Ordinary differential equations, integration

To the best of our knowledge, the first paper devoted to symmetry reduction of the 57/(2) Yang-Mills equations in Minkowski space has been published by Fushchych and Shtelen [27] (see also Ref. 21). They use two conformally invariant ansatzes in order to perform reduction of Eqs. (1) to systems of ordinary differential equations. Integrating the latter yields several exact solutions of Yang-Mills equations (1). 

Orailoglu, A. "A Multirate Ordinary Differential Equation Integrator" University of Illinois Department of Computer Science Report No. UIUCDCS-R-79-959, 1979. 

Gear C W 1966 The numerical integration of ordinary differential equations of various orders ANL 7126... 

Verlet, L. Computer Experiments" on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules. Physical Review 159 (1967) 98-103 Janezic, D., Merzel, F. Split Integration Symplectic Method for Molecular Dynamics Integration. J. Chem. Inf. Comput. Sci. 37 (1997) 1048-1054 McLachlan, R. I. On the Numerical Integration of Ordinary Differential Equations by Symplectic Composition Methods. SIAM J. Sci. Comput. 16 (1995) 151-168... 

W. Gautschi. Numerical integration of ordinary differential equations based on trigonometric polynomials. Numer. Math., 3 381-397, 1961. 

A differential equation for a function that depends on only one variable, often time, is called an ordinary differential equation. The general solution to the differential equation includes many possibilities the boundaiy or initial conditions are needed to specify which of those are desired. If all conditions are at one point, then the problem is an initial valueproblem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinaiy differential equations become two-point boundaiy value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. 

Solving Newton s equation of motion requires a numerical procedure for integrating the differential equation. A standard method for solving ordinary differential equations, such as Newton s equation of motion, is the finite-difference approach. In this approach, the molecular coordinates and velocities at a time it + Ait are obtained (to a sufficient degree of accuracy) from the molecular coordinates and velocities at an earlier time t. The equations are solved on a step-by-step basis. The choice of time interval Ait depends on the properties of the molecular system simulated, and Ait must be significantly smaller than the characteristic time of the motion studied (Section V.B). 

Thus for Hamiltonians of finite dimension the effective action functional can be found by immediately integrating a system of ordinary differential equations. The simplest yet very important case is a bath of two-level systems. 

Solving the gas dynamics expressions of Kuhl et al. (1973) requires numerical integration of ordinary differential equations. Hence, the Kuhl et al. paper was soon followed by various papers in which Kuhl s numerical exact solution was approximated by analytical expressions. 

This result is reminiscent of Equation (1.36). We have replaced the average velocity with the velocity corresponding to a particular streamline. Equation (8.2) is written as a partial differential equation to emphasize the fact that the concentration a = a(r, z) is a function of both r and However, Equation (8.2) can be integrated as though it were an ordinary differential equation. The inlet boundary... 

The preceding equations form a set of algebraic and ordinary differential equations which were integrated numerically using the Gear algorithm (21) because of their nonlinearity and stiffness. The computation time on the CRAY X-MP supercomputer for a typical case in this paper was about 5 min. Further details on the numerical implementation of the algorithm are provided in (Richards, J. R. et al. J. ApdI. Polv. Sci.. in press). 

Thus, in sumnary, the solution of the PEE requires the eval iatlon of the first M moments of the PSD. This can be done by integrating the original PEE to give the following system of ordinary differential equations ... 

The accuracy of this method increases nhen increasing M in equation 33, i. e. the dimension of the system of ordinary differential equations 36. Usually, due to the monomiodal sh ie of the PSD considered in this work, M = 3 provides a satisfactory approxinatlon of the solution for the same reason, a low mmher of quadrature points (<5) is required in the evaluation of the integral terms in equations 20, 21 and 36. 

For models described by a set of ordinary differential equations there are a few modifications we may consider implementing that enhance the performance (robustness) of the Gauss-Newton method. The issues that one needs to address more carefully are (i) numerical instability during the integration of the state and sensitivity equations, (ii) ways to enlarge the region of convergence. 

Briefly the idea behind this method is to delineate families of curves in the x-t plane, called characteristic curves, along which the partial differential equations [(123) and (128)] become a system of ordinary differential equations which could then be integrated with greater ease. However, only hyperbolic partial differential equations possess two families of characteristics curves required by the method. 

Packages to solve boundary value problems are available on the Internet. On the NIST web page http //gams.nist.gov/, choose problem decision tree and then differential and integral equations and then ordinary differential equations and multipoint boundary value problems. On the Netlibweb site http //www.netlib.org/, search on boundary value problem. Any spreadsheet that has an iteration capability can be used with the finite difference method. Some packages for partial differential equations also have a capability for solving one-dimensional boundary value problems [e.g. Comsol Multiphysics (formerly FEMLAB)]. 

This set of ordinary differential equations can be solved using any of the standard methods, and the stability of the integration of these equations is governed by the largest eigenvalue of AA. When Euler s method is used to integrate in time, the equations become... 

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