In this paper we present a Kaleckian-type model of a business cycle based on a nonlinear delay differential equation. A numerical algorithm based on a decomposition scheme is implemented for the approximate solution of the model. The numerical results of the underlying equation show that the business cycle is stable.

The reflection-transmission properties of water waves obliquely incident upon a vortex sheet in water of finite depth are studied. The problem is reduced to that of solving two integral equation. An accurate Galerkin solution is obtained which supports the use of the “variational method” in water wave problems that has recently been questioned by Kirby and Dalyrmple.

Selective withdrawal of a two-layer fluid is considered. The fluid layers are weakly compressible, miscible and viscous and therefore flow rotationally. The lower, denser fluid flows with constant velocity out through one or more drain holes in the bottom of a rectangular tank. The drain is opened impulsively and the subsequent draw-down produces waves in the interface which travel outward to the edges of the tank and are reflected back with a $18{0}^{\circ } $ change of phase. The points on the interface that have the highest absolute gradient form regions of high vorticity in the tank, enabling mixing of the fluids. An inviscid linearized interface is computed and compared to contour plots of density for the viscous solution. The two agree closely at early times in the withdrawal process, but as time increases, nonlinear and viscous effects take over. The time at which the lighter fluid starts to flow out of the tank is dependent on the number of drains, their width, and the fluid flow rate and density, and is investigated here.

A class of discrepancy principles for the choice of parameters for the simplified regularization of ill-posed problems is proposed. This procedure does not require knowledge of the unknown solution, and if the smoothness of the unknown solution is known then the convergence rate obtained is optimal. The results of this paper include the Arcangeli's method considered by Groetsch and Guacaneme (1987) for which the convergence rate was not known and also of a result of Guacaneme (1988) for which there is a gap in the proof.

This paper presents a method for the inverse fractional matching problem. We show that the dual of this inverse problem can be transformed into the circulation flow problem on a directed bipartite graph which can be solved easily. We also give an algorithm to obtain the primal optimum solution of the inverse problem from its dual optimum solution by solving a shortest path problem. Furthermore, we generalize this method to solve the inverse symmetric transportation problem.

The equation modelling the evolution of a foam (a complex porous medium consisting of a set of gas bubbles surrounded by liquid films) is solved numerically. This model is described by the reaction–diffusion differential equation with a free boundary. Two numerical methods, namely the fixed-point and the averaging in time and forward differences in space (the Crank–Nicolson scheme), both in combination with Newton’s method, are proposed for solving the governing equations. The solution of Burgers’ equation is considered as a special case. We present the Crank–Nicolson scheme combined with Newton’s method for the reaction–diffusion differential equation appearing in a foam breaking phenomenon.

In 1984, Elliott and Stenger wrote a joint paper on the approximation of Hilbert transforms over analytic arcs. In the present paper we sharpen the previously obtained results of Elliott and Stenger, and we also obtain formulas for approximating Cauchy integrals over analytic arcs.

In this paper we propose a P1 finite element preconditioning using the so-called ‘hat-function’, to a collocation scheme constructed by quadratic splines for a 2nd-order separable elliptic operator and we show that the resulting preconditioning system of equations is well conditioned with the condition number independent of the number of unknowns.

This note considers an established reaction–diffusion model for a combustion system, in which there are competing endothermic and exothermic reaction pathways. A combustion front is assumed to move at constant speed through the medium. An asymptotic theory is presented for solid fuels in which material diffusion is ignored, and it allows a simple and complete analysis of the approximate system in the phase plane. Both the adiabatic and nonadiabatic cases are discussed.

Sufficent conditions for controllability of nonlinear neutral Volterra integrodifferential systems are established. Controllability of an infinite-delay neutral Volterra system is also considered.

Some corrections of error bounds obtained by Chatelin and Lemordant for the first three terms of the asymptotic case of a strong approximation are given. The error bounds for the approximations of order 2 in the Galerkin method are compared with the Rayleigh quotients constructed with the eigenvectors in the Sloan method. A numerical experiment is also carried out.

In a recent paper estimates of the solutions of two nonlinear differential equations were made by use of the hypercircie method. Here exact solutions are given which are compared with those estimates.

In this paper, we discuss MDP-the moment optimal problem for the first-passage model. A policy improvement iteration algorithm is given for finding the k-moment optimal stationary policy.

A new approach to jump diffusion is introduced, where the jump is treated as a vertical shift of the price (or volatility) function. This method is simpler than the previous methods and it is applied to the portfolio model with a stochastic volatility. Moreover, closed-form solutions for the optimal portfolio are obtained. The optimal closed-form solutions are derived when the value function is not smooth, without relying on the method of viscosity solutions.

This paper studies a system proposed by K. Gopalsamy and P. X. Weng to model a population growth with feedback control and time delays. Sufficient conditions are established under which the positive equilibrium of the system is globally attracting. The conjecture proposed by Gopalsamy and Weng is here confirmed and improved.

We compare six fixed-stepsize fourth-order numerical methods for a number of test problems described by a system of coupled Korteweg–de Vries equations. Particular attention is paid to the ability of these methods to preserve fixed points (solitary waves) and the invariants of the system, and establishing to what extent the conservation of integral invariants is indicative of the solution error for these methods.

We analyse a nonlinear hierarchical size-structured population model with time-dependent individual vital rates. The existence and uniqueness of nonnegative solutions to the model are shown via a comparison principle. Our investigation extends some results in the literature.

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.

An iteration scheme previously obtained by the author is used to study the dependence of criticality on initial data and the parameters in a combustion problem. Numerical results are presented for a slab, a cylinder and a sphere. These are compared with the results of previous workers.

We consider here pulsatile flow in circular tubes of varying cross-section with permeable walls. The fluid exchange across the wall is accounted for by prescribing the normal velocity of the fluid at the wall. A perturbation analysis has been carried out for low Reynolds number flows and for small amplitudes of oscillation. It has been observed that the magnitude of the wall shear stress and the pressure drop decrease as the suction velocity increases. Further, as the Reynolds number is increased, the magnitude of wall shear stress increases in the convergent portion and decreases in the divergent portion of a constricted tube.