Showing papers on "Ordinary differential equation published in 1969"

Block implicit one-step methods

Abstract: A class of one-step methods which obtain a block of r new values at each step are studied. The asymptotic behavior of both implicit and predictor-corrector procedures is examined. 1. Introduction. We shall consider a class of implicit one-step methods for solving ordinary differential equations which generalize the trapezoidal rule. The idea is to determine a block of r new values at each stage, the trapezoidal rule being a case with r = 1. Implicit one-step methods have been studied by Stoller and Morrison (1), Ceschino and Kuntzmann (2) and Butcher (3). For linear problems these methods are quite useful but with the exception of the trapezoidal rule they have not found favor for nonlinear problems because of the relatively great amount of work in- volved in advancing one step. Rosser (4) has suggested obtaining a block of new values simultaneously which makes the implicit methods more competitive. He discusses in detail a procedure which calculates four new values at each stage. In addition to his references to earlier work let us note the procedure of Clippinger and Dimsdale (5)-formula (3) cf. Section 2 below-which obtains two new values at each stage. The methods we study can be described theoretically as block one-step methods. This situation prevails in practice for indefinite integrals and linear problems and also for general problems when we iterate to a fixed accuracy. In Section 2 we show convergence of these methods and study stability for a particular method. As Rosser indicates, one always expects good stability properties and indeed our example is a fourth order procedure which is A-stable. For theoretical purposes the trapezoidal rule can be conveniently regarded as a one-step method but its practicality depends on computing with it as a predictor- corrector procedure. This is what we shall do in the general case. In Section 3 we shall show that a suitable predictor-corrector approach leads to the same asymptotic behavior as iterating to completion. Again we discuss the stability of an example. Some comparative numerical examples are presented in Section 4. 2. Implicit Methods. We wish to approximate the solution of

An A-stable modification of the Adams-Bashforth methods

Abstract: This paper gives new finite difference formulae which are suitable for the numerical integration of stiff systems of ordinary differential equations. The method is exact if the problem is of the type y1 = Py + Q(x) where P is a constant and Q(x) a polynomial of degree q. When P = 0 the method is identical with the Adams-Bashforth formulae.

Oscillation and nonoscillation of solutions of second order linear differential equations with integrable coefficients

Abstract: (1) x"+a(t)x = 0, t > 0, where a(t) is a locally integrable function of t. We call equation (1) oscillatory if all solutions of (1) have arbitrarily large zeros on [0, oo), otherwise, we say equation (1) is nonoscillatory. As a consequence of Sturm's Separation Theorem [21], if one of the solutions of (1) is oscillatory, then all of them are. The same is true for the nonoscillation of (1). The literature on second order linear oscillation is voluminous. The first such result is of course the classical theorem of Sturm which asserts that

An extension of the Eddington approximation.

Abstract: A modification of the Eddington approximation to the equation of radiative transfer is suggested. The basic element of this approach is the derivation of an approximate angular distribution for the specific intensity appropriate to the problem under consideration, rather than an assumption of near isotropy as in the Eddington approximation. The resulting equations retain the simplicity of the Eddington formulation and consist of two coupled first order differential equations for the energy density and radiative flux. Boundary conditions for these equations are derived using arguments from the calculus of variations. The modified formulation reduces to the standard Eddington approximation in those cases for which the latter is an accurate description of radiative transfer. In other instances, the two approximations, while qualitatively similar, give quantitatively different results. Numerical examples indicate that the modified formulation suggested here is substantially more accurate than the standard Eddington approximation. The formulation is given in full geometric generality, but is restricted to grey, steady-state radiative transfer. The basic idea of this approach, however, should also be applicable to time-dependent, multifrequency problems.

Equivalent Realizations of Linear Systems

Abstract: In this paper, we continue an investigation [7] of the case in which the initial system specification is a set of linear first order differential equations as might be derived from a physical description of the system. Since the impulse response matrix is not easily computable from this type of representation, our aim is to develop a theory of equivalence solely in terms of known coefficient matrices. Not surprisingly, it is not possible to include all linear time-variable systems in this theory. However, it is shown that a broad class of systems (including time-invariant and analytic systems) can be delineated which do admit a complete equivalence characterization of this type. In ? 2 some basic system concepts and definitions are briefly reviewed, and in ? 3 the class of "constant rank" system representations is introduced. It is shown that members of this class have a canonical structure whose components are also of constant rank and that the dimensions of these components can be computed explicitly from the given system coefficient matrices and a finite number of their derivatives. In ? 4 a complete equivalence characterization is developed for constant rank systems. Based on this characterization it is shown in ? 5 that a minimal zero-state equivalent of a specified constant rank system representation can be constructed without solving differential equations. The type of realization obtained by this procedure is shown in ? 5 and ? 6 to possess several properties which make it most natural for system analysis or synthesis. It is time-invariant (periodic) when the impulse response matrix is time-invariant (periodic) and, under appropriate conditions, has bounded coefficients and is uniformly asymptotically stable [8] when the impulse response matrix is bounded-input bounded-output stable [8].

Laminar Free‐Convection Heat Transfer from a Needle

Abstract: The partial differential equations for laminar free convection over a needle are reduced to ordinary differential equations by a similarity analysis, and the values of local skin friction, heat transfer for various needles are obtained

Modified quasilinearization technique for the solution of boundary-value problems for ordinary differential equations

Abstract: A new quasilinearization algorithm is presented which essentially eliminates the necessity for computer storage. The representation theorem for the standard quasilinearization procedure is reformulated in terms of the initial value of the solution to a final-value problem, leading to a modification of the successive approximations. Several theorems establishing the convergence properties are proved; as in the original procedure, these convergence properties are both quadratic and monotonic. Finally, the modified approximation scheme is illustrated through several examples.

Stability, consistency and convergence of variable K-step methods for numerical integration of large systems of ordinary differential equations

Abstract: This paper describes a generalization of the Adams method for systems of ordinary differential equations from constant to variable step sizes. This entailed deriving integration formulae and proving the stability, consistency, and convergence of their solutions.

Numerical Investigation of Subharmonic Solutions to Duffing's Equation

Abstract: As far as the author is aware, the analytical or experimental investigation of subharmonic solutions to Buffing's equation (for analytical investigation, e.g. see [6], [10], [11], [12] and, for experimental investigation, e.g. see [5]) has been limited till recently to the equation in which the nonlinear term is small, that is, \p\ <1. Recently, for the strongly nonlinear equation, that is, the equation in which the nonlinear term is not necessarily small, subharmonic solutions have been investigated analytically by P. A. T. Christopher [2, 3, 4] by the use of the method developed by Cesari [1], and numerically by M. E. Levenson [7, 8] by the use of a digital computer and by C. A. Ludeke and J. E. Cornett [9] by the use of an analog computer. Christopher established analytically the existence of a subharmonic solution of order one-third in some region of parameters, but the region of parameters obtained by him does not seem to be large enough for practical use. Numerical investigations by Levenson, Ludeke and Cornett are all based on step-by-step numerical integration of ordinary differential equations and they do

Numerical Integration of the Equations Governing the One‐Dimensional Flow of a Chemically Reactive Gas

Abstract: The concept of parasitic eigenvalues in the numerical solution of sets of ordinary differential equations is introduced The use of an implicit method is proposed for the solution of sets of differential equations containing such parasitic eigenvalues These ideas are then applied to the integration of the equations which govern the one‐dimensional flow of a chemically reactive gas In particular, results are presented for the flow downstream of a normal shock wave and for the flow in a converging‐diverging nozzle The conditions are delineated under which the more complicated computations required by the implicit method appear justified

The Optimal Control of Systems with Transport Lag

Abstract: Publisher Summary This chapter discusses the optimal control of systems with transport lag. The control of dynamical systems described by ordinary differential equations has been extensively studied over the past few decades, and many real systems of practical interest can be adequately represented in this manner. Inherent in this simple description is the assumption that there exists a state of the system, expressible as a vector x of finite, preferable small, dimension, which, together with the differential equation and the future inputs, provides all the information necessary to predict the future behavior of the system. One common form of dynamical system that requires a differential-functional equation description is one in which the propagation of signals from one part of the system to another is slow enough that the propagation time is not negligible. The distinction between the optimal control problem and the so-called optimal feedback control problem seems to be technical rather than fundamental. If the optimal control problem can be solved for any possible state, then the solution can be used to provide the correct current value of the optimal control at any instant, based upon the state of the system at that instant.

The restoration of constraints in holonomic and nonholonomic problems.

Abstract: Constraint restoration in holonomic and nonholonomic problems involving system of algebraic or transcendental equations or first order differential equations

Analytic expressions for bounded solutions of non-linear ordinary differential equations with an irregular type singular point

Abstract: We shall discuss how to construct analytic expressions for bounded solutions of non-linear ordinary differential equations of the form(A) which tend to0 as x approaches the origin along the positive real axis.

On the method of matched asymptotic expansions: Part III: Two boundary-value problems

Abstract: The formal method of matched expansions is applied to two further examples. The first concerns the magnetic field induced by a steady current in a thin toroidal wire. The second, which involves a non-linear ordinary differential equation of the fourth order, has been chosen to resemble the problem of flow past a circular cylinder at small Reynolds numbers. The results of the formal procedure are proved in each case to be expansions of the exact solution.

Periodic solutions of hyperbolic partial differential equations in a strip

Abstract: have been studied by 0. Vejvoda [9], J. K. Hale [5] and P. H. Rabinowitz [8]. L. Cesari [4] and J. K. Hale [5] have also considered the question of existence of periodic solutions in the large for (1.3). The approach in [2], [31, [4] and [5] is based on the method used by these authors for similar problems in ordinary differential equations [6]. The approach and techniques used by Rabinowitz can be considered as an extension of the methods of the theory of elliptic boundary value problems to hyperbolic equations. The results presented in the present paper neither contain the results of the previous authors nor are they contained in their

Oscillation theorems for linear second order ordinary differential equations

Abstract: : The report describes the oscillatory behavior of solutions of the equation (1) x double prime + a(t)x=0, where a(t) is locally integrable on the interval from zero to, but not including, infinity. The main result is an extension of a nonoscillation theorem due to Hartman, the contrapositive of which is a useful criterion for equation (1) to be oscillatory. (Author)

Two-variable expansions and singular perturbation problems.

Abstract: The two-variable, or two-timing, procedure is formulated in a different and more general manner than usual. The procedure is first applied to an ordinary differential equation of the type ordinarily solved by the method of matched asymptotic expansions and is then applied to two wave propagation problems described by partial differential equations. The results are compared with results obtained by the method of matched asymptotic expansions and by a coordinate stretching method.

Stream Equations and Method of Characteristics

Abstract: The method of characteristics is applied to the form of the one dimensional continuity equation that is commonly employed in the analysis of the distribution of dissolved substances, such as BOD and dissolved oxygen, in natural rivers and streams. The resulting pair of ordinary differential equations can be solved numerically using widely available computer programs. This method appears to be superior to methods based on approximations using a rectangular grid in terms of the accuracy obtained and the computer time and memory required.

Asymptotic methods in the theory of ordinary linear differential equations

Abstract: In this paper we consider systems of the form (1)on the semiaxis , where is a column vector with components, is an matrix, and is a parameter. We pose the problem of finding the asymptotic behavior of the solutions of equation (1) as and .

Two problems in singular perturbations of differential equations

Abstract: Let e > 0 be a small real parameter, let y , z be real m-dimensional and n-dimensional vectors respectively and let f , g be respectively real m-dimensional and n-dimensional vector functions of their arguments. This paper aims to discuss the following two problems in singular perturbations.