bluuebunny/crossref_metadata_2025 huggingface dataset:
Impaired flow multi-index transportation problem with axial constraints
Authors: Bandopadhyaya Lakshmisree , Puri M. C.
Published: February 2009

This paper studies the impairing of flows in multi-index transportation problem with axial constraints. For any curtailed flow, the problem is shown to be equivalent to a standard axial sum problem, whose solution can be obtained by known methods. The equivalence is established only for specially defined solutions (referred to as M-feasible solutions) of the standard problem. It is also proved that an optimal solution of the impaired flow problem corresponds to such an M-feasible solution.
Interction of a short-wave field with a dominant long wave in deep water: derivation form Zakharov's spectral formulation
Authors: Craik A. D. D.
Published: February 2009

The leading-order interaction of short gravity waves with a dominant long-wave swell is calculated by means of Zakharov's [7] spectral formulation. Results are obtained both for a discrete train of short waves and for a localised wave packet comprising a spectrum of short waves.

The results for a discrete wavetrain agree with previous work of Longuet-Higgins & Stewart [5], and general agreement is found with parallel work of Grimshaw [4] which employed a very different wave-action approach.
Some remarks on a quasi-steady-state approximation of the Navier-Stokes equation
Authors: Cannon John R. , Knightly George H.
Published: February 2009

A quasi-steady-state apprcncimation to the Navier-Stokes equation is the corresponding equation with nonhomogeneous forcing term f(x, t), but with the term Vt deleted. For solutions that are zero on the boundary, the difference z between the solution of the Navier-Stokes equation and the solution of this quasi-steady-state approximation is estimated in the L2 norm ║z║ with respect to the spatial variables. For sufficiently large viscosity or sufficiently small body force f, the inequality

holds for 0 < t ≤ T and certain real numbres C, β > 0.
Waves below first cutoff in a duct
Authors: Davidson Rodney F.
Published: February 2009

The two-dimensional Helmholtz equation is studied for an infinite region with two semi-infinite plates extending to infinity in opposite directions and a finite duct in the overlapping region. The solution technique leads to coupled Wiener-Hopf equations, and subsequently to an infinite set of simultaneous linear equations. As an example, an asymptotic expansion is calculated and graphed for the case when the duct length divided by duct width is large enough to ensure damping of all but the zero mode wave in the duct.
Boundary value controllability and observability problems for the wave and heat equation
Authors: Werner K.-D.
Published: February 2009

In this paper, we study controllability and observability problems for the wave and heat equation in a spherical region in Rn, where the control enters in the mixed boundary condition. In the main result, we show that all “finite energy” initial states (i.e. (ω0, ν0) ∈ H1(Ω) × L2 (Ω)) can be steered to zero at time T, using a control f ∈ L2 (∂Ω × [0, T]), provided T > 2. On this basis, we use the duality principle to investigate initial observability for the wave equation. Applying the Fourier transform technique, we obtain controllability and observability results for the heat equation.
Assessing regularised solutions
Authors: Lukas M. A.
Published: February 2009

Consider the prototype ill-posed problem of a first kind integral equation ℛ with discrete noisy data di, = f(xi) + εi, i = 1, …, n. Let u0 be the true solution and unα a regularised solution with regularisation parameter α. Under certain assumptions, it is known that if α → 0 but not too quickly as n → ∞, then unα converges to u0. We examine the dependence of the optimal sequence of α and resulting optimal convergence rate on the smoothness of f or u0, the kernel K, the order of regularisation m and the error norm used. Some important implications are made, including the fact that m must be sufficiently high relative to the smoothness of u0 in order to ensure optimal convergence. An optimal filtering criterion is used to determine the order where is the maximum smoothness of u0. Two practical methods for estimating the optimal α, the unbiased risk estimate and generalised cross validation, are also discussed.
A flexible strategy for warring duopolists
Authors: Rickard John , Hathaway Neville
Published: February 2009

The behaviour of duopolists is considered within a framework that allows for flexibility of the adopted strategy against the rival. In a difficult external climate, a firm may concentrate on its own profit, whereas in a more favourable external climate, it may adopt a more aggressive attitude towards the rival. The strategy considered in this paper permits this flexible approach. The market functions are kept general to allow the widest interpretation of the results.
Groups defined on images in fluid diffusion
Authors: Bracken A. J. , Green H. S. , Bass L.
Published: February 2009

A method based on the method of images is described for the solution of the linear equation modelling diffusion and elimination of substrate in a fluid flowing through a chemical reactor of finite length, when the influx of substrate is prescribed at the point of entry and Danckwerts' zero-gradient condition is imposed at the point of exit. The problem is shown to be transformable to an equivalent problem in heat conduction. Associated with the images appearing in the method of solution is a sequence of functions which form a vector space carrying a representation of the Lie group SO(2, 1) generated by three third-order differential operators. The functions are eigenfunctions of one of these operators, with integer-spaced eigenvalues, and they satisfy a third-order recurrence relation which simplifies their successive determination, and hence the determination of the Green's function for the problem, to any desired degree of approximation. Consequently, the method has considerable computational advantages over commonly used methods based on the use of Laplace and related transforms. Associated with these functions is a sequence of polynomials satisfying the same third-order differential equation and recurrence relation. The polynomials are shown to bear a simple relationship to Laguerre polynomials and to satisfy the ordinary diffusion equation, for which SO(2, 1) is therefore revealed as an invariance group. These diffusion polynomials are distinct from the well-known heat polynomials, but a relationship between them is derived. A generalised set of diffusion polynomials, based on the associated Laguerre polynomials, is also described, having similar properties.
A mathematical model for the process of gas exchange in lung capillaries usingnth order one-step kinetics of oxygen uptake by haemoglobin
Authors: Sharan Maithili , Singh M. P. , Singh Balbir
Published: February 2009

A mathematical model is developed for the process of gas exchange in lung capillaries, taking into account the transport mechanisms of molecular diffusion and the facilitated diffusion of the species due to haemoglobin. We have assumed here equilibrium conditions which enable us to neglect advection effects. Thenth order one-step kinetics of oxygen uptake by haemoglobin, proposed by Sharan and Singh [8], have been incorporated. The solution of this coupled nonlinear facilitated diffusion-reaction problem together with the physiologically-relevant boundary conditions is obtained in the closed form.

It is found that about 97.15% of total haemoglobin has combined with oxygen and 2.85% free pigment is left, which is present as carbaminohaemoglobin, met haemoglobin, carboxy haemoglobin etc. It is also shown that the percentage of free haemoglobin at a givenPO2andPCO2is independent of total haemoglobin content present in the blood.

The well-known Hill's empirical relation is deduced from our solution. The results obtained from our model, based on physical formulation, are in good agreement with the documented data [6] and those computed from the Kelman [3] empirical relation.
Dominated strategies in searching for evolutionary stable strategies
Authors: Breen P. , Henderson W.
Published: February 2009

The paper asks and answers the question “When does dominance of a particular strategy play a role in the search for evolutionary stable strategies?” The answer is much less obvious than would appear at first glance.

When there is strict dominance of a pure strategy, it is clear that the dominated strategy should never be employed in any conflict. However, when the dominance is not strict it is less obvious that the strategy should not be used. The research was originally intended to clear up this grey area in the theory of evolutionary stable strategies, but it has turned out to be of more than simply academic interest. The result can be used, with varying degrees of success, to simplify the search procedure for these evolutionary stable strategies when a reward matrix is given.
An application of the homotopy method to the generalised symmetric eigenvalue problem
Authors: Lin Wen-Wei , Lutzer Gerhard
Published: February 2009

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.
Some remarks on Pauli data in quantum mechanics
Authors: Friedman Charles. N.
Published: February 2009

The problem of determining a square integrable function from its modulus and that of its Fourier transform has been considered in an article by Corbett and Hurst ([1]). In this work we point out an error in one of the main results of the cited article concerning the Pauli uniqueness of real states, provide a proof of uniqueness for non-negative states, and present various related examples and discussion of the problem.
Duality between distant point and median of a tree network space
Authors: Kulshrestha D. K.
Published: February 2009

A point on a tree network space is said to be a distant point if it maximises its minimum weighted distance from any of its vertices. The median minimises the sum of its weighted distances from the vertices. In this paper two constrained problems are discussed. The first problem is to maximise the minimum of the weighted distances from the vertices subject to an upper bound value of the sum of the weighted distances from the vertices, while the second problem is to minimise the sum of the weighted distances subject to a lower bound value of the minimum weighted distance to any of its vertices. It is shown that these two constrained problems are duals of each other in a well defined sense.
Constraint qualifications in input optimisation
Authors: Van Rooyen M. , Sears M. , Zlobec S.
Published: February 2009

We introduce assumptions in input optimisation that simplify the necessary conditions for an optimal input. These assumptions, in the context of nonlinear programming, give rise to conceptually new kinds of constraint qualifications.
Boundary conditions optimal control
Authors: Craven B. D.
Published: February 2009

A simple rigorous approach is given to finding boundary conditions for the adjoint differential equation in an optimal control problem. The boundary conditions for a time-optimal problem are calculated from the simpler conditions for a fixed-time problem.
A computational method for combined optimal parameter selection and optimal control problems with general constraints
Authors: Teo K. L. , Goh C. J.
Published: February 2009

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.
An analysis of the impact of T. M. Cherry's work on asymptotic expansions
Authors: Grimshaw R.
Published: February 2009

During the late 1940s, T. M. Cherry published a series of research papers on uniform asymptotic formulae for transition points in ordinary differential equations. This work, together with his research into transonic gas flows, for which it was a necessary precursor, is probably his best known and most widely quoted piece of research. An analysis is made of the impact of Cherry's work on subsequent developments in this field, both in comparison to the work of others and with respect to Cherry's work on other topics.
Mathematical research at the Aeronautical Research Laboratories 1939–1960
Authors: Hurley D. G.
Published: February 2009

The Aeronautical Research Laboratories were established in Australia in 1939 as the CSIR Division of Aeronautics. Mathematicians were amongst the first staff employed, and their number reached a peak in the mid 1950s. They were an able group: in their subsequent careers 12 became Professors, 5 obtained higher doctorates, 6 became Fellows of the Australian Academy of Science and 6 Fellows of the Royal Society. They published over 100 papers, and these are discussed here under 11 separate headings.

The length of discussion given here to the various areas of research is not uniform. I have emphasised those with which I am familiar and those that interest me personally. Nevertheless, I believe the present paper provides an accurate picture of the mathematical research that was carried out at ARL during the period under review, and makes it clear that mathematicians at ARL made substantial contributions to many areas of theoretical aeronautics.
An asymptotic solution of an integro-differential equation arising in magnetic coupling through thin shield walls
Authors: Fitz-Gerald G. F. , McDonald N. A.
Published: February 2009

An equation which has arisen in a study of the magentic coupling through a small rectangular aperture of dimension A × B in a thin shield wall is discussed. The magnetic polarisability of such an aperture in a conducting wall of zero thickness is known to be expressible as RHA3, in which RH is dimensionless and is a function of the aspect ratio α ≡ B/A. An asymptotic solution procedure of a certain variational formulation of this problem is described in the limiting case of large aspect ratio α. Explicit analytical formulase for the leading terms in the expansion are given. These analytical results justify the purely numerical procedures used previously to obtain approximate solutions of this formulation of the problem.
On generalised toroidal-poloidal solutions of vector field equations
Authors: Ivers D. J.
Published: February 2009

The orthogonal coordinate systems ξi(i = 1, 2, 3) are determined, in which the gneralised toroidal and poloidal fields, defined respectively by T{T} = ∇ × {T∇ξ1} and S{S} = ∇ × T{S}, have the following three properties:

GP1 Decoupling of the vector Helmholtz equation: There exist linear differential operators L1 and L2 such that Hu = 0, where H is the vector Helmholtz operator [see equation (1)] and u = T{T} + S {S}, if and only if L1T = 0 and L2S = 0.

GP2 Orthogonality

GP3 Closure: ∇ × S{S} is a T field.

Two choices of T and S fields are considered: type I fields with potentials T and S, which may depend on ξ1, ξ2 and ξ3, and type II fields with ξ1-independent potentials. It is shown that properties GP1–GP3 only hold for type I fields in spherical and cylindrical coordinate systems, and for type II fields in azimuthal and cylindrical coordinate systems with axisymmetric and two-dimensional potentials, respectively. Analogues of GP1 for the vector wave and diffusion equations, and the Navier equation of linear elasticity, are also only true in the same four cases. Generalisations of type I and II T and S fields to arbitrary coordinate systems are indicated.
On the existence and sensitivity analysis of optimal catital accumulation paths in continuous time, infinite horizon models
Authors: Flytzanis E. , Papageorgiou Nikolaos S.
Published: February 2009

In this paper we consider an infinite horizon, continuous time model of economic growth. We prove two theorems; one on the existence of optimal paths of capital accumulation and the other on the dependence of the set of optimal paths on the initial capital stock (sensitivity analysis). In the existence result the underlying technology set is nonconvex and only its “investment’ slices are convex. The proof is direct, without any use of necessary conditions. In the sensitivity analysis, the technology set is convex and so we have that the value function is concave. Then having that, we show that the set of optimal paths is an upper semicaontinuous multifunction of the initial capital stock.
Normal shock reflection-transmission in rubber-like elastic material
Authors: Kosinski S. , Duszczyk B.
Published: February 2009

A finite transverse shock wave propagates through an unbounded medium consisting of two joined incompressible elastic half-spaces of different material properties, in the direction normal to the plane interface. A semi-inverse method is used to examine the reflection-transmission of this wave at the interface. It is found that, depending on the material properties, the reflected wave is either a simple wave or a shock; the transmitted wave is always a shock.
Appropriate initial conditions for asymptotic descriptions of the long term evolution of dynamical systems
Authors: Roberts A. J.
Published: February 2009

A centre manifold or invariant manifold description of the evolution of a dynamical system provides a simplified view of the long term evolution of the system. In this work, I describe a procedure to estimate the appropriate starting position on the manifold which best matches an initial condition off the manifold. I apply the procedure to three examples: a simple dynamical system, a five-equation model of quasi-geostrophic flow, and shear dispersion in a channel. The analysis is also relevant to determining how best to account, within the invariant manifold description, for a small forcing in the full system.
Criticality dependence on data and parameters for a problem in combustion theory, with temperature-dependent conductivity
Authors: Tam K. K.
Published: February 2009

A problem in combustion theory with tenperature-dependent conductivity is considered. It is shown that information regarding criticality dependence on data and parameters can be obtained from a transformed equation in which the conductivity is constant, while the nonlinear source term is modified. Some previous work can then be used in such a study.
An evolutionary monotone follower problem in [0, 1]
Authors: Sun Min
Published: February 2009

We consider in this article an evolutionary monotone follower problem in [0,1]. State processes under consideration are controlled diffusion processes , solutions of dyx(t) = g(yx(t), t)dt + σu(yx(t), t) dwt + dυt with yx(0) = x ∈[0, 1], where the control processes υt are increasing, positive, and adapted. The cost functional is of integral type, with certain explicit cost of control action including the cost of jumps. We shall present some analytic results of the value function, mainly its characterisation, by standard dynamic programming arguments.
A conjugate direction implementation of the BFGS algorithm with automatic scaling
Authors: Coope Ian D.
Published: February 2009

A new implementation of the BFGS algorithm for unconstrained optimisation is reported which utilises a conjugate factorisation of the approximating Hessian matrix. The implementation is especially useful when gradient information is estimated by finite difference formulae and it is well suited to machines which are able to exploit parallel processing.
On nonblocking multiconnection networks composed of digital switching matrices
Authors: Jajszczyk Andrzej
Published: February 2009

In the paper we study the conditions under which multiconnection networks are nonblocking. A multiconnection network deals with the connections of pairs {(T1, T2)} where T1 is a subset of the input terminals and T2 is a subset of the output terminals. We investigate networks composed of digital switching matrices. Such networks can be treated as a very general case encompassing many kinds of networks used in practice as well as studied theoretically.

We present four routing strategies and then develop conditions under which multiconnection networks are nonblocking when each of these strategies is used. We also show that the obtained conditions reduce to known results for some values of network parameters.
Analytic error bounds for approximations of queueing networks with an application to alternate routing
Authors: Van Dijk Nico M.
Published: February 2009

A general condition is provided from which an error bound can be concluded for approximations of queueing networks which are based on modifications of the transition and state space structure. This condition relies upon Markov reward theory and can be verified inductively in concrete situations. The results are illustrated by estimating the accuracy of a simple throughput bound for a closed queueing network with alternate routing and a large finite source input. An explicit error bound for this example is derived which is of order M—1, where M is the number of sources.
Slowly-varying bifurcation theory in dissipative systems
Authors: Denier J. P. , Grimshaw R.
Published: February 2009

Systems of coupled nonlinear differential equations with an externally controlled slowly-varying bifurcation parameter are considered. Canonical equations governing the transition between bifurcated solutions are derived by making use of methods of “steady” bifurcation theory. It is found that, depending on the initial amplitudes, the solutions of the transition equations are either asymptotically equivalent to the bifurcated solutions or the solutions develop algebraic singularities at some positive finite time. These singularities correspond to a transition to a solution of a fully nonlinear problem.
Strong uniqueness in sequential linear programming
Authors: Osborne M. R. , Womersley R. S.
Published: February 2009

It is known that strong uniqueness can be used to prove second order convergence of the generalised Gauss-Newton algorithm. Formally this algorithm includes sequential linear programming as a special case. Here we show that the second order convergence result extends when the sequential linear programming algorithm is formulated appropriately. Also this discussion provides an example which shows that the assumption of Lipschitz continuity is necessary for the second order convergence result based on strong uniqueness.
Propagation of flame fronts
Authors: Smyth N. F.
Published: February 2009

The propagation of a flame front in a combusting gas is considered in the limit in which the width of the reaction-zone is small compared with some overall flow dimension. In this approximation, the front propagates along its normals at a speed dependent on the local curvature of the front and is governed by a nonlinear equivalent of the geometric optics equations. Some exact solutions of this equation are found and a numerical scheme is developed to solve the equation for more complicated geometries.
Nonlinear interaction of positive and negative energy modes in Hamiltonian systems
Authors: Denier J. P. , Grimshaw R. H. J.
Published: February 2009

We consider the nonlinear evolution of a Hamiltonian system as the system passes through a linear resonance (as the system parameters vary). Two cases are considered. In the first case the linearized problem (at resonance) possess a full complement of normal mode solutions. This case is presented in the context of the interaction between modes which may have oppositely signed energy. The second case considered has an additional degeneracy in that the linearized problem (at resonance) has a single normal mode solution.

Both cases are analysed using normal form theory and in both cases the systems governing the transition through resonance are shown to be completely integrable in the classical sense. Possible bifurcations as the resonance is traversed are discussed. Conditions for the existence of algebraic singularities at some finite positive time are also presented.
Functional differential equations determining steady size distributions for populations of cells growing exponentially
Authors: Hall A. J. , Wake G. C.
Published: February 2009

A population of cells growing and dividing often goes through a phase of exponential growth of numbers, during which the size distribution remains steady. In this paper we study the function differential equation governing this steady size distribution in the particular case where the individual cells themselves are growing exponentially in size. A series solution is obtained for the case where the probability of cell division is proportional to any positive power of the cell size, and a method for finding closed-form solutions for a more general class of cell division functions is developed.
An optimal strategy for the control of a train
Authors: Howlett Phil
Published: February 2009

A train travels from one station to the next along a level track. The journey must be completed within a given time and it is desirable to minimise the energy required to drive the train. It has been shown with an appropriate formulation of the problem that an optimal strategy exists and that this strategy must satisfy a Pontryagin type criterion. In this paper the Pontryagin principle will be used to find the nature of the optimal strategy and this information will then be used to determine the precise optimal strategy.
On hearing the shape of an arbitrary doubly-connected region in R2
Authors: Zayed E. M. E.
Published: February 2009

The basic problem in this paper is that of determining the geometry of an arbitrary doubly-connected region in R2 together with an impedance condition on its inner boundary and another impedance condition on its outer boundary, from the complete knowledge of the eigenvalues for the two-dimensional Laplacian using the asymptotic expansion of the spectral function for small positive t.
Duality for fractional minimax programming problems
Authors: Yadav Shri Ram , Mukherjee R. N.
Published: February 2009

Duality theory is discussed for fractional minimax programming problems. Two dual problems are proposed for the minimax fractional problem: minimize maxy∈Υf(x, y)/h(x, y), subject to g(x) ≤ 0. For each dual problem a duality theorm is established. Mainly these are generalisations of the results of Tanimoto [14] for minimax fractional programming problems. It is noteworthy here that these problems are intimately related to a class of nondifferentiable fractional programming problems.
A note on the flow induced by a line sink beneath a free surface
Authors: Hocking G. C. , Forbes L. K.
Published: February 2009

The problem of withdrawing water through a line sink from a region containing an homogeneous fluid beneath a free surface is considered. Assuming steady, irrotational flow of an ideal fluid, solutions with low Froude number containing a stagnation point on the free surface above the sink are sought using a series substitution method. The solutions are shown to exist for a value of the Froude number up to a critical value of about 1.4. No solutions of this type are found for Froude numbers greater than this value.
On waves due to small oscillations of a vertical plate submerged in deep water
Authors: Mandal B. N.
Published: February 2009

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.
On neutral-delay two-species Lotka-Volterra competitive systems
Authors: Kuang Y.
Published: February 2009

The qualitative behavior of positive solutions of the neutral-delay two-species Lotka-Volterra competitive system with several discrete delays is investigated. Sufficient conditions are obtained for the local asymptotic stability of the positive steady state. In fact, some of these sufficient conditions are also necessary except at those critical values. Results on the oscillatory and non-oscillatory characteristics of the positive solutions are also included.
On lognormal random variables: I-the characteristic function
Authors: Leipnik Roy B.
Published: February 2009

The characteristic function of a lognormal random variable is calculated in closed form as a rapidly convergent series of Hermite functions in a logarithmic variable. The series coefficients are Nielsen numbers, defined recursively in terms of Riemann zeta functions. Divergence problems are avoided by deriving a functional differential equation, solving the equation by a de Bruijn integral transform, expanding the resulting reciprocal Gamma function kernel in a series, and then invoking a convergent termwise integration. Applications of the results and methods to the distribution of a sum of independent, not necessarily identical lognormal variables are discussed. The result is that a sum of lognormals is distributed as a sum of products of lognormal distributions. The case of two lognormal variables is outlined in some detail.
Determination of a control parameter in a parabolic partial differential equation
Authors: Cannon J. R. , Lin Yanping , Wang Shingmin
Published: February 2009

The authors consider in this paper the inverse problem of finding a pair of functions (u, p) such that

where F, f, E, s, αi, βi, γi, gi, i = 1, 2, are given functions.

The existence and uniqueness of a smooth global solution pair (u, p) which depends continuously upon the data are demonstrated under certain assumptions on the data.
Determination of a controllable set for a controlled dynamic system
Authors: Peng Shige , Yong Jiongmin
Published: February 2009

The controllable set of a controlled ordinary differential dynamic system to a given set is defined. Under certain reasonable conditions, the controllable set is characterised by a level set of the unique viscosity solution to some Hamilton-Jacobi-Bellman equation. The result is used to determine the asymptotic stable set of nonlinear autonomous differential equations.
Uncertainty principles invariant under the fractional Fourier transform
Authors: Mustard David
Published: February 2009

Uncertainty principles like Heisenberg's assert an inequality obeyed by some measure of joint uncertainty associated with a function and its Fourier transform. The more groups under which that measure is invariant, the more that measure represents an intrinsic property of the underlying object represented by the given function. The Fourier transform is imbedded in a continuous group of operators, the fractional Fourier transforms, but the Heisenberg measure of overall spread turns out not to be invariant under that group. A new family is developed of measures that are invariant under the group of fractional Fourier transforms and that obey associated uncertainty principles. The first member corresponds to Heisenberg's measure but is generally smaller than his although equal to it in special cases.
Some boundary-value problems for nonlinear (N) diffusion and pseudo-plastic flow
Authors: Atkinson C. , Champion C. R.
Published: February 2009

In this article, exact and approximate techniques are used to obtain parameters of interest for two problems involving differential equations of power-law type. The first problem is related to non-linear steady-state diffusion, and is investigated by means of a hodograph transformation and an approximation using a path-independent integral. The second problem involves Poiseuille flow of a pseudo-plasticfluid, and a path-independent integral is derived which yields an exact result for the geometry under consideration.
Control of a surface of discontinuity in continuous thickness
Authors: Barton N. G. , Li C.-H. , Spencer S. J.
Published: February 2009

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.
Exact solutions for redistribution by nonlinear convection-diffusion
Authors: Philip J. R.
Published: February 2009

The nonlinear convection-diffusion equation has been studied for 40 years in the context of nonhysteretic water movement in unsaturated soil. We establish new similarity solutions for instantaneous sources of finite strength redistributed by nonlinear convection-diffusion obeying the dimensionless equation ∂θ/∂t = ∂ (θm ∂θ/∂z) − θm+1 ∂θ/∂z (m ≥ 0). For m = 0 (Burgers’ equation) solutions involve the error function, and for m = 1 Airy functions. Problems 1, 2, and 3 relate, respectively, to the regions 0 ≤ z ≤ ∞, –∞ ≤ z ≤ ∞, and –∞ ≤ z ≤ 0. Solutions for m = 0 have infinite tails, but for m > 0 and finite t, θ > 0 inside, and θ = 0 outside, a finite interval in z. At the slug boundary, θ(z) is tangential to the z-axis for 0 < m < 1; and it meets the axis obliquely for m = 1 and normally for m > 1. Illustrative results are presented. For Problems 1 and 2 (but not 3) finiteness of source strength sets an upper bound on Θ0, the similarity “concentration” at z = 0. The magnitude of convection relative to diffusion increases with Θ0; and apparently the dynamic equilibrium between the two processes, implied by the similarity solutions, ceases to be possible when Θ0 is large enough.
Exact solutions to nonlinear diffusion-convection problems on finite domains
Authors: Sander G. C.
Published: February 2009

New exact solutions are presented for nonlinear diffusion and convection on a finite domain 0 ≤ z ≤ 1. These solutions are developed for the conditions of constant fluxes at both boundaries z = 0 and z = 1. In particular, solutions for the flux QL at the lower boundary z = 1, being a multiple of the flux Qs at the surface z = 0, (that is QL = aQs, where a = constant), are presented. Solutions for any constant, a, are given for an initial condition which is independent of space z. For the special cases (i) a = 1, and (ii) Qs = 0 and hence QL = 0, solutions are given for an initial condition which has an arbitrary dependence on z.
The effect of conductivity on hotspots
Authors: Smyth N. F.
Published: February 2009

In industrial applications of microwave heating, it has been observed that rather than the heating taking place uniformly, regions of high temperature, called hot-spots, tend to form. Depending on the industrial application, these can be either desirable or undesirable, and hence a theoretical understanding of the properties of the material that lead to hotspot formation is necessary. It has been shown in previous studies that hotspot formation is a product of the nonlinear dependence of microwave energy absorption by the material on temperature. It is shown in the present work that the conductivity of the material can have a significant effect on hotspot formation and can, if large enough, stop a hotspot from forming.
The existence of periodic solutions for a class of neutral differential difference equations
Authors: Chen Yongshao
Published: February 2009

In this paper, we study the existence of periodic solutions of the NDDE (neutral differential difference equation):

where τ > 0 and c is a real number. We obtain a sufficient condition under which (*) has at least k nonconstant oscillatory periodic solutions.
Nonlinearly constrained optimal control problems
Authors: Teo K. L. , Wong K. H.
Published: February 2009

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.
Numerical solution of an evolution equation with a positive-type memory term
Authors: McLean W. , Thomée V.
Published: February 2009

We study the numerical solution of an initial-boundary value problem for a Volterra type integro-differential equation, in which the integral operator is a convolution product of a positive-definite kernel and an elliptic partial-differential operator. The equation is discretised in space by the Galerkin finite-element method and in time by finite differences in combination with various quadrature rules which preserve the positive character of the memory term. Special attention is paid to the case of a weakly singular kernel. Error estimates are derived and numerical experiments reported.
Optimal shape design for a nozzle problem
Authors: Butt R.
Published: February 2009

In this paper, a gradient method is developed for the optimal shape design in a nozzle problem described by variational inequalities. It is known that this method can be used for the optimal shape design for systems described by partial differential equations (Pironneau [6]); it is used here for differential inequalities by taking limits of the expression resulting from an approximations scheme. The computations are done by the finite element method; the gradient of the criteria as a function of the coordinates nodes is computed, and the performance criterion is then minimised by the gradient method.
Numerical computation of symmetry-breaking bifurcation points
Authors: Attili B. S.
Published: February 2009

We consider symmetry-breaking bifurcation points which arise in parameter-dependent nonlinear equations of the form f(x, λ) = 0. These types of bifurcation points are connected to pitchfork bifurcation points. A direct method is used to compute such points. Multiple shooting is used to discretise the two-point boundary-value problems to obtain a finite-dimensional problem.
On generalised convex mathematical programming
Authors: Jeyakumar V. , Mond B.
Published: February 2009

The sufficient optimality conditions and duality results have recently been given for the following generalised convex programming problem:

where the funtion f and g satisfy

for some η: X0 × X0 → ℝn

It is shown here that a relaxation defining the above generalised convexity leads to a new class of multi-objective problems which preserves the sufficient optimality and duality results in the scalar case, and avoids the major difficulty of verifying that the inequality holds for the same function η(. , .). Further, this relaxation allows one to treat certain nonlinear multi-objective fractional programming problems and some other classes of nonlinear (composite) problems as special cases.
A model of the injection moulding process
Authors: Whale J. , Fowkes N. , Hocking G. , Hill D.
Published: February 2009

This paper is concerned with the injection moulding process, in which hot molten plastic is injected under high pressure into a thin cold mould. Assuming that the velocity and temperature profiles across the mould maintain their shape, a simple steady state model to describe the behaviour of a Newtonian fluid during the filling stage is developed. Various phenomena of the process are examined, including the formation of a layer of solid plastic along the walls of the mould, and the relationship between the flux of liquid plastic through the mould and the average pressure gradient along the mould. In any given situation, it is shown that there is a range of pressures and injection temperatures which will give satisfactory results.
A note on the convergence of Halley's method for solving operator equations
Authors: Zheng Shiming , Robbie Desmond
Published: February 2009

Halley's method is a famous iteration for solving nonlinear equations. Some Kantorovich-like theorems have been given. The purpose of this note is to relax the region conditions and give another Kantorovich-like theorem for operator equations.
A note on American options with varying exercise price
Authors: Dewynne J. N. , Wilmott P.
Published: February 2009

We examine the valuation of American options in a discrete time setting where the exercise price is known a priori but varies with time. (This is in contrast with the classical Black-Scholes [2] analysis, which lies in a continuous time framework and with constant exercise price.) In particular we consider a time series of exercise prices which are themselves a realisation of the share price random walk — that of the previous year, say.
Stochastic maximum principle for optimal control problem of forward and backward system
Authors: Xu Wensheng
Published: February 2009

The maximum principle for optimal control problems of stochastic systems consisting of forward and backward state variables is proved, under the assumption that the diffusion coefficient does not contain the control variable, but the control domain need not be convex.
Chaos in an anharmonic oscillator
Authors: Christie J. R. , Gopalsamy K.
Published: February 2009

Using Melnikov's method, the existence of chaotic behaviour in the sense of Smale in a particular time-periodically perturbed planar autonomous system of ordinary differential equations is established. Examples of planar autonomous differential systems with homoclinic orbits are provided, and an application to the dynamics of a one-dimensional anharmonic oscillator is given.
On an inequality relating to sum sets
Authors: Pearce C. E. M. , Pecarić J. E.
Published: February 2009

We show how a short and elementary proof can be provided for a recently-published inequality ([6], [4]) which has found a number of applications.
The temporal stability of a developing jet: a model problem
Authors: Stott Jillian A. K. , Denier James P.
Published: February 2009

The temporal instability of a developing swirling incompressible jet is considered. The jet development (in the streamwise direction) is modelled by combining a near-field and far-field approximation to the jet velocity profile into a one parameter family of basic velocity fields. The single parameter in the jet velocity field then allows us to model the radial spreading of the jet and the decay of swirl observed experimentally. Two distinct modes of instability of this model profile are found. The first is that found from a stability analysis of a fully developed swirling jet in the far field whilst the second is relevant to a “top-hat” jet with an imposed rigid body rotation. We demonstrate that the effect of azimuthal swirl is to destabilise both modes of instability. Additionally our results suggest that the near-nozzle modes of instability will dominate; indeed the growth rates of these modes are significantly larger than those found from previous studies of a fully developed jet in the far-field region.
Similarity solutions for a quasilinear parabolic equation
Authors: Guo Jong-Sheng
Published: February 2009

In this paper, we use an ordinary differential equation approach to study the existence of similarity solutions for the equationu1= Δ(uα) + θu–βin Rn× (0, ∞) where β > 0, θ ∈ [0, 1}, andn≥ 1. This includes the slow diffusion equation when α > = 1, and the standard heat equation when α = 1, and the fast diffusion equation when 0 < α < 1. We prove that there are forward self-similar solutions for this equation with initial data of the formc|x|p, wherep= 2/(α + β) if θ = 1;p≥ 0 and 2 + (1 – α)p> 0 if θ = 0, for some positive constantc.
Long wavelength vortices in time-periodic flows
Authors: Bassom Andrew P. , Blennerhassett P. J.
Published: February 2009

The linear stability properties are examined of long wavelength vortex modes in two time-periodic flows. These flows are the motion which is induced by a torsionally oscillating cylinder within a viscous fluid and, second, the flow which results from the sinusoidal heating of an infinite layer of fluid. Previous studies concerning these particular configurations have shown that they are susceptible to vortex motions and linear neutral curves have been computed for wavenumbers near their critical value. These computations become increasingly difficult for long wavelength motions and here we consider such modes using asymptotic methods. These yield simple results which are formally valid for small wavenumbers and we show that the agreement between these asymptotes and numerical solutions is good for surprisingly large wavenumbers. The two problems studied share a number of common features but also have important differences and, between them, our methods and results provide a basis which can be extended for use with other time-periodic flows.
Solution of a Schrödinger equation by iterative refinement
Authors: Kulkarni Rekha P. , Limaye Balmohan V.
Published: February 2009

A simple eigenvalue and a corresponding wavefunction of a Schrödinger operator is initially approximated by the Galerkin method and by the iterated Galerkin method of Sloan. The initial approximation is iteratively refined by employing three schemes: the Rayleigh-Schrödinger scheme, the fixed point scheme and a modification of the fixed point scheme. Under suitable conditions, convergence of these schemes is established by considering error bounds. Numerical results indicate that the modified fixed point scheme along with Sloan's method performs better than the others.
On a model for interference between searching insect parasites
Authors: Pollett P. K.
Published: February 2009

The purpose of this paper is to study a stochastic model which assesses the effect of mutual interference on the searching efficiency in populations of insect parasites. By looking carefully at the assumptions which govern the model, I shall explain why the searching efficiency is of the same order as the total number, N, in the population, a conclusion which is consistent with the predictions of population biologists; previous studies have reached the conclusion that the efficiency is of order . The major results of the paper establish normal approximations for the distribution of the numbers of active parasites. These are valid at all stages of the process, in particular the non-equilibrium phase, where explicit analytic formulae for the state-probabilities are unavailable.
Existence of nonoscillatory solutions of first order nonlinear neutral equations
Authors: Wudu Lu
Published: February 2009

Consider the nonlinear neutral equation

where pi(t), hi(t), gj(t), Q(t) Є C[t0, ∞), limt→∞hi(t) = ∞, limt→∞gj(t) = ∞ i Є Im = {1, 2, …, m}, j Є In = {1, 2, …, n}. We obtain a necessary and sufficient condition (2) for this equation to have a nonoscillatory solution x(t) with limt→∞ inf|x(t)| > 0 (Theorems 5 and 6) or to have a bounded nonoscillatory solution x(t) with limt→∞ inf|x(t)| > 0 (Theorem 7).
Reduction of second order linear dynamical systems, with large dissipation, by state variable transformations
Authors: Leipnik R. B.
Published: February 2009

Linear dynamical systems of the Rayleigh form are transformed by linear state variable transformations , where A and B are chosen to simplify analysis and reduce computing time. In particular, A is essentially a square root of M, and B is a Lyapunov quotient of C by A. Neither K nor C is required to be symmetric, nor is C small. The resulting state-space systems are analysed by factorisation of the evolution matrices into reducible factors. Eigenvectors and eigenvalues are determined by these factors. Conditions for further simplification are derived in terms of Kronecker determinants. These results are compared with classical reductions of Rayleigh, Duncan, and Caughey, which are reviewed at the outset.