Order System Of Differential Equations Research Articles

Microphysical kinetics in phase space: The warm rain process

The budget equations for a spatially homogeneous, isobaric and adiabatic system of cloudy air with only warm rain processes are considered. Together with the corresponding heat equation they define an ordinary, autonomous first order differential equation system. The qualitative theory of ordinary differential equations is applied to this generally nonlinear and stiff system and numerical solutions are determined using an extended GEAR method. Solutions are presented as trajectories in a three-dimensional phase space. This paper studies the basic mathematical characteristics of the system, including fixed points and stability properties of solutions. Some thermodynamic aspects are discussed. In a case study solutions of a specially defined open system are compared with those of the closed system.

Existence and Uniqueness of Solutions to Three-Point Boundary Value Problems Associated with Non-linear First Order Systems of Differential Equations

Existence and Uniqueness of Solutions to Three-Point Boundary Value Problems Associated with Non-linear First Order Systems of Differential Equations

Stability theory for high order equations

A Liapunov type stability theory for high order systems of differential equations is developed. This is done by reduction to the classical case, using the theory of polynomial models.

Solvability of boundary value problems for fourth order systems of differential equations

Solvability of boundary value problems for fourth order systems of differential equations

Zwei trajektorienverfahren zur Lösung nichtlinearer optimierungsaufgaben

In this paper two methods for the solution of nonlinear optimization problems with inequality constraints are proposed. First, a theoretical method is discussed, which bases on the property that a function Fdecreases along a certain F-curve. This F-curve is a solution of a second order differential equation system with appropriate initial conditions. In the other method the F-curves are substituted by simpler curves (F-parable) which play the role of the F-curves in the first method. Some convergence results are discussed.

Collocation for Two-Point Boundary Value Problems Revisited

Collocation methods for two-point boundary value problems for higher order differential equations are considered. By using appropriate monomial bases, we relate these methods to corresponding one-step schemes for first order systems of differential equations. This allows us to present the theory for nonstiff problems in relatively simple terms, refining at the same time some convergence results and discussing stability. No restriction is placed on the meshes used.

Forced Oscillations of Induction Motor

The general mathematical model of asynchronous machines is represented by means of a non linear either 5-th or 6-th order system of differential equations, according to the shaft elasticity is either taken into account or not. These models are sometimes linearized to study particular phaenomena, as for example forced oscillations due to alternative compressors or pumps.The paper investigates on the validity limits of the linearized model and the study tries to predict when their results can be of practical interest for engineering purposes. Therefore, a comparative analysis was made for different motor parameters and for different alternative load torque with reference to the general mathematical model and to the linearized one. On the basis of the 6-th order model, the Authors report the main results obtained into the field and show the validity limits of the linearized mathematical model.

Oscillatory states in the oxidation of carbon‐monoxide on platinum

AbstractThe trend observed on variation of the operating conditions during CO oxidation on platinum in a CSTR is as follows: stable states → simple oscillations → multipeak oscillations → stable states. Analysis of the experimental results indicates that a third or higher order system of differential equations is necessary to describe most of the observations.

A first order system of differential equations for covariant σ models

A first order system of differential equations is obtained for a covariant σ model defined on a Riemannian (or pseudo-Riemannian) manifold of arbitrary dimension n. In the case of compact groups, i.e., SO(n), the first order system coincides with the one yielded by topological arguments. Our considerations hold true also for SO(p,q), p+q=n+1, invariance groups. The scale invariance of the problem is discussed.

On the Lorentz–Dirac equation

We study the problem of a charged particle affected by an external electromagnetic field, adopting the point of view that its evolution is governed by an autonomous ordinary second order system of differential equations on M4 whose acceleration can be expanded into a power series of the charge e and assuming the acceleration satisfies the Lorentz–Dirac equation. Thus, we locate the dynamics of the system in a symplectic structure up to order e3 for external fields satisfying certain weak conditions. This is applied to obtain conserved quantities associated to symmetries of the external field that are also isometries.

The Application of Linear Multistep Methods to Singular Initial Value Problems

A theory for linear multistep schemes applied to the initial value problem for a nonlinear first order system of differential equations with a singularity of the first kind is developed. Predictor-corrector schemes are also considered. The specific examples given are systems derived from partial differential equations in the presence of symmetry.

The application of linear multistep methods to singular initial value problems

A theory for linear multistep schemes applied to the initial value problem for a nonlinear first order system of differential equations with a singularity of the first kind is developed. Predictor-corrector schemes are also considered. The specific examples given are systems derived from partial differential equations in the presence of symmetry.

A refinement process for collocation approximations

In this paper a refinement process is proposed, by means of which an approximate solution of a first order system of differential equations may be improved. The approximate solution should have been obtained with collocation on Gaussian points (cf. Russell [5]). Further, we discuss the equivalence of collocation for second order equations and collocation for the equivalent first order system. As a nice result, it is shown how we can obtain, withk collocation points per subinterval, approximations for the derivative of the solution of the second order equation with an error of orderO (h 2k ) in the nodes, using collocation for the second order equation. Finally, a numerical example is given.

Simplifications in the Determination of the Wavefunction and of the Band Structure for a Semi-infinite Crystal in the Mixed Representation

The one-electron wavefunctions near the surface in a solid may be set up by means of a mixed representation. This representation uses a coordinate normal to the surface, and Fourier components on the bidimensional reciprocal lattice of the surface. In this way the Schrödinger equation may be transformed into a first order system of differential equations with periodic coefficients in only one independent variable. The heart of the method is the determination of a fundamental matrix for the system, and the eigenproblem relevant to this matrix. The matrix must be generally determined by numerical integration of the system through a period. Remarkable simplifications in both the determination of the fundamental matrix and the search for its eigenvalues and eigenvectors can be found by applying symmetry operations. These simplifications may consist of reducing the integration interval or of halving the size of the matrix for which the eigenproblem must be resolved. We examine in detail several cases which occur very often in practical calculations.

A Boundary Value Problem Whose Solution Involves Equations Nonlinear in an Eigenvalue Parameter

The nonseparable partial differential equation for one-dimensional damped wave motion with a variable damping coefficient $\varphi _{xx} - \varphi _{tt} = p(x)\varphi _t $, is solved in series form for the usual mixed boundary value problem on a semi-infinite strip. The solution is assumed to have the form : $\varphi \sum _n c_n \psi _n (x)e^{\lambda _{n^t } } $ from which the differential equation $\psi ''_n - (\lambda _n^2 + \lambda _n p(x))\psi _n = 0$ with $\psi _n ( \pm 1) = 0$ is found. Since the preceding equation is not of the classical Sturm-Liouville type, the existence of eigenvalues, eigenfunctions and properties of the relevant eigenfunction expansion are proved using asymptotic expansions and complex variable techniques. The coefficients in the solution series are found by converting the equation for $\psi _n $ to a first order system of differential equations in which the eigenvalue parameter $\lambda _n $ occurs linearly. The eigenvectors of this system are found to be orthogonal with resp...

Stability of spiral motion

The stability of the motion of particles in spiral orbits is investigated. The force field is represented by the potential functionf=C(expaφ)/rnlimiting case of which is the Newtonian gravitational field (n= 1,a= 0). Particular solutions of the equations of motion are shown to be logarithmic spirals. A linearized stability analysis leads to a fourth order system of differential equations with variable coefficients. Complete solution of this system is obtained showing a rather complex stability situation in which outward motions are always associated with instability while inward motions might be stable or unstable depending on the value of the constantsaandnoccurring in the above potential function.