In this paper, multigrid methods for solving the biharmonic equation using some nonconforming plate elements are considered. An average algorithm is applied to define the transfer operator. A general analysis of convergence is given.

In this article, we use the method of Foias and Temam to show that the strong solutions of the time-dependent magnetohydrodynamics equations in a periodic domain are analytic in time with values in a Gevrey class of functions. As immediate corollaries we find that the solutions are analytic in Hr-norms and that the solutions become smooth immediately after the initial time.

In this paper, we consider denumerable state continuous time Markov decision processes with (possibly unbounded) transition and cost rates under average criterion. We present a set of conditions and prove the existence of both average cost optimal stationary policies and a solution of the average optimality equation under the conditions. The results in this paper are applied to an admission control queue model and controlled birth and death processes.

A model is presented for describing the propagation of a one-dimensional wave of permanent form in a compressible gas in a pipe. Energy is lost to the system through the walls of the pipe, but the combustion wave produces heat through an exothermic chemical reaction. The full set of equations for the model is reduced to a phase-plane system, and it is shown that, for small amplitude waves, a weakly non-linear analysis leads to a temperature profile that is a classical solitary wave. A novel shooting method is developed for the full non-linear problem, and this confirms and extends the solitary-wave solution, up to a value of the temperature amplitude at which the wave begins to develop a shock. The jump conditions across the shock are presented, and numerical integration is used to continue the solution for temperature amplitudes at which a shock is present. An example is given of the extreme situation in which the shock is so strong that all the fuel behind the shock is exhausted.

This paper is devoted to the study of the solvability of certain one-and multidimensional Abel-type integral equations involving the Gauss hypergeometric function as their kernels in the space of summable functions. The multidimensional equations are considered over certain pyramidal domains and the results obtained are used to present the multidimensional pyramidal analogues of generalized fractional calculus operators and their properties.

We consider the nonlinear evolution of a disturbance to a mixing layer, with the base profile given by u0(y) = tanh3y rather than the more usual tanh y, so that the first two derivatives of u0 vanish at y = 0. This flow admits three neutral modes, each of which is singular at the critical layer. Using a non-equilibrium nonlinear critical layer analysis, equations governing the evolution of the disturbance are derived and discussed. We find that the disturbance cannot exist on a linear basis, but that nonlinear effects inside the critical layer do permit the disturbance to exist. We also present results of a direct numerical simulation of this flow and briefly discuss the connection between the theory and the simulation.

In this paper we investigate the boundedness and asymptotic behaviour of the solutions of a class of homogeneous second-order difference equations with a single non-constant coefficient. These equations model, for example, the amplitude of oscillation of the weights on a discretely weighted vibrating string. We present several growth theorems. Two examples are also given.

Many tens of serious incidents involving reactors occur in the developed countries each year. The disaster at the chemical plant in Bhopal, India in 1984 was particularly notable where a thermal runaway process led to more than 3000 tragic fatalities from the cloud of extremely toxic methyl isocyanate that boiled out of a storage tank. This signalled the design of special types of chemical reactors to reduce the risk of thermal runaway by planning (at the design stage) integral safety and thermal stabilization mechanisms. The Endex CSTR (continuously stirred tank reactor) proposed by Gray and Ball [3] involves a reactor in two parts with heat exchange allowed between them. The two parts of the reactor operate side by side in tandem, such that the thermal runaway of one part is offset by an endothermic reaction in the other reactor—hence the term ‘endex’.

It is found that the adiabatic endex system has a large region of parameter space where the operation can be made safe. However adiabatic conditions rely on the continuous supply of reactants to the endothermic side of the reactor, for operation of the system. The risks involved are such that it is always safer to operate batch reactors in a non-adiabatic mode. Thus we consider the limiting case of the approach to adiabatic conditions where although the mathematics produces no oscillatory causes for instability, yet there is a narrow but significant area where the stable solution branch is lost and consequently a persistent and unexpected region of instability in what otherwise appears to be a simple CSTR system.

A numerical method for calculation of the generalized Chakalov-Popoviciu quadrature formulae of Radau and Lobatto type, using the results given for the generalized Chakalov-Popoviciu quadrature formula, is given. Numerical results are included. As an application we discuss the problem of approximating a function f on the finite interval I = [a, b] by a spline function of degree m and variable defects dv, with n (variable) knots, matching as many of the initial moments of f as possible. An analytic formula for the coefficients in the generalized Chakalov-Popoviciu quadrature formula is given.

It is observed (among other things) that a theorem on bilinear and bilateral generating functions, which was given recently in the predecessor of this Journal, does not hold true as stated and proved earlier. Several possible remedies and generalizations, which indeed are relevant to the present investigation of various other results on bilinear and bilateral generating functions, are also considered.

Steady state solutions for spontaneous thermal ignition in a unit sphere are considered. The multiplicity of unstable, intermediate, steady state, temperature profiles is calculated and shown for selected parameter values. The crossing of the temperature profiles corresponding to the unstable, intermediate, steady states is exhibited in a particular case and is proven in general using elementary ideas from analysis. Estimates of the location of crossing points are given.

A differential game model of a technological service industry is reformulated as an equivalent game over a function space by direct substitution of the solutions of the state equations. For this game, Nash equilibria are shown to exist under certain mild assumptions. A generalization is considered in which each firm has a choice of three different objective functions, which may reflect distinct management options in a technological service industry. Nash equilibria for the generalized version exist under similar mild assumptions.

Microwave heating of porous solid materials has received considerable attention in recent years because of its widespread use in industry. In this study, the microwave power absorption term is modelled as the product of an exponential temperature function with function that decays exponentially with distance. The latter describes the penetration of material by the microwaves.

We investigate the phenomena of multiplicity in class A geometries, paying particular attention to how the penetration function affects the behaviour of the system. We explain why the phase-plane techniques which have been used in the case when the penetration term is constant do not extend to non-constant penetration.

We show that a sequence of generalized eigenfunctions of a one-dimensional linear thermoelastic system with Dirichiet-Dirichlet boundary conditions forms a Riesz basis for the state Hilbert space. This develops a parallel result for the same system with Dirichlet-Neumann or Neumann-Dirichlet boundary conditions.

Oxygen index methods have been widely used to measure the flammability of polymeric materials and to investigate the effectiveness of fire-retardants. Using a dynamical systems framework we show how a limiting oxygen index can be identified with an appropriate bifurcation.

The effectiveness of fire-retardants in changing the limiting oxygen index is calculated by unfolding the bifurcation point with a suitable non-dimensionalised variable, which depends upon the mode of action of the additive. In order to use this procedure it is essential the model is non-dimensionalised so as to retain the variables of interest as distinct continuation parameters.