Time Dependent Solutions

Equation (2.67) also follows from the Langevin equation (2.35) if the fluctuations are neglected by putting q (j ) = 0, or letting e - 0. Therefore (2.67) also describes the deterministic path. 

According to (2.65, 66) these points approximately coincide with the maxima/ minima of the stationary probability distribution. In the vicinity of y, i.e. for y = ym + V (i) with small t] (t), 

Under the assumption that the solution y (t) to (2.67) has been found from (2.73), it is then easy to find the explicit solution of (2.68), which assumes the form 

This solution is non-divergent only for K y (0)) 4= 0. Later it will be seen that the solution of (2.68) at an unstable stationary point where K(y = 0 and K (ym) = 7 0, leads to exponential fluctuation enhancement, see (2.93) later in this section. 

2) Secondly, the time dependent solutions of the Fokker-Planck equation (2.24) will be studied. Four cases are considered  

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Gresho, P. M., Lee, R. L. and Sani, R. L., 1980. On the time-dependent solution of the incompressible Navier-Stokes equations in two and three dimensions. In Recent Advances in Numerical Methods in fluids, Ch. 2, Pineridge Press, Swansea, pp. 27-75. 

By assuming the supply air concentration to be zero, since usually there are quite different contaminants in the outside air and from the source, and that the initial concentration also is zero, the time-dependent solution is the following,... 

Thomas L. Saaty, "Time-Dependent Solution of the Many-server Poisson Queue/2 Operations Research, 8, No. 6 (1960). 

We have given an analytical method of deriving a time-dependent solution to our problem that is complicated but illustrates an important method. Frequently, steady state solutions are all that is needed. 

The formulas derived in the time-independent framework can be easily transferred into the corresponding time-dependent solutions. The formulas in the time-independent linear potential model, for example, provide the formulas in the time-dependent quadratic potential model in which the two time-dependent diabatic quadratic potentials are coupled by a constant diabatic coupling [1, 13, 147]. The classically forbidden transitions in the time-independent framework correspond to the diabatically avoided crossing case in the time-dependent framework. One more thing to note is that the nonadiabatic tunneling (NT) type of transition does not show up and only the LZ type appears in the time-dependent problems, since time is unidirectional. 

By applying the Laplace transform to the U-series decay equation, one obtains simple linear equations that can be solved for the Laplace transforms of Ni (the number of nuclei i in the system). By inverting the Laplace transforms using tables, the time-dependent solutions are directly obtained. The Laplace transform for Equation (1) is ... 

Dallimore P.J. and R.F. Holub, General Time-Dependent Solutions for Radon Diffusion from Samples Containing Radium, Report of Investigation 8765, Bureau of Mines, United States Department of the Interior, Denver (1982). 

The Fokker-Planck equation is a partial differential equation. In most cases, its time-dependent solution is not known analytically. Also, if the Fokker-Planck equation has more than one state variable, exact stationary solutions are... 

Before considering the time-dependent solutions, which have to be computed numerically, it is instructive to note that at large times the solution of (55) can approach a steady state, if the boundary conditions are the usual ones n0(x = 0) = constant, na x—> = ) = 0 (Corbett et al., 1986). Setting the right of (55) equal to zero, we find the solution to be... 

M. Berman and R. Kosloff, Comput. Phys. Commun., 63,1 (1991). Time-Dependent Solution... 

The difficulty of evaluating the quantity in Eq. (17) is that it requires the time-dependent solution of the Schrodinger equation in which the Hamiltonian is a function of all the solvent, ion, and metal nuclei, as well as of the metal electrons. In the two-state approximation, the Hamiltonian can be written as... 

As the only explicit time-dependent solution of Cauchy s problem, the Lienard-Wiechert potentials are claimed be inadequate for describing... 

To analyze the dynamics of SHG, we use time-dependent ordinary differential equations. At the beginning, Maxwell s equations governing SHG were studied, and a simple analytical time dependent solutions was found [99]. The classical case of SHG was discussed by Bloembergen [102], and the present-day state in the dynamics of SHG without damping and pumping was clarified... 

Fig. 5.8. Variation in stationary-state intersection relative to the maximum and minimum in the g(ot, 6) = 0 nullcline with the quotient M/K for a system with y = 0.2. (a) Intersection below maximum, M/K = 1.6 a given trajectory moves quickly to the g(a, 0) = 0 nullcline which it then moves along to the stationary-state solution, (b) Intersection above the minimum, M/K = 20 again a given trajectory will approach the stationary state along the g(a, 0) = 0 nullcline. (c) Intersection lying between the extrema, M/K = 5 now the stationary state is not approached and the time-dependent solutions cycle around the phase plane on the g(x, 6) = 0 nullcline (slow motion) with rapid jumps from one branch to the other (fast motion) at the turning points, (d), (e) Schematic representation of the relaxation oscillations for the conditions in (c).

This is a second-order linear partial differential equation. Note that the transport terms (Eq. 22-4) are linear per se, while the reaction term (Eq. 22-5) has been intentionally restricted to a linear expression. For simplicity, nonlinear reaction kinetics (see Section 21.2) will not be discussed here. For the same reason we will not deal with the time-dependent solution of Eq. 22-6 the interested reader is referred to the standard textbooks (e.g. Carslaw and Jaeger, 1959 Crank, 1975). 

Solutions of Eq. 25-46 for nonsteady-state conditions are difficult to obtain analytically, yet numerical procedures are straightforward. For the case of slow reactions, more precisely for Da 1, the solution can be approximated by multiplying the time-dependent solution for a conservative substance with the exponential factor exp(- t). For instance, for the pulse input Eq. 25-20 is modified into ... 

A general time-dependent solution of eqn. (93) does not appear to be feasible. However, for long times Rice [184] has shown that the residual time dependence is of the form... 

A more tractable theory based on the probability that a reactant pair will react at a time t (pass from reactants to products) is that due to Szabo et al. [282]. If the survival probability of a geminate pair of reactants initially formed with separation r0 is p (r0, t) at time t, the average lifetime of the pair is /dr0 p(r0, t)t and this is longer for larger initial separation distances. It provides a convenient and approximate description of the rate at which a reactant pair can disappear, but it does so without the need of a full time-dependent solution of the appropriate equations. Nevertheless, as a means of comparing time-dependent theory and experiment in order to measure the value of unknown parameters, it cannot be regarded as satisfactory. 

In the case of the quasilinear Fokker-Planck equation (2.4), the free energy U defined in terms of the stationary solution by (2.6) is identical with the potential in the deterministic equation (5.2). That identity is often taken for granted when time-dependent solutions have to be constructed for systems of which only the equilibrium distribution is known. We shall now show, however, that it holds only for systems of diffusion type whose Fokker-Planck equation is quasilinear, i.e., of the form (2.4). 

The boundary at oo is of type (i). The boundary at 0 is of type (iii). There is no normalizable stationary distribution, but the time-dependent solution can be found explicitly by Fourier transformation ... 

Discuss the boundaries and verify the result by means of the explicit time-dependent solution. 

Fig. 15. Schematic representation of the time dependent solution viscosity of PAAm.

Use the superposition method to find the time-dependent solution. 

Time-Dependent Solutions. In the time-dependent case, the diffusion equations given by Eq. 6.23 are coupled. However, they can be uncoupled by again using the diagonalization method. 

These equations are solved by separating out the time dependence through the substitutions x(r, t) = x (r)eX1, y(r, t) = y (r)eKl, and diagonalizing the resulting pair of spatially dependent coupled equations. These two separated equations are Helmholtz-type equations whose solutions can be straightforwardly obtained in different coordinate systems.28,49 The complete space-time-dependent solutions are sums of spatial modes or patterns, each with a characteristic temporal behavior. For example, the complete solution on a circle can be written... 

The buildup of the H2 concentration, for any given depth x, starts with all its time derivatives zero at t = 0, increases gradually, and after a depth-dependent induction time becomes linear in t. The unbounded growth can be truncated by allowing the molecules either to dissociate or to diffuse. Dissociation will of course modify the development of the H° distribution molecular diffusion will not. As regards dissociation, there are to date no time-dependent solutions for this problem available presumably if the molecules are immobile, they would show an approach to a flat thermal-equilibrium distribution, which would extend to deeper depths at longer times. The case of diffusion without dissociation will be taken up in the paragraphs to follow. 

The eigenmode expansion was also used to determine the time-dependent solution of the Smoluchowski equations for diverse bistable potentials.185... 

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