solutions are unstable with respect to angular fluctuations of the type $ \Phi (r) e^{ \sqrt{-1} m \theta}$ for some $m$. A relation between $\skew{7}\hat{A}$ and the minimal mode $m$ is given. Similar results are also obtained when $\Om=\R^N$ or $\Om= B_{R_2} (0) \backslash B_{R_1} (0)$ or $\Om= \R^N \backslash B_R (0)$.

We use lubrication theory on the flow equations for nematic liquid crystals to derive a simple model describing the evolution of the film height under gravity, in the case of finite surface “anchoring energy” at the free surface and at the rigid substrate. This means that the molecules of the nematic have a preferred alignment at interfaces, modelled by a single-well potential surface energy (first introduced by Rapini & Papoular [9]). This paper generalises the earlier work of Ben Amar & Cummings, in which the orientation of the nematic liquid crystal molecules is effectively specified at both surfaces (strong anchoring; isotropic surface tension). Additional terms, analogous in some sense to Marangoni terms, are introduced into the PDE governing the film height evolution. The stability of the derived model is considered, and stability criteria are presented and discussed. The existence of static, drop-like solutions to the model is also briefly considered.

We discuss two different approaches for the analysis of the Poisson and of the non-homogeneous biharmonic equations in two dimensions. The first approach yields the solution as an integral in the complex $z$-plane (the physical plane), involving explicitly the given boundary conditions. The second approach yields an integral in the complex $k$-plane (the Fourier plane), involving the Fourier transforms of the given boundary conditions. For simple boundary value problems, such as certain problems formulated in the half complex plane, the first approach is easier. However, for more complicated problems, such as those formulated in the interior of an equilateral triangle, it appears that only the second approach can be used. Furthermore, the second approach also seems more efficient for numerical computations.

This article introduces a new notion of optimal transport (OT) between tensor fields, which are measures whose values are positive semidefinite (PSD) matrices. This “quantum” formulation of optimal transport (Q-OT) corresponds to a relaxed version of the classical Kantorovich transport problem, where the fidelity between the input PSD-valued measures is captured using the geometry of the Von-Neumann quantum entropy. We propose a quantum-entropic regularization of the resulting convex optimization problem, which can be solved efficiently using an iterative scaling algorithm. This method is a generalization of the celebrated Sinkhorn algorithm to the quantum setting of PSD matrices. We extend this formulation and the quantum Sinkhorn algorithm to compute barycentres within a collection of input tensor fields. We illustrate the usefulness of the proposed approach on applications to procedural noise generation, anisotropic meshing, diffusion tensor imaging and spectral texture synthesis.

We relate Kaptsov's method of B-determining equations for finding invariant solutions of PDEs to the nonclassical method and to the method of generalised conditional symmetries. An extension of Kaptsov's method is then used to find new solutions of degenerate diffusion equations.

In this paper, we are interested in the mathematical and numerical study of the Prigozhin model for a growing sandpile. Based on implicit Euler discretization in time, we give a simple improvement of theoretical and numerical analyses of the dual formulation for the problem. By using this model, we also give some application to the Monge–Kantorovich problem for optimal mass transportation.