Simple Ordinary Differential Equation Research Articles

Phase-space solution of the linear mode-conversion problem

The problem of linear mode conversion in a weakly inhomogeneous medium is posed and solved by phase-space methods. The PDEs for the two coupled modes are transformed to a simple first-order ordinary differential equation by a canonical transformation, wherein the two dispersion functions become essentially a locally conjugate pair of coordinates.

Comments on a class of similarity solutions of Einstein's equations relevant to the early Universe.

Comments are made on a class of similarity solutions in general relativity that are spherically symmetric and have a negative pressure as required to account for the matter production of certain current models of the early Universe. This class of solutions is defined by one fairly simple ordinary differential equation. This equation was first derived by Henriksen, Emslie, and Wesson (Phys. Rev. D 27, 1219 (1983)), who studied similarity of the second type, with a finite cosmological constant ..lambda... Now, it is pointed out that the same equation is also derived from similarity of the first type, with a zero ..lambda... The present work extends some of the results found previously and raises new possibilities about how a model Universe may go over from a state with finite ..lambda.. to a state with zero ..lambda...

Hetonic Explosions: The Breakup and Spread of Warm Pools as Explained by Baroclinic Point Vortices

As an alternative to small amplitude continuous theories we study the disintegration of warm pools of fluid on a rotating earth using clouds of baroclinic point vortices, or hetons. The equations describing the motion of the hetons are relatively simple ordinary differential equations whose numerical integration makes it possible to physically interpret the temporal evolution of finite amplitude disturbances. Using experience with small numbers of hetons as a guide, a heuristic model is proposed. This predicts that the number of baroclinic vortices forming at the rim of a circular warm pool is given by the radius of the pool divided by 1.27 times the Rossby radius, while these vortices should expand outward at a rate given by 1.65 times the Rossby radius times the vorticity density of the pool. The first prediction is in agreement with previous laboratory experiments while the second has not previously been reported.

Plane stress crack-line fields for crack growth in an elastic perfectly-plastic material

Mode-I crack growth in an elastic perfectly-plastic material under conditions of generalized plane stress has been investigated. In the plastic loading zone, near the plane of the crack, the stresses and strains have been expanded in powers of the distance, y, to the crack line. Substitution of the expansions in the equilibrium equations, the yield condition and the constitutive equations yields a system of simple ordinary differential equations for the coefficients of the expansions. This system is solvable if it is assumed that the cleavage stress is uniform on the crack line. By matching the relevant stress components and particle velocities to the dominant terms of appropriate elastic fields at the elastic-plastic boundary, a complete solution has been obtained for ϵ y in the plane of the crack. The solution depends on crack-line position and time, and applies from the propagating crack tip up to the moving elastic-plastic boundary. Numerical results are presented for the edge crack geometry.

The Chaotic Hierarchy

Abstract The complexity of dynamical behavior possible in nonlinear (for example, electronic) systems depends only on the number of state variables involved. Single-variable dissipative dynamical systems (like the single-transistor flip-flop) can only possess point attractors. Two-variable systems (like an LC-oscillator) can possess a one-dimensional attractor (limit cycle). Three-variable systems admit two even more complicated types of behavior: a toroidal attractor (of doughnut shape) and a chaotic attractor (which looks like an infinitely often folded sheet). The latter is easier to obtain. In four variables, we analogously have the hyper-toroidal and the hyper-chaotic attractor, respectively; and so forth. In every higher-dimensional case, all of the lower forms are also possible as well as “mixed cases” (like a combined hypertoroidal and chaotic motion, for example). Ten simple ordinary differential equations, most of them easy to implement electronically, are presented to illustrate the hierarchical tree. A second tree, in which one more dimension is needed for every type, is called the weak hierarchy because the chaotic regimes contained cannot be detected physically and numerically. The relationship between the two hierarchies is posed as an open question. It may be approached empirically - using electronic systems, for example.

Convection with heat flux prescribed on the boundaries of the system. I. The effect of temperature dependence of material properties

Abstract We study the bifurcation to steady two-dimensional convection with the heat flux prescribed on the fluid boundaries. The fluid is weakly non-Boussinesq on account of a slight temperature dependence of its material properties. Using expansions in the spirit of shallow water theory based on the preference for large horizontal scales in fixed flux convection, we derive an evolution equation for the horizontal structure of convective cells. In the steady state, this reduces to a simple nonlinear ordinary differential equation. When the horizontal scales of the cells exceed a certain critical size, the bifurcation to steady convection is subcritical and the degree of subcriticality increases with increasing cell size.

A method for solving some classical Yang Mills equations

The symmetry of some Yang Mills solutions dictates a natural set of gauge and Lorentz invariant degrees of freedom. In terms of these the Yang Mills equations decompose into a set of simple ordinary differential equations. Some examples are solved.

Oscillatory and asymptotic properties of differential equations with retarded arguments

We give here some results on oscillatory and asymptotic behavior for differential equations with retarded arguments of arbvitrary order. Lemma 1 establishes a comparison priniple from which we derive the oscillatory anbd asymptotic behavior of the solutions by considering simple ordinary differential equations of the form Y (n)+g(t) Y α=0 whose solutions have known behavior. Several known results for diferential equations with retarded arguments and (cf.[1], [6], [8]and[12])are particular cases of ours. Moreover, even in the case of ordinary differential equations, our results appear as generalizations of other ones (cf.[2-4]and[9-11]).

The modelling and control of water quality in a river system

The paper describes some recent research on the dynamic modelling and control of dissolved oxygen (DO) and biochemical oxygen demand (BOD) in a nontidal river system. A simple ordinary differential equation model for DO-BOD interaction in a single reach of a river has been verified against field data obtained from the River Cam outside Cambridge in Eastern England. The major details of this model are outlined and it is then used as the basis for a series of control studies concerned with the operational regulation of sewage effluent discharges in order to maintain specified levels of in-stream DO in both single and multiple reach river systems.

Satellites and Riemannian geometry

In an axially symmetric three-dimensional Riemann-spacegik(u1,u2)−u3 represents the cyclic parameter-, a gravitational potential ϕ(u1,u2) is given. For all masspoints with equal total energy and equal angular momentum there exists a function Ψ(u1,u2) by means of which the equations of motion can be reduced to a simple ordinary second-order differential equation. The function ϕ can be interpreted as the velocity with which the masspoint moves in the two-dimensional spaceu1,u2.

Mathematics and Statistics for Scientists and Engineers

P. MaCDonald London: Van Nostrand. 1966. Pp. xii + 299. Price £1 12s. 6d. This book is intended to provide a basic course in mathematics and statistics. It is written for engineers and scientists (implying non-mathematicians) and to this end it adopts a conversational rather than rigorous approach. The first part of the book is concerned with calculus which is developed from first principles, through the techniques and applications of differentiation and intergration, to simple ordinary differential equations.

Solutions of a Bethe-Salpeter Equation for Scattering States

A complete set of scattering-state solutions of a Bethe-Salpeter equation is obtained in this paper. This equation is the one for which Wick and Cutkosky succeeded to obtain the bound-state solutions. For bound-state solutions, they have found the same degeneracy as that of the non-relativistic hydrogen atom. This result is very suggestive for the present case, and indeed the scattering-state solutions are obtained without decomposing them into partial waves as is the case for Rutherford scattering. Adopting the integral transformation method, the four dimensional integral equation is reduced to a simple ordinary differential equation. It is interesting to see that this equation is identical with Cutkosky's one for n = 0. Finally by a close inspection of the resulting equation and boundary condition, it is shown that the virtual-state solutions, if any, can be obtained by modifying the inhomogeneous boundary condition for scattering states into a homogeneous one. Furthermore, it is proved that an isolated virtual level, if it really exists, gives rise to a resonance scattering.

CIX. A Technique for rendering approximate solutions to physical problems uniformly valid

Summary A method is described for treating some of the characteristically non-linear problems of physics, in particular those involving a non-linear partial differential equation for which an approximate linearization is permissible everywhere except in a limited region, such as the neighbourhood of (§ 5) a singular characteristic of the approximate solution, or of (§ 6) the point at infinity, where the approximation is valueless. The method involves a transformation of an independent variable, which is determined progressively with successive approximations to the solution : only one step being necessary if a first approximation valid uniformly (even in the critical region) is to be obtained. The method is most easily understood in its application to simple first order ordinary differential equations, which are studied in detail in §§2 and 3 as a preparation for the extension to more complicated problems in §§4, 5 and 6. Physically, the longest section, §6, concerns the “spread” of a progressive wave at ...

Interaction of Progressive Rarefaction Waves

where n is a positive integer, may be integrated readily over a limited region of the X0, t plane. Simple waves of this type may be generated in a tube of uniform cross section by moving a piston at one end of the tube containing the gas. The absence of shocks implies that the piston motion must be such as to increase the volume occupied by the gas and the velocity of the piston must be a monotonic increasing function of the time, however, it may have discontinuities. In a wave produced in this manner, the pressure in the disturbed gas will decrease as one goes from the front of the disturbed region to the rear. Such waves are called rarefaction waves. The method of integration of the hydrodynamics equations is due to Riemann' and involves interchanging the role of the dependent and independent variables. When this is done one is lead to a second order linear partial differential equation. In case (1.1) holds this equation reduces to the Euler-Poisson2 equation whose general solution may be given explicitly in terms of two arbitrary functions. The phenomena occurring, when two waves generated by piston motions at two ends of a tube interact, or, when one such wave meets either a rigid wall or a density discontinuity, may be described by determining these two arbitrary functions. It will be shown in the following that these functions can be determined by solving rather simple ordinary differential equations. Platrier3 has used a similar method for obtaining the interaction of two progressive waves each wave being generated by the motion of a piston subjected