Differential equations evolution

The time evolution of the wavefiinction is described by the differential equation... 

A differential equation for the time evolution of the density operator may be derived by taking the time derivative of equation (Al.6.49) and using the TDSE to replace the time derivative of the wavefiinction with the Hamiltonian operating on the wavefiinction. The result is called the Liouville equation, that is. 

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

The temporal evolution of the variables can be described with the set of differential equations, which corresponds to the scheme (1) ... 

A kinetic study typically prepares some initial Z not equal to and describes the subsequent evolution of each of the concentrations. A basic assumption is that each component evolves according to some differential equation where t represents time. 

Mathematical and Computational Implementation. Solution of the complex systems of partial differential equations governing both the evolution of pollutant concentrations and meteorological variables, eg, winds, requires specialized mathematical techniques. Comparing the two sets of equations governing pollutant dynamics (eq. 5) and meteorology (eqs. 12—14) shows that in both cases they can be put in the form ... 

With the introduction of Gear s algorithm (25) for integration of stiff differential equations, the complete set of continuity equations describing the evolution of radical and molecular species can be solved even with a personal computer. Many models incorporating radical reactions have been pubHshed. 

The energy conservation equation is not normally solved as given in (9.4). Instead, an evolution equation for internal energy is used [9]. First an evolution equation for the kinetic energy is derived by taking the dot product of the momentum balance equation with the velocity and integrating the resulting differential equation. The differential equation is... 

In the finite-difference appntach, the partial differential equation for the conduction of heat in solids is replaced by a set of algebraic equations of temperature differences between discrete points in the slab. Actually, the wall is divided into a number of individual layers, and for each, the energy conserva-tk>n equation is applied. This leads to a set of linear equations, which are explicitly or implicitly solved. This approach allows the calculation of the time evolution of temperatures in the wall, surface temperatures, and heat fluxes. The temporal and spatial resolution can be selected individually, although the computation time increa.ses linearly for high resolutions. The method easily can be expanded to the two- and three-dimensional cases by dividing the wall into individual elements rather than layers. 

Its temporal evolution is specified by an autonomous system of N, possibly coupled, ordinary first-order differential equations ... 

The development of the differential equations which describe the evolution of particle size and molecular weight properties during the course of the polymerization is based on the so-called "population balance" approach, a quite general model framework which will be described shortly. Symbols which will be used in the subsections to follow are all defined in the nomenclature. 

Differentiating equation (II-8) with respect to time and using Leibnitz s rule, one can obtain the evolution of P(t) with time. A Laplace transform analysis will finally yield (58,59) ... 

Vi A and B. This interaction is responsible for a bi-exponential evolution of their polarization which is accounted for by two simultaneous differential equations called Solomon equations... 

The differential equations Eqs. (10) and (29)3, which represent the heat transfer in a heat-flow calorimeter, indicate explicitly that the data obtained with calorimeters of this type are related to the kinetics of the thermal phenomenon under investigation. A thermogram is the representation, as a function of time, of the heat evolution in the calorimeter cell, but this representation is distorted by the thermal inertia of the calorimeter (48). It could be concluded from this observation that in order to improve heat-flow calorimeters, one should construct instruments, with a small... 

Analytical solutions of the self-preserving distribution do exist for some coalescence kernels, and such behavior is sometimes seen in practice (see Fig. 40). For most practical applications, numerical solutions to the population balance are necessary. Several numerical solution techniques have been proposed. It is usual to break the size range into discrete intervals and then solve the series of ordinary differential equations that result. A geometric discretization reduces the number of size intervals (and equations) that are required. Litster, Smit and Hounslow (1995) give a general discretized population balance for nucleation, growth and coalescence. Figure 41 illustrates the evolution of the size distribution for coalescence alone, based on the kernel of Ennis Adetayo (1994). 

The book explains how to solve coupled systems of ordinary differential equations of the kind that commonly arise in the quantitative description of the evolution of environmental properties. All of the computations that I shall describe can be performed on a personal computer, and all of the programs can be written in such familiar languages as BASIC, PASCAL, or FORTRAN. My goal is to teach the methods of computational simulation of environmental change, and so I do not favor the use of professionally developed black-box programs. 

In the following section, we only consider the integration of the equation of linear motion Eq. (20) the procedure for the equation of rotational motion, Eq. (21), will be completely analogous. Mathematically, Eq. (20) represents an initial-value ordinary differential equation. The evolution of particle positions and velocities can be traced by using any kind of method for ordinary differential equations. The simplest method is the first-order integrating scheme, which calculates the values at a time t + 5t from the initial values at time t (which are indicated by the superscript 0 ) via ... 

Consequently, Bakker (1996) described the concentration evolution of a single layer subjected to the vorticity field of a single CSV by means of a onedimensional differential equation where both the nondimensional time and the nondimensional spatial coordinate contain the exponential shrinking rate. In this respect, the CSV approach differs from the various Bourne models in which the successive generation of several multiple-layer stacks is required and vortex age is a crucial element. 

In the Heisenberg picture the operators themselves depend explicitly on the time and the time evolution of the system is determined by a differential equation for the operators. The time-dependent Heisenberg operator AH(t) is obtained from the corresponding Schrodinger operator As by the unitary transformation... 

These differential equations depend on the entire probability density function / (x, t) for x(t). The evolution with time of the probability density function can, in principle, be solved with Kolmogorov s forward equation (Jazwinski, 1970), although this equation has been solved only in a few simple cases (Bancha-Reid, 1960). The implementation of practical algorithms for the computation of the estimate and its error covariance requires methods that do not depend on knowing p(x, t). 

The differential equation (186) describes the evolution of the B-particle once the moments (187) and (188) are known. Clearly, it is only in these latter expressions that the fluid properties... 

Solving the forward problem of the isotopic and chemical evolution of n reservoir exchanging a radioactive and its daughter isotope requires the solution of 3n— 1 differential equations (the minus one stems from the closure condition). The parameters are n (n — 1) independent flux factors k for the stable isotope N and n (n — 1) independent M/N fractionation factors D. In addition, the n values of R y the n values of Rh and the n—1 allotments x of the stable isotope among the reservoirs must be assumed at some time, preferably at the beginning of the evolution (e.g., 4.5 Ga ago), or in the modern times, in which case integration is carried out backwards in time. 

The theory of linear differential equations indicates that long-term evolution depends on the boundary conditions and the determinant of the coefficients preceding the second spatial derivatives (which can actually be considered as effective diffusion coefficients). Such a system is likely to be highly non-linear. One extreme case, however, is particularly interesting in demonstrating how periodic patterns of precipitation can be arrived at. We assume that (i) species i diffuses very fast and dC /dp is large so that P is small and (ii) that species j is much less mobile and P is large. The... 

Kinetic methods describing the evolution of distributions of molecules by systems of kinetic differential equations (obeying either the classic mass action law of chemical kinetics or the generalized Smoluchowski coagulation process). 

As a first approach, the experimental quenching rate constant fcq is assumed to be time-independent. According to the simplified Scheme 4.1, the time evolution of the concentration of M following a b-pulse excitation obeys the following differential equation ... 

The time evolution of the fluorescence intensity of the monomer M and the excimer E following a d-pulse excitation can be obtained from the differential equations expressing the evolution of the species. These equations are written according to the kinetic in Scheme 4.5 where kM and kE are reciprocals of the excited-state lifetimes of the monomer and the excimer, respectively, and ki and k i are the rate constants for the excimer formation and dissociation processes, respectively. Note that this scheme is equivalent to Scheme 4.3 where (MQ) = (MM) = E and in which the formation of products is ignored. 

The time evolution of the fluorescence intensity of the acidic form AH and the basic form A following b-pulsc excitation can be obtained from the differential equations expressing the evolution of the species. These equations are written according to Scheme 4.6 ... 

These equations are a set of nonlinear first-order ordinary differential equations that describe the evolution of the n species as a function of time starting from a set of initial conditions... 

This first order differential equation now governs the evolution of an initial field. For a finite step length the propagation is approximated by... 

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