Case Of Variable Coefficients Research Articles

Oscillation and nonoscillation for systems of two first-order linear differential equations with delay

Functional differential equations with delay provide a mathematical model for a physical or biology system in which the rate of change of the system depends upon its past history. In the literature, there are several papers [14] concerning the study of oscillatory behavior of solutions of first order differential equations with time delay. But there are few results [S] about the oscillation of linear system of first order equation with delay. In this paper, we study the oscillatory behavior of linear system with delay. First, in Section 2, we consider the case of constant coefficients in details. We show that the delay can cause or destroy the oscillations. We present constructive procedure to generate or destroy oscillations of a system. In section 3 we consider the case of variable coefficients and obtain some results analogous to the result in Section 2.

Initial boundary value problems for the method of lines

This paper treats the stability of the initial boundary value problem for the method of lines applied to hyperbolic and parabolic partial differential equations in one space dimension. The theory treats the case of variable coefficients and allows for very general boundary conditions. Several examples are given which illustrate the theory. The theory is analogous to that developed by Gustafsson, Kreiss, and Sundström for finite-difference methods.

The cubic spline solution of practical problems modelled by hyperbolic partial differential equations

An approximation is developed for a general one-dimensional hyperbolic partial differential equation with constant coefficients and function value boundary conditions. The time derivative is replaced by a finite difference representation and the space derivative by a cubic spline. As expected, a three-level finite difference formula in time is obtained giving the solution at each succeeding time level. The spline approximation produces a spline function which can be used on each time level to obtain the solution at any points intermediate to the mesh points. The numerical scheme is extended to the more general variable-coefficient case and for derivative boundary conditions. Truncation errors and stability criteria are produced, and the scheme is rigorously tested on practical problems, a comparison being made with a more well-known fully implicit finite difference scheme.

A bi-directional mixing model for lead isotope evolution

This paper explores the characteristics of two-reservoir evolutionary models for radiogenic isotopes. It extends the studies of Russell (1972) to the case of bi-directional transport. Like that earlier study, it is based on the hypothesis that the characteristic lead isotope patterns of oceanic basalts and conformable ores betray the existence of at least two major geochemical systems of global extent and great age. The lead-207 deficiency of the oceanic basalt leads is considered to be a fundamental characteristic of the systems. Analytical solutions are obtained for both the case of constant and variable transport rate coefficients. It is shown that both will predict isotopic patterns in accord with those observed, but that the time-variable transport coefficients provide quantitative agreement over a much wider range of parameters. The model of Armstrong (1968) is shown to correspond to a special case of the analytical solutions here derived; study of this case confirms that the Armstrong model cannot provide for the lead-207 discrepancy mentioned above. Otherwise numerical agreement with that model seems acceptable. The most remarkable feature of the models studied is the constancy of the value obtained for the beginning time of evolution of the two systems. Constant rate coefficients give times in the range 3100 ± 300 m.y., and variable rate coefficients give times in the range 3100 ± 200 m.y. This value is compared with the youngest ages of volcanism observed for the moon and the oldest ages for the earth, as discussed by Whetherill (1972). The model also provides information about the distribution of Th/U in an evolving earth.

A Mixed Problem for Hyperbolic Equations of Second Order with a First Order Derivative Boundary Condition

On the mixed problems for hyperbolic equations of second order, only the problems with the Dirichlet condition and with the Neumann condition are studied satisfactorily. Concerning the problems with the other boundary conditions Agmon [1] contains the results on more general boundary conditions in the case when the domain is a half space and the coefficients are constant. And the author showed the not well-posedness in L 2-sense of the problem for the wave equation with an oblique derivative boundary condition [7]. In this paper we extend the results for second order equations of Agmon's paper to the case of variable coefficients and a general domain.

Stochastic models for tracer experiments in the circulation: II. Serial random walks

The theory of the simple one-dimensional random walk has been extended to the case of variable diffusion coefficient D( x). An approximate solution of the diffusion equation with drift has been obtained which is applicable for the case of a quasi-normal spatial distribution C( x). This includes cases where D( x) is a slowly varying function and particularly cases early in time following initial introduction of a delta function distribution into the system. A relation is derived through which higher moments of C( x) can be obtained from lower moments and their integrals. A numerical iteration method is described by which the evolution of C( x) can be characterized through the evolution of its moments. The approximate effect of an upstream reflecting barrier on the distribution is discussed. The approximate effect of diffuse rather than discrete absorption at the terminus of the system is also estimated.

The numerical solution of linear recurrence relations

By analysing the effect of rounding error on linear recurrence relations having exponential-type complementary solutions, the distribution of two-point boundary conditions most likely to specify a well-conditioned problem is deduced. An intuitive explanation is derived by extending the Bernouilli method of polynomial factorisation to the case of variable coefficients. The replacement of specific boundary conditions by a knowledge of the relative behaviour of the particular and complementary solutions is discussed, and illustrated by examples.

Diffusion of Small Particles in a Nonuniform Atmosphere

The downward motion of small particles in an isothermal turbulence-free atmosphere has been studied for the case of variable diffusion coefficient, utilizing two models. In the first of these, the diffusion coefficient has been taken as an exponential function of altitude, while in the second the variation has been assumed parabolic. Analytic solutions have been obtained for one dimension and three dimensions for both models. The calculations, which have been restricted to the exponential case, show the development of a profile of constant shape which moves with the same velocity, independent of the mass of the particles. These results, which are derivable from our solution, are in agreement with the numerical work of Banister and Davis. It is also shown that the qualitative nature of the results for late time is independent of the initial distribution.

Stochastic models for tracer experiments in the circulation: Parallel random walks

Abstract The theory of the simple one-dimensional random walk has been extended to the case of variable diffusion coefficient D(x). An approximate solution of the diffusion equation with drift has been obtained which is applicable for the case of a quasi-normal spatial distribution C(x). This includes cases where D(x) is a slowly varying function and particularly cases early in time following initial introduction of a delta function distribution into the system. A relation is derived through which higher moments of C(x) can be obtained from lower moments and their integrals. A numerical iteration method is described by which the evolution of C(x) can be characterized through the evolution of its moments. The approximate effect of an upstream reflecting barrier on the distribution is discussed. The approximate effect of diffuse rather than discrete absorption at the terminus of the system is also estimated.

Conditionally stable difference approximations for the wave-operator

A mixed problem with simple boundary conditions is considered for the two-dimensional wave-operator and for the corresponding system of first order equations. Some particular difference approximations to the problem are studied, and their stability and convergence are proved by means of energy inequalities, when the mesh ratio satisfies certain necessary and sufficient conditions. The case of variable coefficients and additional lower order terms is also covered.