We study a problem related to the spin-coating process in which a fluid coats a rotating surface. Our interest lies in the contact-line region for which we propose a simplified travelling wave approximation. We construct solutions to this problem by a shooting method that matches solution branches in the contact-line region and in the interior of the droplet. Furthermore, we prove uniqueness and qualitative properties of the solution connected to the fourth-order nature of the equation, such as a global maximum in the film height close to the contact line, elevated from the average height of the film.

The high Reynolds-number structure of the laminar, chaotic and turbulent attractors is investigated in a two-dimensional Kolmogorov flow. The laminar attractors include the families of multi-phased travelling waves and quasi-periodic standing waves both of which form the backbone of the transition to a turbulent flow. At leading order, each laminar attractor under study is obtained by solving the Euler equations on a manifold subject to the appropriate periodicity and symmetry conditions. The manifold is determined by a finite number of vorticity equations, these being required to suppress the secular terms at the next order. Our results show that, for the multi-phased travelling waves, the first phase velocity can be determined by an integral conservation law for kinetic energy and the subsequent phase velocities can be evaluated by a non-linear eigenvalue problem. The results also reveal that whereas viscosity determines the smallest scales and controls the amplitude of the flow, the inertial terms govern the shape and form of the flow. The comparison of our analytical predictions for evaluating the stable single-phased travelling wave with the direct numerical simulation of the Navier–Stokes equations has been undertaken, the agreement being excellent. For sufficiently high Reynolds number, after the bifurcation to chaotic flow, all of the multi-phased travelling waves and quasi-periodic standing waves become unstable non-wandering sets. Based on the above new findings for these unstable non-wandering sets and other travelling and standing waves of this kind in phase space, necessary conditions for the invariant manifolds of the chaotic and turbulent attractors are obtained, these necessary conditions being conjectured to be also sufficient.

We present a detailed derivation and analysis of a model consisting of seven coupled delay differential equations for louse-borne relapsing fever (LBRF), a disease transmitted from human to human by the body louse Pediculus humanus humanus. Delays model the latency stages of LBRF in humans and lice, which vary in duration from individual to individual, and are therefore modelled using distributed delays with relatively general kernels. A particular feature of the transmission of LBRF to a human is that it involves the death of the louse, usually by crushing which has the effect of releasing the infected body fluids of the dead louse onto the hosts skin. Careful attention is paid to this aspect. We obtain results on existence, positivity, boundedness, linear and nonlinear stability, and persistence. We also derive a basic reproduction number R0 for the model and discuss its dependence on the model parameters. Our analysis of the model suggests that effective louse control without crushing should be the best strategy for LBRF eradication. We conclude that simple measures and precautions should, in general, be sufficient to facilitate disease eradication.

This paper proposes the use of a generalized finite difference method for the numerical simulation of free surface single phase flows during mould filling process which are common in some industrial processes particularly in the area of metal casting. A novel and efficient idea for the computation of the normal vectors for free surface flows is introduced and presented for the first time. The incompressible Navier–Stokes equations are numerically solved by the well-known Chorin's projection method. After we showed the main ideas behind the meshless approach, some numerical results in two and three dimensions are presented corresponding to mould filling process simulation.

We study inversion of the spherical Radon transform with centres on a sphere (the data acquisition set). Such inversions are essential in various image reconstruction problems arising in medical, radar and sonar imaging. In the case of radially incomplete data, we show that the spherical Radon transform can be uniquely inverted recovering the image function in spherical shells. Our result is valid when the support of the image function is inside the data acquisition sphere, outside that sphere, as well as on both sides of the sphere. Furthermore, in addition to the uniqueness result, our method of proof provides reconstruction formulas for all those cases. We present a robust computational algorithm and demonstrate its accuracy and efficiency on several numerical examples.

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 introduce a multi-species chemotaxis type system admitting an arbitrarily large number of population species, all of which are attracted versus repelled by a single chemical substance. The production versus destruction rates of the chemotactic substance by the species is described by a probability measure. For such a model, we investigate the variational structures, in particular, we prove the existence of Lyapunov functionals, we establish duality properties as well as a logarithmic Hardy–Littlewood–Sobolev type inequality for the associated free energy. The latter inequality provides the optimal critical value for the conserved total population mass.

We present the numerical solution of two-point boundary value problems for a third-order linear PDE, representing a linear evolution in one space dimension. To our knowledge, the numerical evaluation of the solution so far could only be obtained by a time-stepping scheme, that must also take into account the issue, generically non-trivial, of the imposition of the boundary conditions. Instead of computing the evolution numerically, we evaluate the novel solution representation formula obtained by the unified transform, also known as Fokas transform. This representation involves complex line integrals, but in order to evaluate these integrals numerically, it is necessary to deform the integration contours using appropriate deformation mappings. We formulate a strategy to implement effectively this deformation, which allows us to obtain accurate numerical results.