The motion of a two-dimensional bubble rising at a constant velocity U in an inclined tube of width H is considered. The bubble extends downwards without limit, and is bounded on the right by a wall of the tube, and on the left by a free surface. The same flow configuration describes also a jet emerging from a nozzle and falling down along an inclined wall. The acceleration of gravity g and the surface tension T are included in the free surface condition. The problem is characterized by the Froude number the angle β between the left wall and the horizontal, and the angle γ between the free surface and the right wall at the separation point. Numerical solutions are obtained via series truncation for all values of 0 < β < π. The results extend previous calculations of Vanden-Broeck [12–14] for β = π/2 and of Couët and Strumolo [3] for 0 < β < π/2. It is found that the behavior of the solutions depends on whether 0 < β 2π/3 or 2π/3 ≤ β < π. When T = 0, it is shown that there is a critical value F of Froude number for each 0 < β 2π/3 such that solutions with γ = 0, π/3 and π - β occur for F > Fc F = Fc and F < Fc respectively, and that all solutions are characterized by γ = 0 for 2π/3 ≤ β < π. When a small amount of surface tension T is included in the free surface condition, it is found that for each 0 < β < π there exists an infinite discrete set of values of F for which γ = π - β. A particular value F* of the Froude number for which T = 0 and γ = π - β is selected by taking the limit as T approaches zero. The numerical values of F* and the corresponding free surface profiles are found to be in good agreement with experimental data for bubbles rising in an inclined tube when 0 < β < π/2.

This note introduces quasi-local-balance for discrete-time Markov chains with absorbing states. From quasi-local-balance product-form quasi-stationary distributions are derived by analogy with product-form stationary distributions for Markov chains that satisfy local balance.

The population dynamics for the replicator equation has been well studied in continuous time, but there is less work that explicitly considers the evolution in discrete time. The discrete-time dynamics can often be justified indirectly by establishing the relevant evolutionary dynamics for the corresponding continuous-time system, and then appealing to an appropriate approximation property. In this paper we study the discrete-time system directly, and establish basic stability results for the evolution of a population defined by a positive definite system matrix, where the population is disrupted by random perturbations to the genotype distribution either through migration or mutation, in each successive generation.

Here we consider a particular class of stochastic geometric programs in which the randomness occurs in the decision variables. Specifically we analyse a program in which we specify a joint normal probability for the dicision variables and require the constraint set to be satisfied in the chance constrained mode. A numerical example is given to illustrate the approach.

Nonlocal calculus is often overlooked in the mathematics curriculum. In this paper we present an interesting new class of nonlocal problems that arise from modelling the growth and division of cells, especially cancer cells, as they progress through the cell cycle. The cellular biomass is assumed to be unstructured in size or position, and its evolution governed by a time-dependent system of ordinary differential equations with multiple time delays. The system is linear and taken to be autonomous. As a result, it is possible to reduce its solution to that of a nonlinear matrix eigenvalue problem. This method is illustrated by considering case studies, including a model of the cell cycle developed recently by Simms, Bean and Koeber. The paper concludes by explaining how asymptotic expressions for the distribution of cells across the compartments can be determined and used to assess the impact of different chemotherapeutic agents.

We continue our study of the adaptation from spherical to doubly periodic slot domains of the poloidal-toroidal representation of vector fields. Building on the successful construction of an orthogonal quinquepartite decomposition of doubly periodic vector fields of arbitrary divergence with integral representations for the projections of known vector fields and equivalent scalar representations for unknown vector fields (Part 1), we now present a decomposition of vector field equations into an equivalent set of scalar field equations. The Stokes equations for slow viscous incompressible fluid flow in an arbitrary force field are treated as an example, and for them the application of the decomposition uncouples the conservation of momentum equation from the conservation of mass constraint. The resulting scalar equations are then solved by elementary methods. The extension to generalised Stokes equations resulting from the application of various time discretisation schemes to the Navier-Stokes equations is also solved.

We would like to present a method to compute the incompatibility operator in any system of curvilinear coordinates (components). The procedure is independent of the metric in the sense that the expression can be obtained by means of the basis vectors only, which are first defined as normal or tangential to the domain boundary, and then extended to the whole domain. It is an intrinsic method, to some extent, since the chosen curvilinear system depends solely on the geometry of the domain boundary. As an application, the in-extenso expression of incompatibility in a spherical system is given.

Two-dimensional free-surface flows produced by a submerged source in a fluid of infinite depth are considered. It is assumed that the point on the free surface just above the source is a stagnation point and that the fluid outside two shear layers is at rest. The free-surface profile and the shape of the shear layers are determined numerically by using a series-truncation method. It is shown that there is a solution for each value of the Froude number F > 0. When F tends to infinity, the flow also describes a thin jet impinging in a fluid at rest.

A general theory is given for autonomous perturbations of non-linear autonomous second order oscillators. It is found using a multiple scales method. A central part of it requires computation of Fourier coefficients for representation of the underlying oscillations, and these coefficients are found as convergent expansions in a suitable parameter.

We propose a Legendre–Laguerre spectral approximation to price the European and double barrier options in the time-fractional framework. By choosing an appropriate basis function, the spectral discretization is used for the approximation of the spatial derivatives of the time-fractional Black–Scholes equation. For the time discretization, we consider the popular $L1$ finite difference approximation, which converges with order $\mathcal {O}((\Delta \tau )^{2-\alpha })$ for functions which are twice continuously differentiable. However, when using the $L1$ scheme for problems with nonsmooth initial data, only the first-order accuracy in time is achieved. This low-order accuracy is also observed when solving the time-fractional Black–Scholes European and barrier option pricing problems for which the payoffs are all nonsmooth. To increase the temporal convergence rate, we therefore consider a Richardson extrapolation method, which when combined with the spectral approximation in space, exhibits higher order convergence such that high accuracies over the whole discretization grid are obtained. Compared with the traditional finite difference scheme, numerical examples clearly indicate that the spectral approximation converges exponentially over a small number of grid points. Also, as demonstrated, such high accuracies can be achieved in much fewer time steps using the extrapolation approach.

Stokes’ axisymmetrical translational motion of a slip sphere, located anywhere on the diameter of a virtual spherical fluid ‘cell’, is investigated. The fluid is micropolar and flows are parallel to the line connecting the two centres. An infinite-series solution is presented for the stream function, pressure field, vorticity, microrotation component, shear stress and couple stress of the flow. Basset-type slip boundary conditions on the sphere surface are used for velocity and microrotation. The Happel and Kuwabara boundary conditions are used on the fictitious surface of the cell model. Numerical results for the normalized drag force acting on the sphere are obtained with excellent convergence for various values of the volume fraction, the relative distance between the centre of the sphere and the virtual envelope, the vortex viscosity parameter and the slip coefficients of the sphere. In the special case when the spherical particle is in the concentric position with the cell surface, the numerical values of the normalized drag force agree with the available values in the literature.

The homotopy analysis method (HAM) is applied to a nonlinear ordinary differential equation (ODE) emerging from a closure model of the von Kármán–Howarth equation which models the decay of isotropic turbulence. In the infinite Reynolds number limit, the von Kármán–Howarth equation admits a symmetry reduction leading to the aforementioned one-parameter ODE. Though the latter equation is not fully integrable, it can be integrated once for two particular parameter values and, for one of these values, the relevant boundary conditions can also be satisfied. The key result of this paper is that for the generic case, HAM is employed such that solutions for arbitrary parameter values are derived. We obtain explicit analytical solutions by recursive formulas with constant coefficients, using some transformations of variables in order to express the solutions in polynomial form. We also prove that the Loitsyansky invariant is a conservation law for the asymptotic form of the original equation.

It has been known for a long time that the equivariant $2+1$ wave map into the $2$-sphere blows up if the initial data are chosen appropriately. Here, we present numerical evidence for the stability of the blow-up phenomenon under explicit violations of equivariance.

We consider a theoretical model for the flow of Newtonian fluid through a long flexible-walled channel which is formed from four compliant and rigid compartments arranged alternately in series. We drive the flow using a fixed upstream flux and derive a spatially one-dimensional model using a flow profile assumption. The compliant compartments of the channel are assumed subject to a large external pressure, so the system admits a highly collapsed steady state. Using both a global (linear) stability eigensolver and fully nonlinear simulations, we show that these highly collapsed steady states admit a primary global oscillatory instability similar to observations in a single channel. We also show that in some regions of the parameter space the system admits a secondary mode of instability which can interact with the primary mode and lead to significant changes in the structure of the neutral stability curves. Finally, we apply the predictions of this model to the flow of blood through the central retinal vein and examine the conditions required for the onset of self-excited oscillation. We show that the neutral stability curve of the primary mode of instability discussed above agrees well with canine experimental measurements of the onset of retinal venous pulsation, although there is a large discrepancy in the oscillation frequency.

The Maxwell-Dirac equations model an electron in an electromagnetic field. The two equations are coupled via the Dirac current which acts as a source in the Maxwell equation, resulting in a nonlinear system of partial differential equations (PDE's). Well-behaved solutions, within reasonable Sobolev spaces, have been shown to exist globally as recently as 1997 [12]. Exact solutions have not been found—except in some simple cases.

We have shown analytically in [6, 18] that any spherical solution surrounds a Coulomb field and any cylindrical solution surrounds a central charged wire; and in [3] and [19] that in any stationary case, the surrounding electron field must be equal and opposite to the central (external) field. Here we extend the numerical solutions in [6] to a family of orbits all of which are well-behaved numerical solutions satisfying the analytic results in [6] and [11]. These solutions die off exponentially with increasing distance from the central axis of symmetry. The results in [18] can be extended in the same way. A third case is included, with dependence on z only yielding a related fourth-order ordinary differential equation (ODE) [3].

Some optimal choices for a parameter of the Dai–Liao conjugate gradient method are proposed by conducting matrix analyses of the method. More precisely, first the $\ell _{1}$ and $\ell _{\infty }$ norm condition numbers of the search direction matrix are minimized, yielding two adaptive choices for the Dai–Liao parameter. Then we show that a recent formula for computing this parameter which guarantees the descent property can be considered as a minimizer of the spectral condition number as well as the well-known measure function for a symmetrized version of the search direction matrix. Brief convergence analyses are also carried out. Finally, some numerical experiments on a set of test problems related to constrained and unconstrained testing environment, are conducted using a well-known performance profile.

Methods for integral equations are used to derive upper and lower pointwise bounds for the solution of a nonlinear boundary value problem arising in the steady-state finite cable model of cell membranes. Test calculations are performed to illustrate the results and the accuracy achieved is significantly better than that obtained previously by other 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.

Sufficient conditions are established for all solutions of the linear system

to be oscillatory, where qij ∈(−∞, ∞), τij ∈ (0, ∞), i, j = 1, 2, …, n.

The paper uses the factorisation method to discuss solutions of period three for the difference equation

which has been proposed as a simple mathematical model for the effect of frequency dependent selection in genetics. Numerical values are obtained for the critical values of a at which solutions of period three first appear. In addition, the interval in which stable solutions are possible has been determined. Exact solutions are given for the case a = 4 and these have been used to check the results.