Laplace transformation evolution times

Solving equation (1-8) (using Laplace transform techniques) yields the time evolution of the current of a spherical electrode ... 

Its poles are determined to any order of by expansion of M. However, even in the lowest order in the inverse Laplace transformation, which restores the time kinetics of Kemni, keeps all powers to Jf (t/xj. This is why the theory expounded in the preceding section described the long-time kinetics of the process, while the conventional time-dependent perturbation theory of Dirac [121] holds only in a short time interval after interaction has been switched on. By keeping terms of higher order in i, we describe the whole time evolution to a better accuracy. 

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) ... 

Using deterministic kinetics, one can force-fit the time evolution of one species—for example, eh but then those of other yields (e.g., OH) will be inconsistent. Stochastic kinetics can predict the evolutions of radicals correctly and relate these to scavenging yields via Laplace transforms. 

The main lines of the Prigogine theory14-16-17 are presented in this section. A perturbation calculation is employed to study the IV-body problem. We are interested in the asymptotic solution of the Liouville equation in the limit of a large system. The resolvent method is used (the resolvent is the Laplace transform of the evolution operator of the N particles). We recall the equation of evolution for the distribution function of the velocities. It contains, first, a part which describes the destruction of the initial correlations this process is achieved after a finite time if the correlations have a finite range. The other part is a collision term which expresses the variation of the distribution function at time t in terms of the value of this function at time t, where t > t t—Tc. This expresses the fact that the system has a memory because of the finite duration of the collisions which renders the equations non-instantaneous. 

Tavare and Garside ( ) developed a method to employ the time evolution of the CSD in a seeded isothermal batch crystallizer to estimate both growth and nucleation kinetics. In this method, a distinction is made between the seed (S) crystals and those which have nucleated (N crystals). The moment transformation of the population balance model is used to represent the N crystals. A supersaturation balance is written in terms of both the N and S crystals. Experimental size distribution data is used along with a parameter estimation technique to obtain the kinetic constants. The parameter estimation involves a Laplace transform of the experimentally determined size distribution data followed a linear least square analysis. Depending on the form of the nucleation equation employed four, six or eight parameters will be estimated. A nonlinear method of parameter estimation employing desupersaturation curve data has been developed by Witkowki et al (S5). 

The exact time evolution within subspace O follows from the convolution theorem of the Fourier-Laplace transform, i.e.,... 

Up to now we have discussed only the first equation of the set (3.135), which describes the quenching of reactant A. However, the physical picture outlined above would be incomplete if the time evolution of the product, B, was not taken into account. The integral equations (3.135) solved by the Laplace transformation result in the following ... 

These equations are in the form of convolutions and can be solved straightforwardly by the Laplace transform. The ensuing time evolution is consistent with our physical expectations in all parameter regions some results are shown in Fig. 5. 

First, there are terms of the form <5S(12)[z —0L (12 z)] 5S(12)>o. From its definition in (9.12), we see that 55(12) is the deviation of the reactive operator from its velocity average. The correlation function above characterizes the time evolution (Laplace transformed) of these fluctuations. If the chemical reaction is slow, we expect that perturbations of the velocity distribution induced by the reaction will be small hence such contributions may be safely neglected in this limit. This argument may be made more formal using limiting procedures analogous to those described in Section V. In principle, one may also use this term to introduce a modification to in S(/- 2) due to velocity relaxation effects. This will lead to some effective reactive collision frequency in place of k p. 

The Fourier-Laplace transform of the time derivative of the evolution operator (see Appendix A) is also worth of interesf ... 

Equation [10] can be solved using Laplace transform techniques to give the time evolution of the cmrent, i(t), subject to the boimdary conditions described resulting in eqn [14] ... 

Equations for planar (Tj in Table 15.1) and spherical (Tj 2 in Table 15.1) electrodes can be solved using the Laplace transform technique to give, after considering Eq. (15.2), the time evolution of the current (Q. 

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