Showing papers on "Exact differential equation published in 1974"

Time-dependent projection-operator approach to master equations for coupled systems

Abstract: In this paper we derive master equations for two or more systems coupled to each other, perhaps strongly, by using a generalization of the usual projection-operator technique to include time-dependent projection operators. The coupled systems may be either similar or dissimilar and classical or quantum mechanical. Whereas the customary approaches to coupled systems are best able to treat situations in which some of the systems are "baths" with a specified density operator or phase-space probability density, our approach allows us to treat situations where it is necessary or convenient to treat the coupled systems on an equal footing. In our scheme the "relevant" part of the full density operator is considered to be the uncorrelated part of the full density operator and is a symmetric functional of the reduced density operators of each of the coupled subsystems. The "irrelevant" part of the density operator is then the part describing correlations between the coupled systems. Our formalism is particularly useful where systems are coupled to one another predominantly in a self-consistent fashion. First, we develop exact master equations for two coupled systems, taking as our prototype the dynamical problem of quantum optics, where a spatially extended collection of two-level atoms interact with a multimode optical field. We then generalize our results to $N$ coupled systems, taking as our prototype the kinetics of a classical nonideal gas interacting through two-body forces, and derive exact master equations for the system. We then consider as examples several approximate theories resulting from our exact equations. In the case of the imperfect gas we investigate the low-density limit and show how Bogoliubov's form of the Boltzmann equation emerges from our formalism, as well as corrections due to Klimontovich. We consider as special cases of our exact quantum-optical equations the equations in the first Born approximation, with and without memory, and show how several existing quantum-optical master equations are contained in our general results. As a second example in quantum optics, we consider the case where the predominant behavior of the system is described by the self-consistent-field or coupled Bloch and Maxwell equations and derive a first-order perturbation description for deviations from self-consistent-field behavior.

On the Strong and Weak Limit-Point Classification of Second-Order Differential Expressions

Abstract: : The concept of limit-point for a formally self-adjoint ordinary differential operator of the second order is associated with the existence of a unique self-adjoint extension. A concept of strongly limit-point has also been defined, and this paper considers the circumstances under which the two concepts are or are not equivalent. (Author)

On the Fokker-Planck equation for the linear chain

Abstract: The Fokker-Planck equation with retarded friction, derived approximately from the Liouville equation, is considered for the Rubin model of a heavy particle, mass M, in a harmonic lattice of light particles, mass m. We find that the solution is inconsistent with Rubin's exact results, except when both the exact and present results are truncated at lowest order in the mass ratio m/M. This casts some doubt on the validity of retarded Fokker-Planck equation for more general situations. We also consider the so-called initial value term in the statistical mechanical derivation, and find it slowly decaying in time, contrary to what is usually assumed.

A Differential Game of Evasion with Nonlinear Control

Abstract: A differential game of evasion in which the state is governed by an nth order differential equation is studied. The equation is linear in the state and its derivatives. Control is exercised through a nonlinear forcing term. A sufficient condition for evasion is given and the evasion strategy is obtained as a generalized solution of a Volterra integral equation. The generalized solution is approximated by ordinary measurable functions by constructing a sliding regime.

4.—A Cauchy Problem for an Ordinary Integro-differential Equation

Abstract: In this paper we study an ordinary second-order integro-differential equation (IDE) on a finite closed interval. We demonstrate the equivalence of this equation to a certain integral equation, and deduce that the homogeneous IDE may have either 2 or 3 linearly independent solutions, depending on the value of a parameter λ. We study a Cauchy problem for the IDE, both by this integral equation approach and by an independent approach, based on the perturbation theory for linear operators. We give necessary and sufficient conditions for the Cauchy problem to be solvable for arbitrary right-hand sides—these conditions again depend on λ—and specify the behaviour of the IDE when these conditions are not satisfied. At the end of the paper some examples are given of the type of behaviour described.

A note on pairs of solutions of a nonlinear sturm-liouville problem

Abstract: Earlier qualitative theorems are used to study the structure of the set of solutions of a family of ordinary differential equations. In particular lower bounds for the number of solutions having different numbers of nodes is obtained.

Nonperturbative approach to the theory of transition probabilities by the stroboscopic method

Abstract: The exact differential equations with periodic coefficients characterizing the transition in a two-level quantum system are solved by the stroboscopic method which is a nonperturbative approach. This method is based on an averaging over the fluctuations in the system. The solution is unitary and is valid for large times also, as opposed to the short-time validity of the perturbative solution. It should also be noted that this method gives Rabi's well-known solution to the two-level system at resonance by the so-called transformation to the rotating coordinate frame.

Exponential fitting of matricial multistep methods for ordinary differential equations

Abstract: We study a class of explicit or implicit multistep integration formulas for solving N X N systems of ordinary differential equations. The coefficients of these formulas are diagonal matrices of order N, depending on a diagonal matrix of parameters Q of the same order. By definition, the formulas considered here are exact with respect to y -Dy + ?(x, y) provided Q hD, h is the integration step, and P belongs to a certain class of polynomials in the independent variable x. For arbitrary step number k > 1, the coefficients of the formulas are given explicitly as functions of Q. The present formulas are generalizations of the Adams methods (Q = 0) and of the backward differentiation formulas (Q = + -o). For arbitrary Q they are fitted exponentially at Q in a matricial sense. The implicit formulas are unconditionally fixed-h stable. We give two different algorithmic implementations of the methods in question. The first is based on implicit formulas alone and utilizes the Newton-Raphson method; it is well suited for stiff problems. The second implementation is a predictor-corrector approach. An error analysis is carried out for arbitrarily large Q. Finally, results of numerical test calculations are presented.

An extrapolation step-size monitor for solving ordinary differential equations

Abstract: A step-size monitor is presented for use in numerically solving ordinary differential equations by extrapolation methods. The monitor uses the information present in the extrapolation lozenge to determine the “optimal” step-size and order. This allows the monitor to adjust both the order and step-size to the local behavior of the solution in a reasonably “optimal” fashion. The monitor is particularly useful when obtaining low-precision solutions which require radical step-size changes. The results of this monitor are compared quite favorably with previous proposals.

Stability of bounded and unbounded sets for ordinary differential equations

Abstract: The stability properties of subsets of Rn are examined using a family of Liapunov functions and the invariance properties of the sets.

An investigation of the accuracy of the Merkel equation for evaporative cooling tower calculations. Waste heat management

Abstract: The exact differential equations governing heat and mass transfer and air flow in an evaporative, natural-draft cooling tower are presented. The Merkel equation is then derived starting from this exact formulation and showing all the approximations involved. The Merkel formulation lumps the sensible and the latent heat transfer together and considers a single enthalpy-difference driving force for the total heat transfer. The effect of the approximations inherent in the Merkel equation is investigated and analyzed by a series of parametric numerical calculations of cooling tower performance under various ambient conditions and load conditions.

A Bifurcation Problem for a Fourth Order Nonlinear Ordinary Differential Equation

Abstract: In this paper a boundary value problem for a certain fourth order nonlinear ordinary differential equation is discussed. In particular, it is shown that if the eigenvalue parameter is greater than the nth eigenvalue of the linearized problem then the nonlinear problem has at least n distinct pairs of nontrivial solutions. Results are obtained on the position of the zeros of these solutions and bounds on the solution are obtained.