We prove the existence and regularity of the solution of an initial boundary value problem for viscous incompressible non-homogeneous fluids, using a semi-Galerkin approximation and so-called compatibility conditions.

An expression for the impluse due to a vapour (cavitation) bubble is obtained in terms of an integral over a nearby boundary. Examples for a point source near a free surface, rigid boundary, inertial boundary and a fluid of different density are considered. It appears that the sign of the impluse determines the direction a cavitation bubble will migrate and the direction of the high speed liquid jet during the collapse phase. The theory may explain recent observations on buoyant bubbles near an interface between two fluids of different densities.

In this paper, we consider a class of combined optimal parameter selection and optimal control problems with general constraints. The first aim is to provide a unified approach to the numerical solution of this general class of optimisation problems by using the control parametrisation technique. This approach is supported by some convergence results. The second aim is to show that several different classes of optimal control problems can all be transformed into special cases of the problem considered in this paper. For illustration, four numerical examples are presented.

This note gives a necessary and sufficient condition that a compressible, isotropic elastic material should admit non-trivial states of finite anti-plane shear.

We study flows in physical networks with a potential function defined over the nodes and a flow defined over the arcs. The networks have the further property that the flow on an arc a is a given increasing function of the difference in potential between its initial and terminal node. An example is the equilibrium flow in water-supply pipe networks where the potential is the head and the Hazen-Williams rule gives the flow as a numerical factor ka times the head difference to a power s > 0 (and s ≅ 0.54). In the pipe-network problem with Hazen-Williams nonlinearities, having the same s > 0 on each arc, given the consumptions and supplies, the power usage is a decreasing function of the conductivity factors ka. There is also a converse to this. Approximately stated, it is: if every relationship between flow and head difference is not a power law, with the same s on each arc, given at least 6 pipes, one can arrange (lengths of) them so that Braess's paradox occurs, i.e. one can increase the conductivity of an individual pipe yet require more power to maintain the same consumptions.

It is shown that if intraspecific self-regulating negative feedback effects are strong enough such that a nontrivial steady state of a two species system is locally asymptotically stable, then time delays in the positive feedback as well as in other interspecific interactions cannot destabilise the system and hence delay induced instability leading to persistent oscillations is impossible whatever the magnitude of the time delays. A method is also proposed for an estimate of decay rate of perturbations.

In this paper, we determine the optimal controlled approximation rates from certain bivariate splines on regular meshes.

When an object is heated by microwaves, isolated regions of excessive heating can often occur. The present paper investigates such hotspots by both perturbation and numerical means. For quite normal materials, it is shown that small temperature anomalies can grow to form hotspots. Furthermore, such effects do not need to be associated with thermal runaway.

A number of Kuhn-Tucker type sufficient optimality criteria for a class of variational problems under weaker invexity assumptions are presented. As an application of these optimality results, various Mond-Weir type duality results are proved under a variety of generalised invexity assumptions. These results generalise many well-known duality results of variational problems and also give a dynamic analogue of certain corresponding (static) results relating to duality with generalised invexity in mathematical programming.

The method of the Lie theory of extended groups has recently been formulated for Hamiltonian mechanics in a manner which is consistent with the results obtained using the Newtonian equation of motion. Here the method is applied to the three-dimensional time-independent harmonic oscillator and to the classical Kepler problem. The expected constants of motion are obtained. Previously unobserved relations between generators and invariants are also noticed.

For a set function G on an atomless finite measure space (X, , m), we define the subgradient, conjugate set of and conjugate functional of G. It is proved that a minimization problem of set function G has an optimal solution if and only if the Lagrangian on × L1(X, , m)has a saddle point (Ω0, f0) such that

where f0 is an element of the conjugate set (for the definition, see the later context).

Using a parameterisation of general self-adjoint boundary conditions in terms of Lagrange planes we propose a scheme for factorising the matrix Schrödinger operator and hence construct a Darboux transformation, an interesting feature of which is that the matrix potential and boundary conditions are altered under the transformation. We present a solution of the inverse problem in the case of general boundary conditions using a Marchenko equation and discuss the specialisation to the case of a graph with trivial compact part, that is, with diagonal matrix potential.

We carry out a study of the peristaltic motion of an incompressible micropolar fluid in a two-dimensional channel. The effects of viscoelastic wall properties and micropolar fluid parameters on the flow are investigated using the equations of the fluid as well as of the deformable boundaries. A perturbation technique is used to determine flow characteristics. The velocity profile is presented and discussed briefly. We find the critical values of the parameters involving wall characteristics, which cause mean flow reversal.

Short-crested waves are defined as propagating surface gravity waves which are doubly-periodic in the horizontal plane. Linearly, the short-crested wave system we consider occurs when two progressive wavetrains of equal amplitude and frequency are propagating at an angle to each other.

Solutions are calculated via a computer-generated perturbation expansion in wave steepness. Harmonic resonance affects the solutions but Padé approximants can be used to estimate wave properties such as maximum wave steepness, frequency, kinetic energy and potential energy.

The force exerted by waves being reflected by a seawall is also calculated. Our results for the maximum depth-integrated onshore wave force in the standing wave limit are compared with experiment. The maximum force exerted on a seawail occurs for a steep wave in shallow water incident at an oblique angle. Results are given for this maximum force.

We extend the concept of V-pseudo-invexity and V-quasi-invexity of multi-objective programming to the case of nonsmooth multi-objective programming problems. The generalised subgradient Kuhn-Tucker conditions are shown to be sufficient for a weak minimum of a multi-objective programming problem under certain assumptions. Duality results are also obtained.

The problem of determining a square integrable function from both its modulus and the modulus of its Fourier transform is studied. It is shown that for a large class of real functions the function is uniquely determined from this data. We also construct fundamental subsets of functions that are not uniquely determined. In quantum mechanical language, bound states are uniquely determined by their position and momentum distributions but, in general, scattering states are not.

The singularity subtraction technique described by Kantorovich and Krylov in [11] is designed to reduce or overcome the effect of a weakly singular kernel in the numerical solution of integral equations. First, the equation is rearranged in such a way that the singularity of the kernel is at least partially cancelled by the smoothness of the solution, and then numerical integration is applied. We present convergence results and error bounds under general conditions on the nature of the singularity and the numerical integration procedure. Numerical examples demonstrate the benefit of the singularity subtraction technique.

Given a Fredhoim integral equation of the second kind, which is defined over a certain region ⊆ R2, we define and , two different numerical approximations to its solution, using the collocation and iterated collocation methods respectively. We describe without proof some known results concerning the general convergence properties of and when the kernel and solution of the integral equation are smooth. Then, we prove rigorously order of convergence estimates for and which are applicable in the practically siginificant case when is a rectangle, and the kernel of the integral equation is weakly singular. These estimates are illustrated by the numerical solution of a two dimensional weakly singular equation which arises in electrical engineering.

The classic problem, first treated by Taylor [18], of the dispersion of inert soluble matter in fluid flow continues to attract attention from researchers describing the approach to the asymptotic state [5, 17]. The present article considers some of the complications caused when the solute is chemically active, Dispersing chemically active solutes occur in diverse fields such as chromatography, chemical engineering and environmental fluid mechanics.

The asymptotic large-time analysis of Chatwin [5] is re-worked to handle the case of reactive solutes dispersing in parallel flow. Matching between moderate and large-time solutions requires consideration of the integral moments of the reactive contaminant could, and the Aris method of moments is therefore invoked and modified for reaction effects. The results are applied in detail to the outstanding paractical example—the chemical flow reactor (a device used to measure reaction rates for chemical reactions taking place between fluids). For this case, the paper provides a practical alternative to recent variable diffusion coefficient studies [6, 7, 15], and presents further results for the concentration distribution and for the limiting behaviour under weak and vigorous recactions at the boundary of the flow.

This paper examines the control of an interface between a suspension of sedimenting particles in liquid and a bed of dense-packed particles at the bottom of the suspension. The problem arises in the operation of continuous thickeners (e.g. in mineral processing) and is here mathematically described by a first order inhomogeneous partial differential equation for the concentration C(x, t) of particles. The controlled variable is the height H* of the bed, and the control variables are the volume fluxes injected at the feed level and removed at the bed. A strategy to control the interface is devised, and control is confirmed and demonstrated by a series of numerical experiments.