The convergence and stability analysis of a simple explicit finite difference method is studied in this paper. Conditional convergence and stability theorems for this method are given. We have also proved that this scheme is stable in a much stronger sense.

In the continuous casting of steel, many problems, such as surface cracks in solidified steel and breakouts of molten steel from the bottom of moulds, frequently occur in practice. It is believed that the occurrence of these problems is directly related to the events in the mould, especially the transfer of heat from the strand surface across the lubricating mould powder and its interface with the mould wall to the mould cooling-water. However, as far as the authors are aware, there is no published work dealing with heat transfer across both the lubricating layer and the interface. Generally, a parameter representing the average overall heat transfer coefficient between the strand surface and the mould cooling-water is employed, instead of including the lubricating layer, the mould wall and their interface in the computation region. The existing treatment consequently does not permit analysis of some of the more important phenomena, such as the effect of mould powder properties and interface thermal contact resistance on the solidification of steel. In this paper, a novel finite element model is developed and the heat transfer across the interface between the lubricating layer and the mould wall is simulated by introducing a new type of element, referred to as the thermal contact element. The proposed model is used to investigate the effect of various casting parameters on heat transfer from the molten steel to the cooling-water. The results indicate that the thermal contact resistance between the mould wall and the mould powder is a key factor which dominates the thickness of the solidified steel shell and the heat extraction rate from the mould wall.

Para-Bose coherent states, defined as “displaced” ground states, are obtained using the differential operator representation for the annihilation operator.

A set of easily verifiable sufficient conditions are obtained for the existence of a globally asymptotically stable periodic solution in a Lotka-Volterra system with periodic coefficients.

A class of optimal control models which involve different weightings in the integrand of the objective function is considered. The motivation for considering this class of problems is that this type of objective function is used to account for eccentric movement in biomechanical models. The computation of these optimal control problems using control parametrization directly is difficult, firstly because of ill-conditioning, and secondly because the objective function is not differentiable. A method for smoothing the integrand is presented with convergence results. An example is computed which shows favourable computational improvements.

This paper develops a simple model for the containment of oil behind a boom in water. The flow of water beneath the oil is assumed two dimensional (horizontal and vertical) and perpendicular to the boom. We look for steady solutions and assume the oil is so viscous that the fluid velocity within the oil is zero. We are able to calculate what shape the oilslick will form and under which circumstances the boom will be successful (that is, no oil escapes under the boom) based on the predicted depth of the slick at the boom.

We discuss general, time-dependent, linear systems of second-order ordinary differential equations. A study is made of the similarities and discrepancies between the inverse problem of Lagrangian mechanics on the one hand, and the search for linear dynamical symmetries on the other hand.

The problem of passing from an L∞ function to a Wiener-Hopf factorization is considered. It is shown that a small L∞ perturbation which does not change the factorization indices will lead to small Lp (1 < p < ∞) perturbations in the Wiener-Hopf factors, but can lead to large L∞ perturbations, unless the derivatives are controlled during the perturbation.

It is well known that n-process, n-commodity (or square), models of productive single-product industries have positive solutions to their price and quantity systems if the rates of profit and growth lie in appropriate non-negative intervals. On the other hand, negative prices and quantities can occur in formal solutions of models of square, productive, multiple-product industries even when the rates of profit and growth are less than their respective maximum positive values. It is shown in this paper that these differences can be attributed to the presence in joint production of dominance, in either row or column versions. Results on positive solutions to the price (respectively, quantity) system are derived in terms of the absence of column (respectively, row) dominance of the net output matrix. As the concepts of row and column dominance are defined in terms of linear inequalities, the basic mathematical results to be applied are theorems of the alternative.

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.

The Zakharov-Shabat scattering transform is an exact solution technique for the nonlinear Schrödinger equation, which describes the time evolution of weakly nonlinear wave trains. Envelope soliton and uniform wave train solutions of the nonlinear Schrödinger equation are separable in scattering transform space. The scattering transform is a potential analysis and synthesis technique for natural wave trains. Discrete versions of the direct and inverse scattering transform are presented, together with proven algorithms for their numerical computation from typical ocean wave records. The consequences of discrete resolution are considered.

This paper is concerned with surface water waves produced by small oscillations of a thin vertical plate submerged in deep water. Green's integral theorem in the fluid region is used in a suitable manner to obtain the amplitude for the radiated waves at infinity. Particular results for roll and sway of the plate, and for a line source in the presence of a fixed vertical plate, are deduced.

The homotopy method is used to find all eigenpairs of a generalised symmetric eigenvalue problem Ax = λBx with positive definite B. The determination of n eigenpairs (x, λ) is reduced to curve tracing of n disjoint smooth curves in Rn × R × [0, 1]. The method might be attractive if A and B are sparse symmetric. In this paper it is shown that the method will work for almost all symmetric tridiagonal matrices A and B.

We study certain types of composite nonsmooth minimization problems by introducing a general smooth approximation method. Under various conditions we derive bounds on error estimates of the functional values of original objective function at an approximate optimal solution and at the optimal solution. Finally, we obtain second-order necessary optimality conditions for the smooth approximation prob lems using a recently introduced generalized second-order directional derivative.

We present a new derivation of the polynomial identities satisfied by certain matrices A with entries Aij (i, j = 1,…, n) from the universal enveloping algebra of a semi-simple Lie algebra. These polynomial identities are exhibited in a representation-independent way as p(A) = 0 where p(x) (herein called the characteristic polynomial of A) is a polynomial with coefficients from the centre Z of the universal enveloping algebra. The minimum polynomial identity m(A) = 0 of the matrix A over Z is also obtained and it is shown that p(x) and m(x) possess properties analogous to the characteristic and minimum polynomials respectively of a matrix with numerical entries. Acting on a representation (finite or infinite dimensional) admitting an infinitesimal character these polynomial identities may be expressed in a useful factored form. Our results include the characteristic identities of Bracken and Green [1] as a special case and show that these latter identities hold also in infinite dimensional representations.

In this paper all three-stage third order (explicit) Runge-Kutta-Nyström (R-K-N) methods for y″ = f (x, y, y′) are presented. While determining particular methods we require that when these methods are applied to the test equation: y″ − (α + β) y ′ + αβy = 0, the measure of the relative error F, introduced by Rutishauser [4], should not deteriorate in the case of equal eigenvalues (β → α). Further, we require that when these methods are applied to special differential equations y″ = f(x, y) they should possess either of the two properties: (P1) a method remains of order three but is two-stage, (P2) a method remains three-stage but attains order four. We present new R-K-N methods which are stabilized in the sense of Ruthishauser [4] and which possess the property (P1). (There does not exist any three-stage third order R-K-N method which is stabilized and which possesses the property (P2).)