In this paper, we develop a duality theory for the Apex dual in the case of primal constraints. As suggested by Duffin in [4], the objective function in this framework is a weighted average of the Legendre-Lagrangian function evaluated at key points. We show that whenever this new dual is feasible there is no duality gap for this dual, and moreover, no duality gap for both the Lagrangian and Wolfe duals too. We conclude with an outline of an algorithm to solve constrained minimization problems in the Apex framework.

Analytical, approximate and numerical methods are used to study the Neumann boundary value problem

− uxx + q 2 u = u 2(1 + sin x), for 0 < x < π,

subject to ux (0) = 0, ux (π) = 0,

for q 2 ∈ (0,∞). Asymptotic approximations to (1) are found for q 2 small and q 2 large. In the case where q 2 is large u(x) ≈ 3qδ(x − π/2). When q 2 = 0 we show that the only possible solution is u ≡ 0. However, there exist non-zero solutions for q 2 > 0 as well as the trivial solution u ≡ 0. To O(q 4) in the q 2 small case u(x) = q 2π(π + 2)−1, so that bifurcation occurs about the trivial solution branch u ≡ 0 at the first eigenvalue λ0 = 0 and in the direction of the first eigenfunction ξ0 = constant.

We obtain a bifurcation diagram for (1), which confirms that there exists a positive solution for q 2 ∈ (0, 10). Symmetry-breaking bifurcations and blow-up behaviour occur on certain regions of the diagram. We show that all non-trival solutions to the problem must be positive.

The formal outer solution u = q 2 û appears to satisfy û = û 2(1 + sin x), so that û ≡ 0 and û = (1 + sin x)−1 are possible limit solutions. However, in the non-trivial case ûx (0) = −1 and ûx(π) = 1; this means that û does not satisfy the boundary conditions required for a solution of (1). This behaviour usually implies that for q 2 large a boundary layer exists near x = 0 (and one near x = π), which corrects the slope. However, we find no evidence for such a solution structure, and only find perturbations in the direction of a delta function about u ≡ 0. We show using the monotone convergence theorem for quadratic forms that the inverse of the operator on the left-hand side of (1) is strongly convergent as q 2 → ∞. We show that strong convergence of the operator is sufficient to stop outer-layer behaviour occurring.

A model is presented for describing the propagation of a one-dimensional wave of permanent form in a compressible gas in a pipe. Energy is lost to the system through the walls of the pipe, but the combustion wave produces heat through an exothermic chemical reaction. The full set of equations for the model is reduced to a phase-plane system, and it is shown that, for small amplitude waves, a weakly non-linear analysis leads to a temperature profile that is a classical solitary wave. A novel shooting method is developed for the full non-linear problem, and this confirms and extends the solitary-wave solution, up to a value of the temperature amplitude at which the wave begins to develop a shock. The jump conditions across the shock are presented, and numerical integration is used to continue the solution for temperature amplitudes at which a shock is present. An example is given of the extreme situation in which the shock is so strong that all the fuel behind the shock is exhausted.

We consider the Volterra-Lotka equations for two competing species in which the right-hand sides are periodic in time. Using topological degree, we show that conditions recently given by K. Gopalsamy, which imply the existence of a periodic solution with positive components, also imply the uniqueness and asymptotic stability of the solution. We also give optimal upper and lower bounds for the components of the solution.

A Galerkin-Petrov method for the approximate solution of the complete singular integral equation with Cauchy kernel, based upon the use of two sets of orthogonal polynomials, is considered. The principal result of this paper proves convergence of the approximate solutions to the exact solution making use of a convergence theorem previously given by the author. In conclusion, some related topics such as a first iterate of the approximate solution and a discretized Galerkin-Petrov method are considered. The paper extends to a much more general equation many results obtained by other authors in particular cases.

In this paper we consider an initial value problem for systems of impulsive differential-difference equations is considered. Making use of the method of comparison and differential inequalities for piecewise continuous functions, sufficient conditions for practical stability of the solutions of such systems are obtained. Applications to population dynamics are also given.

A dam is considered with independently and identically distributed inputs occurring in a renewal process, and in particular a Poisson process, with a general release rate r(·) depending on the content. This is related to a GI/G/1 queue with service times dependent on the waiting time. Some results are obtained for the limiting content distribution when it exists; these are more complete for some special release rates, such as r(x) = μxα and r(x) = a + μx, and particular input size distributions.

The well-known Wilkinson expressions for the first derivatives of (ordinary) eigen-values and eigenvectors of simple matrices, in terms of the set of eigenvalues and eigenvectors, are redifferentiated and combined to obtain partial differential equations for the eigenvalues. Analogous expressions are obtained for the first derivatives of generalised eigenvalues and eigenvectors of simple pairs of matrices (A, B), defined by . Again, redifferentiation and combination yields slightly more complicated partial differential equations for the generalised eigenvalues. When the matrices depend on a few parameters θ1, θ2, …, the resulting differential equations for the eigenvalues, with those parameters as independent variables, can easily be derived. These parametric equations are explicit representations of analytic perturbation results of Kato, expressed by him as rather abstract complex matrix integrals. Connections with bounds for eigenvalues derived by Stewart and Sun can also be made. Two applications are exhibited, the first being to a broken symmetry problem, the second being to working out the second-order perturbations for a classical problem in the theory of waves in cold plasmas.

A fourth-order nonlinear evolution equation is derived for a wave propagating at the interface of two superposed fluids of infinite depths in the presence of a basic current shear. On the basis of this equation a stability analysis is made for a uniform wave train. Discussions are given for both an air-water interface and a Boussinesq approximation. Significant deviations are noticed from the results obtained from the third-order evolution equation, which is the nonlinear Schrödinger equation. In the Boussinesq approximation, it has been possible to compare the present results with the exact numerical analysis of Pullin and Grimshaw [12], and they are found to agree rather favourably.

A general theory is given for autonomous perturbations of non-linear autonomous second order oscillators. It is found using a multiple scales method. A central part of it requires computation of Fourier coefficients for representation of the underlying oscillations, and these coefficients are found as convergent expansions in a suitable parameter.

Iterative methods for solving systems of linear equations may be accelerated by coarse mesh rebalance techniques. The iterative technique, the Method of Implicit Non-stationary Iteration (MINI), is examined through a local-mode Fourier analysis and compared to relaxation techniques as a potential candidate for such acceleration. Results of a global-mode Fourier analysis for MINI, relaxation methods, and the conjugate gradient method are reported for two test problems.

The paper is concerned with formation of singularities in a density stratified fluid subject to a monochromatic point source of frequency σ. The frequency of the source is assumed to be such that the steady-oscillation equation is hyperbolic in the neighbourhood of the source and degenerates at a critical level. We obtain asymptotic formulae demonstrating how the solution diverges as t → ∞ on the characteristic surface emanating from the source. It is shown that, at points of the surface that belong to the critical level, the solution behaves as t⅔ exp {i(σt + π/2)} as t → ∞, whereas its large time behaviour at the other points of the surface is given by t½ exp {i(σt + π/2 ± π/4)}.

The paper is mainly concerned with the difference equation

where k and m are parameters, with k > 0. This equation arises from a method proposed for solving a cubic equation by iteration and represents a standardised form of the general problem. In using the above equation it is essential to know when the iteration process converges and this is discussed by means of the usual stability criterion. Critical values are obtained for the occurrence of solutions with period two and period three and the stability of these solutions is also examined. This was done by considering the changes as k increases, for a give value of m, which makes it effectively a one-parameter problem, and it turns out that the change with k can differ strongly from the usual behaviour for a one-parameter difference equation. For m = 2, for example it appears that the usual picture of stable 2-cycle solutions giving way to stable 4-cycle solutions is valid for smaller values of k but the situation is recersed for larger values of k where stable 4-cycle solutions precede stable 2-cycle solutions. Similar anomalies arise for the 3-cycle solutions.

The parameters describing the trapping kinetics of a linear model for diffusion, in solids involving a captured immobile phase of the diffusing entity, can be determined by measuring mean residence times for matter in the systems and evaluating the exponents for the final exponential decay rates of the diffusing entity from various shaped solids. The mean residence time for matter in a given region can be expressed in terms of a “torsion parameter” S which in the case of Dirichlet boundary conditions and cylindrical geometries, coincides with the torsional rigidity of the cylinder. The final decay rate is given by the first eigenvalue μ of a Helmholtz problem. Expressions and inequalities are derived for these parameters S and μ for general linear boundary conditions and for geometries relevant to diffusion experiments.

A linear programming model for optimally assigning diameters to a gas pipeline network is discussed. Computational results for a real life situation are presented, and certain properties that have to be satisfied by an optimal assignment are derived.

One of the most succesful algorithims for nonlinear least squares calculations is that associated with the names of Levenberg, Marquardt, and Morrison. This algorithim gives a method which depends nonlinearly on a parameter γ for computing the correction to the current point. In this paper an attempt is made to give a rule for choosing γ which (a) permits a satisfactory convergence theorem to be proved, and (b) is capable of satisfactory computer implementation. It is beleieved that the stated aims have been met with reasonable success. The convergence theorem is both simple and global in character, and a computer code is given which appears to be at least competitive with existing alternatives.

An integro-differential equation of Prandtl's type and a collocation method as well as a collocation-quadrature method for its approximate solution is studied in weighted spaces of continuous functions.

In this note we consider various theoretical aspects of the problem of least-squares approximation subject to constraints on the range of the approximating polynomial. The problem is treated from an optimization theory viewpoint. Rice's parameter space procedure is discussed.

If a finite segment of a spectrum is known, the determination of the finite object function in image space (or the full spectrum in frequency space) is a fundamental problem in image analysis. Gerchberg's method, which solves this problem, can be formulated as a fixed point iteration. This and other related algorithms are shown to be equivalent to a steepest descent method applied to the minimization of an appropriate functional for the Fourier Inversion Problem. Optimal steepest descent and conjugate gradient methods are derived. Numerical results from the application of these techniques are presented. The regularization of the problem and control of noise growth in the iteration are also discussed.