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.

Two-dimensional gravity-capillary solitary waves propagating at the surface of a fluid of infinite depth are considered. The effects of gravity and of variable surface tension are included in the free-surface boundary condition. The numerical results extend the constant surface tension results of Vanden-Broeck and Dias to situations where the surface tension varies along the free surface.

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. The nth 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 given PO2 and PCO2 is 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.

We derive local transformations mapping radially symmetric nonlinear diffusion equations with power law or exponential diffusivities into themselves or into other equations of a similar form. Both discrete and continuous transformations are considered. For the cases in which a continuous transformation exists, many additional forms of group-invariant solution may be constructed; some of these solutions may be written in closed form. Related invariance properties of some multidimensional diffusion equations are also exploited.

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

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.

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.

We discuss some propositions of Holmes and Manning relating to the evolution of price in a cobweb market approaching equilibrium. We find in particular that the detailed behaviour of the linear model is quite typical of nonlinear cobweb models.

The Faà di Bruno formulæ for higher-order derivatives of a composite function are important in analysis for a variety of applications. There is a substantial literature on the univariate case, but despite significant applications the multivariate case has until recently received limited study. We present a succinct result which is a natural generalization of the univariate version. The derivation makes use of an explicit integralform of the remainder term for multivariate Taylor expansions.

The high Reynolds number flow past a circular cylinder with a trailing wake region is considered when the wake region is bounded and contains uniform vorticity. The formulation allows only for a single vortex pair trapped behind the cylinder, but calculates solutions over a range of values of vorticity. The separation point and length of the region are determined as outputs. It was found that using this numerical method there is an upper bound on the vorticity for which solutions can be calculated for a given arclength of the cavity. In some cases with shorter cavities, the limiting solutions coincide with the formation of a stagnation point in the outer flow at both separation from the cylinder and reattachment at the end of the cavity.

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.

Daniel et al. [6] analysed the singularity structure of the continuum limit of the one-dimensional anisotropic Heisenberg spin chain in a transverse field and determined the conditions under which the system is nonintegrable and exhibits chaos. We investigate the governing differential equations for symmetries and find the associated first integrals. Our results complement the results of Daniel et al.

A pair of multi-objective programming problems is shown to be symmetric dual by associating a vector-valued infinite game to the given pair. This symmetric dual pair seems to be more general than those studied in the literature.

The wave motion of magnetohydrodynamic (MHD) systems can be quite complicated. In order to study the motion of waves in a perfectly conducting fluid under the influence of an external magnetic field in a stratified medium, we make the simplifying assumption that the pressure is constant (to first order). This is the simplest form of the equations with variable coefficients and is not strongly propagative. Alfven waves are still present. The system is further simplified by assuming that the external field is parallel to the boundary. The Green's function for the operator is constructed and then the spectral family is constructed in terms of generalized eigenfunctions, giving four families of propagating waves, including waves “trapped” in the boundary layer. These trapped waves are interesting, since they are not the relics of surface waves, which do not exist in this context when the boundary layer shrinks to zero thickness no matter what (maximal energy preserving) boundary condition is chosen. We conjecture a similar structure for the full MHD problem.

The Bonhoeffer Van der Pol system is a planar autonomous nonlinear system of differential equations which has been invoked as a qualitative model of physiological states in a nerve membrane. It contains three independent parameters and previous work has only studied a small portion of the parameter space, that part which is thought to be of physiological relevance. Here we give a complete study of the full parameter space, using both theoretical results and numerical solutions.