A Fredholm operator exists which maps the solutions of a system of linear partial differential equations of the form ∂u/∂t = DLu + Au coupled by a matrix A onto those solutions of a similar system coupled by a matrix B which have the same initial values. The kernels of this operator satisfy a hyperbolic system of equations. Since these equations are independent of the linear partial differential operator L, the same operator serves as a mapping for a large class of equations. If B is chosen diagonal, the solutions of a coupled system with matrix A may be obtained from the uncoupled system with matrix B.

For a three-dimensional gravity capillary wave packet in the presence of a thin thermocline in deep water two coupled nonlinear evolution equations correct to fourth order in wave steepness are obtained. Reducing these two equations to a single equation for oblique plane wave perturbation, the stability of a uniform gravity-capillary wave train is investigated. The stability and instability regions are identified. Expressions for the maximum growth rate of instability and the wavenumber at marginal stability are obtained. The results are shown graphically.

A quasi-Newton method (QNM) in infinite-dimensional spaces for identifying parameters involved in distributed parameter systems is presented in this paper. Next, the linear convergence of a sequence generated by the QNM algorithm is also proved. We apply the QNM algorithm to an identification problem for a nonlinear parabolic partial differential equation to illustrate the efficiency of the QNM algorithm.

DeVore-Gopengauz-type operators have attracted some interest over the recent years. Here we investigate their relationship to shape preservation. We construct certain positive convolution-type operators Hn, s, j which leave the cones of j-convex functions invariant and give Timan-type inequalities for these. We also consider Boolean sum modifications of the operators Hn, s, j show that they basically have the same shape preservation behavior while interpolating at the endpoints of [−1, 1], and also satisfy Telyakovskiῐ- and DeVore-Gopengauz-type inequalities involving the first and second order moduli of continuity, respectively. Our results thus generalize related results by Lorentz and Zeller, Shvedov, Beatson, DeVore, Yu and Leviatan.

In management science and system engineering, problems with two incommensurate objectives are often detected. Bicriterion optimization finds an optimal solution for the problems. In this paper it is shown that bicriterion discrete optimal control problems can be solved by using a parametric optimization technique with relaxed convexity assumptions. Some necessary optimality conditions for discrete optimal control problems subject to a linear state difference equation are derived. It is shown that in this case no adjoint equation is required.

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.

The main ideas of Hopf bifurcation theory and its relevance to the development of periodic motions of an autonomous system depending on a parameter are presented, and an algorithm for the computation of the orbits is described. It is then shown that a model system for the motion of a wheelset can be cast in the form amenable to Hopf bifurcation theory. Numerical results for the period and amplitudes of the lateral and yaw motions are obtained in terms of the forward speed of the wheelset, and the wheel-rail profile parameters.

It is found that the period of oscillation decreases while the lateral and yaw motion amplitudes increase as the forward speed increases, for any given rail and wheel profile. While the effect of wheel curvature on the lateral motions seems to be non-existent, its effect on the yaw motion amplitude and the period is to increase them very slightly as the wheel profile changes from a conical to a curved profile. On the other hand, the effect of rail curvature on the lateral amplitude, for instance, is significant; the larger the curvature the smaller the amplitude for a given forward speed.

We consider the stability of the zero solution of a system of impulsive functional-differential equations. By means of piecewise continuous functions, which are generalizations of classical Lyapunov functions, and using a technique due to Razumikhin, sufficient conditions are found for stability, uniform stability and asymptotical stability of the zero solution of these equations. Applications to impulsive population dynamics are also discussed.

The results obtained by A. J. Roberts and N. Ujević in a recent paper are generalised. A number of inequalities for functions whose derivatives are either functions of bounded variation or Lipschitzian functions or R-integrable functions are derived. Also, some error estimates for the derived formulae are obtained.

Members of an hierarchy of integrable nonlinear evolution equations, related to the well-known linearizable diffusion equation which has the diffusivity form as the reciprocal of the square of the concentration, are adapted to derive a new integrable nonlinear equation which models the surface evolution of an arbitrarily-oriented theoretical anisotropic material by the concomitant action of evaporation-condensation and surface diffusion. The constitutive relations are explicitly formulated and these show that the theoretical anisotropic material behaves like a liquid crystal. The integrable nonlinear equation may be used to advantage as test cases for numerical schemes. Its form has many attributes of the nonlinear governing equation for an isotropic material. Closed-form solutions are constructed for the evolution of a ramped surface by concomitant evaporation-condensation and surface diffusion.

Many gravity driven flows can be modelled as homogeneous layers of inviscid fluid with a hydrostatic pressure distribution. There are examples throughout oceanography, meteorology, and many engineering applications, yet there are areas which require further investigation. Analytical and numerical results for two-layer shallow-water formulations of time dependent gravity currents travelling in one spatial dimension are presented. Model equations for three physical limits are developed from the hydraulic equations, and numerical solutions are produced using a relaxation scheme for conservation laws developed recently by S. Jin and X. Zin [6]. Hyperbolicity of the model equations is examined in conjunction with the stability Froude number, and shock formation at the interface of the two layers is investigated using the theory of weakly nonlinear hyperbolic waves.

Mathematical theories describing chaotic behaviour in physical systems are introduced by developing and reviewing applications to optical fibres. A theory is presented for laser light propagating in a loop formed by an optical fibre and an optical coupler. As the light traverses the fibre it suffers an attenuation and is subjected to a phase shift which will have a component proportional to the light intensity via the nonlinear optics Kerr effect. At each pass through the coupler, an extra fraction of laser light is coupled into the loop. The mathematical formulation leads to a two-dimensional map having a clear physical and geometrical interpretation. The complete solution is given in the linear regime and the onset of nonlinear behaviour is investigated as the laser power is increased. A variety of transitions is obtained including period doubling and iteration onto a strange attractor.

An analytical expression for the voltage response to current stimulation at relatively short and long times is used to obtain estimates of the passive electrical constants of a neuron that is electrotonically coupled at the soma and dendritic terminals to other neurons in a neural network.

Wavelet systems with a maximum number of balanced vanishing moments are known to be extremely useful in a variety of applications such as image and video compression. Tian and Wells recently created a family of such wavelet systems, called the biorthogonal Coifman wavelets, which have proved valuable in both mathematics and applications. The purpose of this work is to establish along with direct proofs a very neat extension of Tian and Wells' family of biorthogonal Coifman wavelets by recovering other “missing” members of the biorthogonal Coifman wavelet systems.

We study an optimal control problem for a quasilinear parabolic equation which has delays in the highest order spatial derivative terms. The cost functional is Lagrange type and some terminal state constraints are presented. A Pontryagin-type maximum principle is derived.

Optimality conditions via subdifferentiability and generalised Charnes-Cooper transformation are obtained for a continuous-time nonlinear fractional programming problem. Perturbation functions play a key role in the development. A dual problem is presented and certain duality results are obtained.

A generalised form of the Reynolds equation for two symmetrical surfaces is derived by considering slip at the bearing surfaces. This equation is then used to study the effects of velocity-slip for the lubrication of journal bearings using half-Sommerfeld boundary conditions. Expressions for pressure and load capacity and the coefficient of friction are obtained and numerically analysed for various parameters. It is found that the load capacity decreases with slip. This is unfavourable for lubrication. The coefficient of friction decreases with a high viscous layer and increases with slip.

In the present paper questions related to stability and boundedness with respect to manifolds of solutions of impulsive differential-difference equations are considered. The investigations are carried out by means of piecewise-continuous functions which are analogues of the classical Lyapunov's functions. By means of a vectorial comparison equation and differential inequalities for piecewise-continuous functions, theorems are proved on stability and boundedness with respect to manifolds of solutions of impulsive differential-difference equations with impulse effect at fixed moments.

Criteria for guaranteeing the existence, uniqueness and asymptotic stability (in the sense of Liapunov) of periodic solutions of a forced Liénard-type equation under certain assumptions are presented. These criteria are obtained by application of the Manásevich–Mawhin continuation theorem, Floquet theory, Liapunov stability theory and some analysis techniques. An example is provided to demonstrate the applicability of our results.

For the completely stiff real homogeneous system

where e is a small positive parameter, a method is given for the construction of a basis for the solution space.

If A has n linearly independent eigenvector functions, then there exists a choice of these, {si}, with corresponding eigenvalue functions {λi}, such that there is a local basis for solution, that takes the form

where vi is a vector that tends to zero with e. In general, a basis of this form exists only on an interval in which the distinct eigenvalues have their real parts ordered. A construction is provided for continuing any solution across the boundaries of any such interval. These results are proved for a finite or infinite interval for which there are only a finite number of points at which the ordering of the real parts of eigenvalues changes.