Abstract: Several numerical methods for solving systems of ordinary differential equations are presented, including multistep methods and single step methods. Particular emphasis is given to the application of these methods to problems in dynamical astronomy.

Abstract: Random decrement signatures of structures vibrating in a random environment are studied through use of computer-generated and experimental data. Statistical properties obtained indicate that these signatures are stable in form and scale and hence, should have wide application in one-line failure detection and damping measurement. On-line procedures are described and equations for estimating record-length requirements to obtain signatures of a prescribed precision are given.

Abstract: n(lg2 n?2) multiplications and divisions are necessary to compute the set of elementary symmetric functions inn indeterminates. This lower bound and similar ones for the computational complexity of various evaluation and interpolation problems are obtained by introducing ideas and results from algebraic geometry.

Abstract: It has been shown by this investigator and numerous others [6], [7], [8] that exterior boundary value problems involving localized inhomogeneous media are most conveniently solved using finite difference or finite element techniques together with integral equations or harmonic expansions, which satisfy the radiation conditions. The methods result in large matrices that are partly full and partly sparse; and methods to solve them, such as iteration or banded matrix methods are not very satisfactory. The unimoment method alleviates the difficulties by decoupling exterior problems from the interior boundary value problems. This is done by solving the interior problem many times so that N linearly independent solutions are generated. The continuity conditions are then enforced by a linear combination of the N independent solutions, which may be done by solving much smaller matrices. Methods of generating solutions of the interior problems are discussed.

Abstract: The method of modal expansions is a fundamental technique for solving the electromagnetic boundary value problems. As its practical alogrithm, the point-matching method has been used but frequently failed. A new computer-aided procedure based on the method of modal expansions is presented and applied to the scattering from a periodic structure composed of a perfectly conducting surface. As an example, the scattering from the grating with a sinusoidal height profile is calculated. It is demonstrated numerically that the new procedure leads to desirable results, while the point-matching method does not. Several physical characteristics on the reflection grating including the diffraction anomalies are explored.

Abstract: Quadrature formulas suitable for evaluation of improper integrals such as $$\int\limits_{ - 1}^1 {f(x)(1 - x)^{ - \alpha } (1 + x)^{ - \beta } dx,\alpha ,\beta< 1} $$ are obtained by means of variable transformations ?=tanhu and ?=erfu, and subsequent use of trapezoidal quadrature rule. Error analysis is carried out by the method of contour integral, and the results are confirmed on several concrete examples. Similar formulas are also obtained to accelerate the convergence of infinite integrals $$\int\limits_\infty ^\infty {f(x)dx} $$ by means of variable transformations ?=sinhu and ?=tanu.

Abstract: An efficient numerical method for calculating plane, axisymmetric, and fully three-dimensional blunt-body flow is presented. It is a second-order-accurate, time-dependent finite-volume procedure that solves the Euler equations in integral conservation-law form. These equations are written with respect to a Cartesian coordinate system in which an embedded mesh adjusts in time to the motion of the bow shock that is automatically captured as part of the weak solution. With such an adjusting mesh, oscillations in flow properties near the shock are shown to be virtually eliminated. The scheme uses a time-splitting concept that accelerates the convergence appreciably. Comparisons are made between computed and experimental results.

Abstract: In this paper we study from the numerical point of view elliptic free boundary problems in the theory of fluid flow through porous media by a new method.

Abstract: An investigation is made into the evolution, from a sinusoidal initial wave train, of long periodic waves of small but finite amplitude propagating in one direction over water in a uniform channel. The spatially periodic surface displacement is expanded in a Fourier series with time-dependent coefficients. Equations for the Fourier coefficients are derived from three sources, namely the Korteweg–de Vries equation, the regularized long-wave equation proposed by Benjamin, Bona & Mahony (1972) and the relevant nonlinear boundary-value problem for Laplace's equation. Solutions are found by analytical and by numerical methods, and the three models of the system are compared. The surface displacement is found to take the form of an almost linear superposition of wave trains of the same wavelength as the initial wave train.

Abstract: A Newton-type algorithm has been presented elsewhere for solving non-linear inequalities of the formf(x)?0,g(x)=0, and quadratic convergence has been proved under very strong hypotheses. In this paper we show that the same results hold under a considerable weakening of the hypotheses.

Abstract: The finite difference formulation and method of solution is presented for a wide variety of fluid flow problems with associated heat transfer. Only a few direct results from these formulations are given as examples, since the book is intended primarily to serve a discussion of the techniques and as a starting point for further investigations; however, the formulations are sufficiently complete that a workable computer program may be written from them. In the appendixes a number of topics are discussed which are of interest with respect to the finite difference equations presented. These include a very rapid method for solving certain sets of linear algebraic equations, a discussion of numerical stability, the inherent error in flow rate for confined flow problems, and a method for obtaining high accuracy with a relatively small number of mesh points.

Abstract: The author develops numerical methods based on the virtual-casing principle for solving some problems relating to the equilibrium of plasma in axially symmetric toroidal configurations with a transverse cross-section of arbitrary shape. Methods are presented for calculating maintaining magnetic fields on the basis of a known equilibrium field at the plasma boundary, for extending numerically the equilibrium magnetic field beyond the plasma boundary, and for solving the direct equilibrium problem in the case of a quasi-uniform current. A method for obtaining some non-linear solutions of the equilibrium equations is discussed. The paper also goes into the question of calculating an iron-free inductor and the question of determining the inductance of a toroidal superconductor of arbitrary transverse cross-section. The methods described are illustrated by magnetic-field calculations for configurations of the "finger-ring" Tokamak type and the "Ohkawa doublet" type.

Abstract: The solution of two-dimensional transient seepage problems under the influence of both electrokinetic and hydrodynamic forces is found using the finite element method. The governing differential equations account for both the time-dependent consolidation and the time-dependent storage of electricity of the soil media. Solutions to problems are primarily presented as the movement of the phreatic surface with time, but pore water velocities, hydraulic, and electric potentials at the nodes are also calculated in the program. A practical problem, where electroosmosis was used because normal dewatering methods have proven ineffective, was analyzed using the numerical model. The results obtained compared favorably with measurements made in the field. The main advantage of the numerical model is that the electrode configuration can easily be changed to enable the most efficient arrangement to be determined.

Abstract: A numerical scheme of integration is devised to integrate the unsteady laminar boundary‐layer equations with partly reversed flows. The neighborhood of the vanishing skin friction which has been considered up to now as the location of separation, is found to be nonsingular while a typical Goldstein‐type of a traveling singularity is discovered farther downstream. This singularity is interpreted according to the theory of Sears and Telionis as unsteady boundary‐layer separation. The features of the flow in the neighborhood of the singularity are investigated.

Abstract: Two classes of high order finite difference methods for first kind Volterra integral equations are constructed. The methods are shown to be convergent and numerically stable.

Abstract: The modal analysis is given of a new optical glass-fibre waveguide configuration having a core with an equilateral triangular-shaped cross-section. Calculations are based on the point-matching method, which is known to be inaccurate if sharp corners exist in the boundary geometry. In view of this, two numerical methods are derived to check and correct the resulting eigenvalues and their fields. These new methods involve, respectively, a varia-tional formula and a harmonic-correction integral. A comparison of computations shows that the lower-order point-matched eigenvalue solutions are sufficiently accurate for the constraints laid down by manufacturing tolerances, and apart from regions near to the dielectric interface, the field plots require only a small correction. Close to the apexes, however, the mathematically exact correction procedure also becomes numerically unstable. Modal charts, field and power-density plots, and estimates of cladding field leakage, are given, and degenerate modes and symmetry properties are discussed. Experimental data available confirm the usefulness of the analysis, and it is concluded that the new triangular-cored guide may be operated in a very similar manner to the circular-cored monomode fibre.

Abstract: A modification of the classical Rayleigh technique utilizing the fast Fourier transform (FFT) is developed for electromagnetic scattering and is referred to as the Rayleigh-FFT approach. Numerical scattering results from a sinusoidal surface are compared with calculations obtained by a Rayleigh perturbation approach, physical optics, and a rigorous integral equation method. These comparisons together with self-consistent error criteria are used to define the circumstances under which the Rayleigh-FFT approach is valid. For perfectly conducting sinusoidal surfaces, the method is valid at normal incidence when the maximum surface slope is less than about 0.6 ( \lsim 31\deg ) but no limit on surface height is apparent. Slope restrictions are explained by the inherent Rayleigh error since maximum errors occur in surface troughs as expected. The Rayleigh-FFT approach is increasingly reliable, for a given geometry, when the rough half-space tends toward dielectric from perfectly conducting. Perturbation solutions to Rayleigh's technique are shown to be extremely limited. Physical optics is valid when the minimum radius of curvature of the surface is greater than a wavelength. The Rayleigh-FFT method has been extended to obtain valid scattering results from arbitrary irregular periodic structures composed of a rough layer over a rough half-space.

Abstract: Lyapunov methods and equations of parabolic type.- Multiple solutions of nonlinear partial differential equations.- Fading memory and functional-differential equations.- Singular perturbation by a quasilinear operator.- Remarks on branching from multiple eigenvalues.- Asymptotic analysis of a class of nonlinear integral equations.- Remarks about bifurcation and stability of quasi-periodic solutions which bifurcate from periodic solutions of the Navier Stokes equations.- Bifurcation of periodic solutions into invariant tori: The work of Ruelle and Takens.- Ergodic theory and statistical mechanics of non-equilibrium processes.- Mathematical problems in theoretical biology.- On predator-prey equations simulating an immune response.- Bifurcation theory for gradient systems.- Six lectures on the transition to instability.- Groundwater flow as a singular perturbation problem and remarks about numerical methods.- Nonlinear problems in nuclear reactor analysis.- Some non-linear problems in statistical mechanics and biology.- Stability properties and periodic behavior of controlled biochemical systems.

Abstract: A numerical method is developed to determine the nonlinear dynamic responses of thin, elastic, rectangular plates subjected to pulse-type uniform pressure loads. The nonlinear plate theory used in this study may be identified as the dynamic von Karman theory. The numerical method is based on finite-difference approximations of the differential equations using central difference formulations. A special form of Gaussian elimination is used to solve the system of algebraic equation resulting from the finite-difference formulation. A stability criterion is developed and checked empirically. The convergence of the solution is examined. Four sets of boundary conditions are considered. The use of the method is demonstrated by specific example problems and the results are compared with other approximate solutions.