In this paper, finite phase semi-Markov processes are introduced. By introducing variables and a simple transformation, every finite phase semi-Markov process can be transformed to a finite Markov chain which is called its associated Markov chain. A consequence of this is that every phase semi-Markovian switching system may be equivalently expressed as its associated Markovian switching system. Existing results for Markovian switching systems may then be applied to analyze phase semi-Markovian switching systems. In the following, we obtain asymptotic stability for the distribution of nonlinear stochastic systems with semi-Markovian switching. The results can also be extended to general semi-Markovian switching systems. Finally, an example is given to illustrate the feasibility and effectiveness of the theoretical results obtained.

Sufficient conditions are obtained for the existence of a globally attracting positive periodic solution of the mutualism model

where ri, Ki, αi ∈ C(R, R+) and αi > Ki, i = 1, 2, τi, σi ∈ C(R, R+), i = 1, 2, and ri, Ki, αi, τi, σi (i = 1, 2) and functions of period ω > 0.

The Kelvin–Helmholtz flow is a shearing instability that occurs at the interface between two fluids moving with different speeds. Here, the two fluids are each of finite depth, but are highly viscous. Consequently, their motion is caused by the horizontal speeds of the two walls above and below each fluid layer. The motion of the fluids is assumed to be governed by the Stokes approximation for slow viscous flow, and the fluid motion is thus responsible for movement of the interface between them. A linearized solution is presented, from which the decay rate and the group speed of the wave system may be obtained. The nonlinear equations are solved using a novel spectral representation for the streamfunctions in each of the two fluid layers, and the exact boundary conditions are applied at the unknown interface location. Results are presented for the wave profiles, and the behaviour of the curvature of the interface is discussed. These results are compared to the Boussinesq–Stokes approximation which is also solved by a novel spectral technique, and agreement between the results supports the numerical calculations.

In a paper by Teo and Jennings, a constraint transcription is used together with the concept of control parametrisation to devise a computational algorithm for solving a class of optimal control problems involving terminal and continuous state constraints of inequality type. The aim of this paper is to extend the results to a more general class of constrained optimal control problems, where the problem is also subject to terminal equality state constraints. For illustration, a numerical example is included.

In this paper, we present a method for the construction of a robust observer-based H∞ controller for an uncertain time-delay system. Cases of both single and multiple delays are considered. The parameter uncertainties are time-varying and norm-bounded. Observer and controller are designed to be such that the uncertain system is stable and a disturbance attenuation is guaranteed, regardless of the uncertainties. It has been shown that the above problem can be solved in terms of two linear matrix inequalities (LMIs). Finally, an illustrative example is given to show the effectiveness of the proposed techniques.

The properties of static spherically symmetric black holes, which carry electric and magnetic charges, and which are coupled to the dilaton in the presence of a cosmological constant, A, are reviewed.

Withdrawal flow through a point sink on the bottom of a fluid of finite depth is considered. The fluid is at rest at infinity, and a stagnation point is present at the free surface, directly above the point sink. Numerical solutions are computed by means of the method of fundamental solutions, and it is observed that flows of this type are apparently possible for Froude number less than about 1.5. Relationships to previous work are discussed.

We consider several methods for solving the linear equations arising from finite difference discretizations of the Stokes equations. The two best methods, one presented here for the first time, apparently, and a second, presented by Bramble and Pasciak, are shown to have computational effort that grows slowly with the number of grid points. The methods work with second-order accurate discretizations. Computational results are shown for both the Stokes equations and incompressible Navier-Stokes equations at low Reynolds number.

Modified versions of the Euler midpoint formula are given for functions whose derivatives are either functions of bounded variation, Lipschitzian functions or functions in Lp-spaces. The results are applied to quadrature formulae.

In this paper various two-dimensional motions are determined for waves in a stratified region of infinite total depth with a free surface containing two superposed liquids, allowing for the effects of surface and interfacial tension. The fundamental set of wave-source potentials for the two layers is used to construct the set of slope potentials that produce discontinuous free-surface and interface slopes. The latter potentials are then utilized to obtain the potentials for waves due to both heaving vertical plates and incident progressive waves against a vertical wall. The underlying assumption of small time-harmonic motion pertains, described by a pair of velocity potentials for the two layers satisfying coupled linearized boundary-value problems, and all solutions are obtained in terms of their matching basic solutions. The technique for applying Green's theorem in the two layers is developed for use with the wave-source potentials, which themselves are found to obey a generalised reciprocity principle. Familiar results for a single liquid of infinite depth are hereby extended, but the new feature emerges of there being two types of progressive waves in all solutions. For ease of presentation the solutions are obtained for a particular relationship between surface and interfacial tension.

It has been known for some time that if a certain non-degeneracy condition is satisfied then the successive solution estimates x(r) produced by barrier function techniques lie on a smooth trajectory. Accordingly, extrapolation methods can be used to calculate x(0). In this paper we analyse the situation further treating the special case of the log barrier function. If the non-degeneracy assumption is not satisfied then the approach to x(0) is like r½ rather than like r which would be expected in the non-degenerate case. A measure of sensitivity is introduced which becomes large when the non- degeneracy assumption is close to violation, and it is shown that this sensitivity measure is related to the growth of dix/dri with respect to i for fixed r small enough on the solution trajectory. With this information it is possible to analyse the extrapolation procedure and to predict the number of stages of extrapolation which are useful.

A general analytic approach is proposed for nonlinear eigenvalue problems governed by nonlinear differential equations with variable coefficients. This approach is based on the homotopy analysis method for strongly nonlinear problems. As an example, a beam with arbitrary variable cross section acted on by a compressive axial load is used to show its validity and effectiveness. This approach provides us with great freedom to transfer the original nonlinear buckling equation with variable coefficients into an infinite number of linear differential equations with constant coefficients that are much easier to solve. More importantly, it provides us with a convenient way to guarantee the convergence of solution series. As an example, the beam displacement and the critical buckling load can be obtained for arbitrary variable cross sections. The influence of nonuniformity of moment of inertia is investigated in detail and the optimal distributions of moment of inertia are studied. It is found that the critical buckling load of a beam with the optimal distribution of moment of inertia can be approximately 21–22% larger than that of a uniform beam with the same average moment of inertia. Mathematically, this approach is rather general and thus can be used to solve many other linear/nonlinear differential equations with variable coefficients.

This paper deals the global existence and blow-up properties of the following non-Newton polytropic filtration system with nonlocal source, Under appropriate hypotheses, we prove that the solution either exists globally or blows up in finite time depending on the initial data and the relations between αβ and mn(p−1)(q−1). In the special case, α=n(q−1), β=m(p−1), we also give a criteria for the solution to exist globally or blow up in finite time, which depends on a,b and ζ(x),ϑ(x) as defined in our main results.

A boundary integral procedure for the solution of an important class of problems in anisotropic elasticity is outlined. Specific numerical examples are considered in order to provide a comparison with the standard boundary integral method.

From a knowledge of the eigenvalue spectrum of the Laplacian on a domain, one may extract information on the geometry and boundary conditions by analysing the asymptotic expansion of a spectral function. Explicit calculations are performed for isosceles right-angle triangles with Dirichlet or Neumann boundary conditions, yielding in particular the corner angle terms. In three dimensions, right prisms are dealt with, including the solid vertex terms.

A rapid spherical harmonic calculation method is used for the design of Nuclear Magnetic Resonance shim coils. The aim is to design each shim such that it generates a field described purely by a single spherical harmonic. By applying simulated annealing techniques, coil arrangements are produced through the optimal positioning of current-carrying circular arc conductors of rectangular cross-section. This involves minimizing the undesirable harmonics in relation to a target harmonic. The design method is flexible enough to be applied for the production of coil arrangements that generate fields consisting significantly of either zonal or tesseral harmonics. Results are presented for several coil designs which generate tesseral harmonics of degree one.

We discuss a model of a burning process, essentially due to Sal'nikov, in which a substrate undergoes a two-stage decay through some intermediate chemical to form a final product. The second stage of the process occurs at a temperature-sensitive rate, and is also responsible for the production of heat. The effects of thermal conduction are included, and the intermediate chemical is assumed to be capable of diffusion through the decomposing substrate. The governing equations thus form a reaction-diffusion system, and spatially inhomogeneous behaviour is therefore possible.

This paper is concerned with stationary patterns of temperature and chemical concentration in the model. A numerical method for the solution of the governing equations is outlined, and makes use of a Fourier-series representation of the pattern. The question of the stability of these patterns is discussed in detail, and a linearised solution is presented, which is valid for patterns of very small amplitude. The results of accurate solutions to the fully non-linear equations are discussed, and compared with the predictions of the linearised theory. Parameter regions in which there exists genuine nonuniqueness of solutions are identified.

Steady magnetohydrodynamic flow of an incompressible micropolar fluid through a pipe of circular cross-section is studied by considering Hall and ionic effects. The fluid motion is due to a constant pressure gradient, and an external uniform magnetic field directed perpendicular to the flow direction is applied. Expressions for the velocity, microrotation, skin friction and flow rate are obtained. The effects of the micropolar parameter, magnetic parameter, Hall parameter and ion-slip parameter on the velocity, microrotation, skin friction and flow rate are discussed.

We construct games of chance from simpler games of chance. We show that it may happen that the simpler games of chance are fair or unfavourable to a player and yet the new combined game is favourable—this is a counter-intuitive phenomenon known as Parrondo's paradox. We observe that all of the games in question are random walks in periodic environments (RWPE) when viewed on the proper time scale. Consequently, we use RWPE techniques to derive conditions under which Parrondo's paradox occurs.

A numerical solution of the RLW equation is presented using a cubic spline collocation method. Basic cubic spline relations are outlined and incorporated into the numerical solution procedure. Two test problems are studied to show the robustness of the proposed procedure.