The nonlinear and weakly dispersive Serre equations contain higher-order dispersive terms. These include mixed spatial and temporal derivative flux terms which are difficult to handle numerically. These terms can be replaced by an alternative combination of equivalent temporal and spatial terms, so that the Serre equations can be written in conservation law form. The water depth and new conserved quantities are evolved using a second-order finite-volume scheme. The remaining primitive variable, the depth-averaged horizontal velocity, is obtained by solving a second-order elliptic equation using simple finite differences. Using an analytical solution and simulating the dam-break problem, the proposed scheme is shown to be accurate, simple to implement and stable for a range of problems, including flows with steep gradients. It is only slightly more computationally expensive than solving the shallow water wave equations.

Modelling fluid turbulence is perhaps one of the hardest problems in Applied Mathematics. In a recent paper, the author argued that the classical Navier–Stokes equation is not sufficient to describe the transition to turbulence, but that a Reiner–Rivlin type equation is needed instead. This is explored here for the simplest of all viscous fluid flows, the Couette flow, which is a simple shear between two moving plates. It is found that at high wavenumbers, the transition to unstable flow at the critical Reynolds number is characterized by a large number of eigenvalues of the Orr–Sommerfeld equation moving into the unstable zone essentially simultaneously. This would generate high-dimensional chaos almost immediately, and is a suggested mechanism for the transition to turbulence. Stability zones are illustrated for the flow, and a simple asymptotic solution confirms some of the features of these numerical results.

The maximal return and optimal leverage of a constant proportion debt obligation with finite termination and two boundaries are analysed by numerically solving Hamilton–Jacobi–Bellman equations. We discuss the probabilities of the asset value reaching the upper or lower bound under the optimal control and the optimal control problem with a time-varying boundary. Furthermore, we also analyse the relationship between the optimal return, the optimal policy and different parameters.

Large-scale low-pressure systems in the atmosphere are occasionally observed to possess Kelvin–Helmholtz fingers, and an example is shown in this paper. However, these structures are hundreds of kilometres long, so that they are necessarily affected strongly by nonlinearity. They are evidently unstable and are observed to dissipate after a few days.

A model for this phenomenon is presented here, based on the usual $f$-plane equations of meteorology, assuming an atmosphere governed by the ideal gas law. Large-amplitude perturbations are accounted for, by retaining the equations in their nonlinear forms, and these are then solved numerically using a spectral method. Finger formation is modelled as an initial perturbation to the $n$th Fourier mode, and the numerical results show that the fingers grow in time, developing structures that depend on the particular mode. Results are presented and discussed, and are also compared with the predictions of the ${\it\beta}$-plane theory, in which changes of the Coriolis acceleration with latitude are included. An idealized vortex in the northern hemisphere is considered, but the results are at least in qualitative agreement with an observation of such systems in the southern hemisphere.

The scaled boundary finite element method (SBFEM) is a semi-analytical computational method initially developed in the 1990s. It has been widely applied in the fields of solid mechanics, oceanic, geotechnical, hydraulic, electromagnetic and acoustic engineering problems. Most of the published work on SBFEM has focused on its theoretical development and practical applications, but, so far, no explicit discussion on the numerical stability and accuracy of its solution has been systematically documented. However, for a reliable engineering application, the inherent numerical problems associated with SBFEM solution procedures require thorough analysis in terms of its causes and the corresponding remedies. This study investigates the numerical performance of SBFEM with respect to matrix manipulation techniques and their properties. Some illustrative examples are given to identify reasons for possible numerical difficulties, and corresponding solution schemes are proposed to overcome these problems.

The most fundamental characteristic of a physical system can often be deduced from its behaviour under discrete symmetry transformations, such as time reversal, parity and chirality. Here, we review some of the basic symmetry properties of the relativistic quantum theories for free electrons in ($2+1$)- and ($1+1$)-dimensional spacetime. Additional flavour degrees of freedom are necessary to properly define symmetry operations in ($2+1$) dimensions, and are generally present in physical realizations of such systems, for example in single sheets of graphite. We find that there exist two possibilities for defining any flavour-coupling discrete symmetry operation of the two-flavour ($2+1$)-dimensional Dirac theory. Some physical implications of this previously unnoticed duplicity are discussed.

Safety issues for the use of products containing nanoparticles need to be considered, since these nanoparticles may break through human skin to damage cells. In this paper, applied mathematical techniques are used to model the penetration of a spherical gold nanoparticle into an assumed circular hole in a lipid bilayer. The 6–12 Lennard-Jones potential is employed, and the total molecular interaction energy is obtained using the continuous approximation. Nanoparticles of three different radii, namely, 10, 15 and 20 Å, are studied, which are initiated at rest, confined to the axis of the hole. A similar behaviour for these three cases is observed. The critical hole radii at which these nanoparticles enter the bilayer are 12.65, 17.62 and 22.60 Å, respectively. Further, once the hole radii become larger than 20.79, 23.14 and 27.02 Å, respectively, the gold nanoparticles tend to remain at the mid-plane of the bilayer, and do not pass through the bilayer.

We study a cell growth model with a division function that models cells which divide only after they have reached a certain minimum size. In contrast to the cases studied in the literature, the determination of the steady size distribution entails an eigenvalue that is not known explicitly, but is defined through a continuity condition. We show that there is a steady size distribution solution to this problem.

British put options are financial derivatives with an early exercise feature whereby on payoff, the holder receives the best prediction of the European put payoff under the hypothesis that the true drift of the stock price is equal to a contract drift. In this paper, we derive simple analytic approximations for the optimal exercise boundary and the option valuation, valid for short expiry times – which is a common feature of most options traded in the market. Empirical results show that the approximations provide accurate results for expiries of at least up to two months.

The steady, axisymmetric flow induced by a point sink (or source) submerged in an inviscid fluid of infinite depth is computed and the resulting deformation of the free surface is obtained. The effect of surface tension on the free surface is determined and is the new component of this work. The maximum Froude numbers at which steady solutions exist are computed. It is found that the determining factor in reaching the critical flow changes as more surface tension is included. If there is zero or a very small amount of surface tension, the limiting factor appears to be the formation of small wavelets on the free surface; but, as the surface tension increases, this is replaced by a tendency for the lowest point on the free surface to descend sharply as the Froude number is increased.

We investigate the dynamics of a susceptible infected recovered (SIR) epidemic model on small networks with different topologies, as a stepping stone to determining how the structure of a contact network impacts the transmission of infection through a population. For an SIR model on a network of $N$ nodes, there are $3^{N}$ configurations that the network can be in. To simplify the analysis, we group the states together based on the number of nodes in each infection state and the symmetries of the network. We derive analytical expressions for the final epidemic size of an SIR model on small networks composed of three or four nodes with different topological structures. Differential equations which describe the transition of the network between states are also derived and solved numerically to confirm our analysis. A stochastic SIR model is numerically simulated on each of the small networks with the same initial conditions and infection parameters to confirm our results independently. We show that the structure of the network, degree of the initial infectious node, number of initial infectious nodes and the transmission rate all significantly impact the final epidemic size of an SIR model on small networks.

We investigate the mechanics of a nano logic gate, comprising a metallofullerene which is located inside a square-shaped single-walled carbon nanotorus involving non-metallic, single-walled carbon nanotubes with perfect nanotoroidal corners. These are highly novel and speculative nanodevices whose construction, no doubt, involves many technical challenges. The energy for the system is obtained from the 6–12 Lennard-Jones potential with the continuous approximation. Our approach shows that there is not much difference between the energy when the metallofullerene is located in the tubes compared to when it is at the corners, and therefore the metallofullerene may be controlled by a small voltage. By applying two voltage inputs to produce external electric fields, one for the left–right motion and the other for the top–bottom motion, the metallofullerene can be moved to one of the four corners. Assuming that at the four corners there are charge detectors, the proposed device can be designed as a logic gate with different signals corresponding to particular gates.

The main purpose of this paper is to present a novel analytical approach for pricing discretely sampled gamma swaps, defined in terms of weighted variance swaps of the underlying asset, based on Heston’s two-factor stochastic volatility model. The closed-form formula obtained in this paper is in a much simpler form than those proposed in the literature, which substantially reduces the computational burden and can be implemented efficiently. The solution procedure presented in this paper can be adopted to derive closed-form solutions for pricing various types of weighted variance swaps, such as self-quantoed variance and entropy swaps. Most interestingly, we discuss the validity of the current solutions in the parameter space, and provide market practitioners with some remarks for trading these types of weighted variance swaps.

The lower part of the spectrum of the Helmholtz equation for a heterogeneous system in a finite region in $d$ dimensions, where the solutions to the corresponding homogeneous system are known, can be systematically approximated by means of iterative methods. These methods only require the specification of an arbitrary ansatz and converge to the desired solution, regardless of the strength of the inhomogeneities, provided the ansatz has a finite overlap with it. In this paper, different boundary conditions at the borders of the domain are assumed, and some applications are used to illustrate the methods.

We derive an analytical solution for the value of Parisian up-and-in calls by using the “moving window” technique for pricing European-style Parisian up-and-out calls. Our pricing formula can be applied to both European-style and American-style Parisian up-and-in calls, due to the fact that with an “in” barrier, the option holder cannot do or decide on anything before the option is activated, and once the option is activated it is just a plain vanilla call, which could be of American style or European style.

We investigate two routing problems that arise when order pickers traverse an aisle in a warehouse. The routing problems can be viewed as Euclidean travelling salesman problems with points on two parallel lines. We show that if the order picker traverses only a section of the aisle and then returns, then an optimal solution can be found in linear time, and if the order picker traverses the entire aisle, then an optimal solution can be found in quadratic time. Moreover, we show how to approximate the routing cost in linear time by computing a minimum spanning tree for the points on the parallel lines.

We consider a coupled, logistic predator–prey system with delay. Mainly, by choosing the delay time ${\it\tau}$ as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay time ${\it\tau}$ passes some critical values. Based on the normal-form theory and the centre manifold theorem, we also derive formulae to obtain the direction, stability and the period of the bifurcating periodic solution at critical values of ${\it\tau}$. Finally, numerical simulations are investigated to support our theoretical results.

Nanospace governs the dynamics of physical, chemical, material and biological systems, and the facility to model it with analytical formulae provides an essential tool to address some of the worlds’ key problems such as gas purification, separation and storage. This paper aims to provide some analytical models to exploit building blocks representing various geometric shapes that describe nanostructures. In order to formulate the various building blocks, we use the continuous approximation which assumes a uniform distribution of atoms on their surfaces. We then calculate the potential energy of the van der Waals interaction between an atom and the structure to evaluate the location of the atom where the potential energy is at its minimum. We provide applications of the analytical models for some real structures where more than one type of building block is required.