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 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.

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.

In a glass furnace solid batches of material are fed into a chamber and radiation heating applied. An individual batch is melted over the course of several minutes to form molten glass. A travelling front within the batch designates the progress of the melting, a process characterized by multiple radiation reflections. This results in an effective conductivity within the melting zone that is significantly larger than that in the unmelted batch. Approximations based on these disparate conductivities enable accurate explicit expressions for the almost constant melting front speed and the associated temperature profile to be derived. Our results compare favourably with existing numerical simulations of the process, with the advantage of being both analytic and relatively simple. These predictions may be useful in suggesting how a furnace might be most effectively controlled under varying batch conditions, as well as ensuring the quality of the glass sheets produced.

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.

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.

A general condition is provided from which an error bound can be concluded for approximations of queueing networks which are based on modifications of the transition and state space structure. This condition relies upon Markov reward theory and can be verified inductively in concrete situations. The results are illustrated by estimating the accuracy of a simple throughput bound for a closed queueing network with alternate routing and a large finite source input. An explicit error bound for this example is derived which is of order M—1, where M is the number of sources.

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 give a simple and transparent proof for the square-root method of solving the continuous-time algebraic Riccati equation. We examine some benefits of combining the square-root method with other solution methods. The iterative square-root method is also discussed. Finally, paradigm numerical examples are given to compare the square-root method with the Schur method.

Sufficient conditions are established for all solutions of the linear system

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

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.

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.

A sharp L2 inequality of Ostrowski type is established, which provides a generalization of some previous results and gives some other interesting results as special cases. Applications in numerical integration are also given.

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 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.

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.

In a classical paper, E. R. Love considered a certain function defined by a singular integral which is harmonic outside a circular disk. Love's objective was to derive a simple integral equation whose solution leads to a useful formula for the capacitance of the condenser consisting of two parallel circular plates. We close a gap in Love's derivation by finding a new nonsingular representation of Love's singular integral which permits one to draw the required conclusions about its boundary values and thereby establishes the correctness of Love's expression for the capacitance.

Entirely elementary methods are employed to determine explicit formulae for the coefficients of commuting ordinary differential operators of orders six and nine which correspond to an elliptic curve. These formulae come from solving the nonlinear ordinary differential equations which are equivalent to the commutativity condition. Most solutions turn out to be rational expressions in one or two arbitrary functions and their derivatives. The corresponding Burchnall-Chaundy curves are computed.

In this paper we derive extremality and comparison results for explicit and implicit initial and boundary value problems of first-order differential equations. Both the differential equations and the boundary conditions may involve discontinuities.

In this paper, we consider an insurance company whose surplus (reserve) is modeled by a jump diffusion risk process. The insurance company can invest part of its surplus in n risky assets and purchase proportional reinsurance for claims. Our main goal is to find an optimal investment and proportional reinsurance policy which minimizes the ruin probability. We apply stochastic control theory to solve this problem. We obtain the closed form expression for the minimal ruin probability, optimal investment and proportional reinsurance policy. We find that the minimal ruin probability satisfies the Lundberg equality. We also investigate the effects of the diffusion volatility parameter, the market price of risk and the correlation coefficient on the minimal ruin probability, optimal investment and proportional reinsurance policy through numerical calculations.