The hybrid numerical-asymptotic (HNA) approach aims to reduce the computational cost of conventional numerical methods for high-frequency wave scattering problems by enriching the numerical approximation space with oscillatory basis functions, chosen based on partial knowledge of the high-frequency solution asymptotics. In this paper, we propose a new methodology for the treatment of shadow boundary effects in HNA boundary element methods, using the classical geometrical theory of diffraction phase functions combined with mesh refinement. We develop our methodology in the context of scattering by a class of sound-soft non-convex polygons, presenting a rigorous numerical analysis (supported by numerical results) which proves the effectiveness of our HNA approximation space at high frequencies. Our analysis is based on a study of certain approximation properties of the Fresnel integral and related functions, which govern the shadow boundary behaviour.

An analysis is undertaken of the formation and stability of localised patterns in a 1D Schanckenberg model, with source terms in both the activator and inhibitor fields. The aim is to illustrate the connection between semi-strong asymptotic analysis and the theory of localised pattern formation within a pinning region created by a subcritical Turing bifurcation. A two-parameter bifurcation diagram of homogeneous, periodic and localised patterns is obtained numerically. A natural asymptotic scaling for semi-strong interaction theory is found where an activator source term \[a = O(\varepsilon )\] and the inhibitor source \[b = O({\varepsilon ^2})\], with ε2 being the diffusion ratio. The theory predicts a fold of spike solutions leading to onset of localised patterns upon increase of b from zero. Non-local eigenvalue arguments show that both branches emanating from the fold are unstable, with the higher intensity branch becoming stable through a Hopf bifurcation as b increases beyond the \[O(\varepsilon )\] regime. All analytical results are found to agree with numerics. In particular, the asymptotic expression for the fold is found to be accurate beyond its region of validity, and its extension into the pinning region is found to form the low b boundary of the so-called homoclinic snaking region. Further numerical results point to both sub and supercritical Hopf bifurcation and novel spikeinsertion dynamics.

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.

The Lie symmetry classification of the known three-component reaction–diffusion system modelling the spread of an initially localized population of farmers into a region occupied by hunter-gatherers is derived. The Lie symmetries obtained for reducing the system in question to systems of ordinary differential equations (ODEs) and constructing exact solutions are applied. Several exact solutions of travelling front type are also found, their properties are identified and biological interpretation is discussed.

When a thin layer of normal (non-superconducting) material is placed between layers of superconducting material, a superconducting-normal-superconducting junction is formed. This paper considers a model for the junction based on the Ginzburg–Landau equations as the thickness of the normal layer tends to zero. The model is first derived formally by averaging the unknown variables in the normal layer. Rigorous convergence is then established, as well as an estimate for the order of convergence. Numerical results are shown for one-dimensional junctions.

Ovsiannikov's method of partially invariant solutions of differential equations can be considered to be a special case of the method of differential constraints introduced by Yanenko and by Olver and Rosenau. Differential constraints are used to construct non-reducible partially invariant solutions of the boundary layer or Prandtl equations.

We introduce a general framework for modelling the dynamics of the propensity to offend in a population of (possibly interacting) agents. We consider that each agent has an ‘honesty index’ which parameterizes his probability of abiding by the law. This probability also depends on a composite parameter associated to the attractiveness of the crime outcome and of the crime setting (the context which makes a crime more or less likely to occur, such as the presence or not of a guardian). Within this framework we explore some consequences of the working hypothesis that punishment has a deterrent effect, assuming that, after a criminal act, an agent's honesty index may increase if he is caught and decrease otherwise. We provide both analytical and numerical results. We show that in the space of parameters characterizing the probability of punishment, there are two ‘phases’: one corresponding to a population with a low crime rate and the other to a population with a large crime rate. We speculate on the possible existence of a self-organized state in which, due to the society reaction against crime activities, the population dynamics would be stabilized on the critical line, leading to a wide distribution of propensities to offend in the population. In view of empirical works on the causes of the recent evolution of crime rates in developed countries, we discuss how changes of socio-economic conditions may affect the model parameters, and hence the crime rate in the population. We suggest possible extensions of the model that will allow us to take into account more realistic features.

The study of radially symmetric motion is important for the theory of explosion waves. We construct rigorously self-similar entropy solutions to Riemann initial-boundary value problems for the radially symmetric relativistic Euler equations. We use the assumption of self-similarity to reduce the relativistic Euler equations to a system of nonlinear ordinary differential equations, from which we obtain detailed structures of solutions besides their existence. For the ultra-relativistic Euler equations, we also obtain the uniqueness of the self-similar entropy solution to the Riemann initial-boundary value problems.

The paper is concerned with heat and sweat transport in porous textile media with a non-local thermal radiation and phase change. The model, based on a combination of these classical heat transfer mechanisms (convection, conduction and radiation), is governed by a nonlinear, degenerate and strongly coupled parabolic system. The thermal radiative flow is described by a radiation transport equation and characterized by the thermal absorptivity and emissivity of fibre. A conservative boundary condition is introduced to describe the radiative heat flux interacting with environment. With the conservative boundary condition, we prove the global existence of positive/non-negative weak solutions of a nonlinear parabolic system. A typical clothing assembly with a polyester batting material sandwiched in two laminated covers is investigated numerically. Numerical results show that the contribution of radiative heat transfer is comparable with that of conduction/convection in the sweating system.

A new control strategy for a class of piecewise deterministic processes (PDP) is presented. In this class, PDP stochastic processes consist of ordinary differential equations that are subject to random switches corresponding to a discrete Markov process. The proposed strategy aims at controlling the probability density function (PDF) of the PDP. The optimal control formulation is based on the hyperbolic Fokker–Planck system that governs the time evolution of the PDF of the PDP and on tracking objectives of terminal configuration with a target PDF. The corresponding optimization problems are formulated as a sequence of open-loop hyperbolic optimality systems following a model predictive control framework. These systems are discretized by first-order schemes that guarantee positivity and conservativeness of the numerical PDF solution. The effectiveness of the proposed computational control framework is validated considering PDP with dichotomic noise.

We study an inverse boundary problem for the diffusion equation in ℝ2. Our motivation is that this equation is an approximation of the linear transport equation and describes light propagation in highly scattering media. The diffusion equation in the frequency domain is the nonself-adjoint elliptic equation div(D grad u) - (cμa + iω0) u = 0; ω0 ≠ 0, where D and μa are the diffusion and absorption coefficients. The inverse problem is the reconstruction of D and μa inside a bounded domain using only measurements at the boundary. In the two-dimensional case we prove that the Dirichlet-to-Neumann map, corresponding to any one positive frequency ω0, determines uniquely both the diffusion and the absorption coefficients, provided they are sufficiently slowly-varying. In the null-background case we estimate analytically how large these coefficients can be to guarantee uniqueness of the reconstruction.

If a body enters a viscous-inviscid fluid layer near a wall, then significant effects can be felt from the presence of incident vorticity, viscous forces and nonlinear forces. The focus here is on the response in the outer edge of such a wall layer. Nonlinear two-dimensional unsteady behaviour is examined through modelling, computation and analysis applied for a thin body travelling streamwise upstream or downstream or staying still relative to the wall. The wall layer with its balance between inviscid and viscous effects interacts freely with the body movement, causing relatively high magnitudes of pressure on top of the body and nonlinear responses in the gap between the body and the wall. The study finds explicit solutions for the motion of the body, separation of the flow arising near the wall and possible instabilities occurring over the length scale of any short body.

We study functions whose time-frequency content are concentrated in a compact region in phase space using time-frequency localization operators as a main tool. We obtain approximation inequalities for such functions using a finite linear combination of eigenfunctions of these operators, as well as a local Gabor system covering the region of interest. These would allow the construction of modified time-frequency dictionaries concentrated in the region.

We deal with a variational approach to the inverse crack problem, that is the detection and reconstruction of cracks, and other defects, inside a conducting body by performing boundary measurements of current and voltage type. We formulate such an inverse problem in a free-discontinuity problems framework and propose a novel method for the numerical reconstruction of the cracks by the available boundary data. The proposed method is amenable to numerical computations and it is justified by a convergence analysis, as the error on the measurements goes to zero. We further notice that we use the Γ-convergence approximation of the Mumford–Shah functional due to Ambrosio and Tortorelli as the required regularization term.

We consider an American call option and let C(S, T0) be the price of an option corresponding to asset price S at some time T0 prior to the expiration time TF . We analyze C(S, T0) in various asymptotic limits. These include situations where the interest and dividend rates are large or small, compared to the volatility of the asset. We also analyze the optimal exercise boundary for the option. We use perturbation methods to analyze either the PDE that C(S, T0) satisfies, or a nonlinear integral equation that is satisfied by the optimal exercise boundary.

In this paper, we propose an integral equation approach for pricing an American-style Parisian up-and-out call option under the Black–Scholes framework. The main difficulty of pricing this option lies in the determination of its optimal exercise price, which is a three-dimensional surface, instead of a two-dimensional (2-D) curve as is the case for a “one-touch” barrier option. In our approach, we first reduce the 3-D pricing problem to a 2-D one by using the “moving window” technique developed by Zhu and Chen (2013, Pricing Parisian and Parasian options analytically. Journal of Economic Dynamics and Control, 37(4): 875-896), then apply the Fourier sine transform to the 2-D problem to obtain two coupled integral equations in terms of two unknown quantities: the option price at the asset barrier and the optimal exercise price. Once the integral equations are solved numerically by using an iterative procedure, the calculation of the option price and the hedging parameters is straightforward from their integral representations. Our approach is validated by a comparison between our results and those of the trusted finite difference method. Numerical results are also provided to show some interesting features of the prices of American-style Parisian up-and-out call options and the behaviour of the associated optimal exercise boundaries.

This work studies functional difference equations of the second order with a potential belonging to a special class of meromorphic functions. The equations depend on a spectral parameter. Consideration of this type of equations is motivated by applications in diffraction theory and by construction of eigenfunctions for the Laplace operator in angular domains. In particular, such eigenfunctions describe eigenoscillations of acoustic waves in angular domains with ‘semitransparent’ boundary conditions. For negative values of the spectral parameter, we study essential and discrete spectrum of the equations and describe properties of the corresponding solutions. The study is based on the reduction of the functional difference equations to integral equations with a symmetric kernel. A sufficient condition is formulated for the potential that ensures existence of the discrete spectrum. The obtained results are applied for studying the behaviour of eigenfunctions for the Laplace operator in adjacent angular domains with the Robin-type boundary conditions on their common boundary. At infinity, the eigenfunctions vanish exponentially as was expected. However, the rate of such decay depends on the observation direction. In particular, in a vicinity of some directions, the regime of decay is switched from one to another and such asymptotic behaviour is described by a Fresnel-type integral.

We present a general procedure for recursively improving the invariance of a numerical integrator under a symmetry group. If h is a symmetry, we construct the adjoint method h−1h. In each time step we apply either the original method or the adjoint method, according to a prescription based on the Thue–Morse sequence. The outcome is a solution sequence which displays progressively smaller symmetry errors, to any desired order in the time-step. The method can also be used to force the solution to stay close to a desired submanifold of phase space, while retaining structural properties of the original method.

This paper proposes the use of a variational framework to model fluid wetting dynamics. The central problem of infinite energy dissipation for a moving contact line is dealt with explicitly rather than by introducing a specific microscopic mechanism which removes it. We analyze this modelling approach in the context of the quasi-steady limit, where contact line motion is slower than bulk relaxation. We find that global effects enter into Tanner-type laws which relate line velocity to apparent contact angle through the role that energy dissipation plays in the bulk of the fluid. A comparison is made to the dynamics of lubrication equations that include attractive and repulsive intermolecular interactions. A Galerkin-type approximation method is introduced which leads to reduced-dimensional dynamical descriptions. Computations are conducted using these low-dimensional approximations, and a substantial connection to lubrication equation dynamics is found.

A mushy region is assumed to consist of a fine mixture of two distinct phases separated by free boundaries. A method of multiple scales, with restrictions on the form of the microscopic free boundaries, is used to derive a macroscopic model for the mushy region. The final model depends both on the microscopic structure and on how the free-boundary temperature varies with curvature (Gibbs–Thomson effect), kinetic undercooling, or, for an alloy, composition.