Differential equations method

The Poisson equation has been used for both molecular mechanics and quantum mechanical descriptions of solvation. It can be solved directly using numerical differential equation methods, such as the finite element or finite difference methods, but these calculations can be CPU-intensive. A more efficient quantum mechanical formulation is referred to as a self-consistent reaction field calculation (SCRF) as described below. 

Dimensional Analysis is.a method by which the variables characterizing a phenomenon may be related. Accdg to Eschbach (Ref 2)> it is fundamentally identical with the analysis of physical equations, and in particular, with the analysis of physical differential equations. Methods of Lord Rayleigh and of E. Buckingham are used in ballistics, thermodynamics and fluid mechanics... 

A.3.C Second-Order Ordinary Differential Equation Methods of solving differential equations of the type... 

DIFFERENTIAL EQUATIONS Differential equation Method of solution... 

This chapter focuses on the stability properties of networks or arrays of coupled monostable units or cells. We consider two types of coupling, namely diffusive coupling and photochemical coupling. The two main concerns are how the topology of the network connectivity and how spatial inhomogeneities in the array affect instabilities. Spatially discrete systems or networks of coupled cells are described by sets of ordinary differential equations. Methods to determine the stability of stationary states of ODEs are well developed. 

An analytical method provides the solution in closed form , i.e. a formula can be given for the time evolution of the concentrations. This possibility only exists for a rather limited class of differential equations, even in the special subclass of kinetic differential equations. Methods of how to derive solutions can be found in the book by Kamke (1959) or in problems books like those by Filippov (1979), Matveev (1983), or Krasnov et al. (1978). The special case of kinetic differential equations is treated by Rodiguin Rodiguina (1964) who published the explicit solution for many first order reactions, and by Szabo (1969), who collected results for second order reactions too (together with realistic chemical examples). 

As in the case of homogeneous systems, there are two kinds of stochastic descriptions for reaction-diffusion systems as well the master equation approach and the stochastic differential equation method. Until now we have dealt with the first approach however, stochastic partial differential equations are also used extensively. Most often partial differential equations are supplemented with a term describing fluctuations. In particular, time-dependent Ginzburg-Landau equations describe the behaviour of the system in the vicinity of critical points (Haken, 1977 Nitzan, 1978 Suzuki, 1984). A usual formulation of the equation is ... 

Equations 79a, 79b, and 79c are a coupled set of second-order differential equations. Methods exist to solve these equations numerically only for small timesteps. The timestep can not be greater than about 1 fs, which is about a tenth of the C—H bond oscillation time. Because such extremely small timesteps are required, an upper Unfit of several nanoseconds is imposed on current MD simifiations (191). 

R. CouRANT, D. Hilbert, Partial Differential Equations Methods of Mathematical Physics, Vol. II), pp. 97-105, Interscience Publishers, New York, 1962. 

As discussed previously, the relation between stress and strain for linear viscoelastic materials involves time and higher derivatives of both stress and strain. While the differential equation method can be quite general, a hereditary integral method has proved to be appealing in many situations. This hereditary integral equation approach is attributed to Boltzman and was only one of his many accomplishments. In the late nineteenth century, when the method was first introduced, considerable controversy arose over the procedure. Now, it is the method of choice for the mathematical expression of viscoelastic constitutive (stress-strain) equations. For an excellent discussion of these efforts of Boltzman, see Markovitz (1977). 

The first order exluded volume effects were correctly treated by Grihley in 1953 (2) by a differential equation method. In this method, a hierarchy of equations is derived for tlffi distribution function of one end of a polymer chain in reference to the other end. The probability function satisfies a diffusion equation in the absence of interactions between chain segments. In the presence of interactions, the diffuaon equation is added a correction term throng which excluded volume effects are found. 

Liang, K., Jian-Jun, W. (2010). The solutionfor mobile robot path planning based on partial differential equations method. Paperpresented at the International Conference on Electrical and Control Engineering (ICECE) 2010. Wuhan, China. 

---