Laplace transform pairs

Table 3.1 gives further Laplace transforms of common functions (called Laplace transform pairs). 

In practice, inverse transformation is most easily achieved by using partial fractions to break down solutions into standard components, and then use tables of Laplace transform pairs, as given in Table 3.1. 

This transform is widely used to formulate semi-Markov stochastic models, where t and / (t) are the random variable and its probability density function, respectively. In Table F.l, we briefly report some Laplace transform pairs. 

Another method of getting the numerical inverse is by the Fourier series approximation. The Laplace transform pairs are given as in Eqs. 9.1 and 9.3, written again for convenience... 

Using the well-known Laplace transform pair,... 

Table 3.1 lists some important Laplace transform pairs that occur in the solution of linear differential equations. For a more extensive list of transforms, see Dyke (1999). 

A special situation arises in the limit of small scavenger concentration. Mozumder (1971) collected evidence from diverse experiments, ranging from thermal to photochemical to radiation-chemical, to show that in all these cases the scavenging probability varied as cs1/2 in the limit of small scavenger concentration. Thus, importantly, the square root law has nothing to do with the specificity of the reaction, but is a general property of diffusion-dominated reaction. For the case of an isolated e-ion pair, comparing the t—°° limit of Eq. (7.28) followed by Laplace transformation with the cs 0 limit of the WAS Eq. (7.26), Mozumder derived... 

The limitation of the prescribed diffusion approach was removed, for an isolated ion-pair, by Abell et al. (1972). They noted the equivalence of the Laplace transform of the diffusion equation in the absence of scavenger (Eq. 7.30) and the steady-state equation in the presence of a scavenger with the initial e-ion distribution appearing as the source term (Eq. 7.29 with dP/dt = 0). Here, the Laplace transform of a function/(t) is defined by... 

An interesting approach has been employed in paper [74] to find the distribution f(li, l2) of copolymer chains for numbers l and h of monomeric units Mi and M2. This distribution is evidently equivalent to the SCD, because the pair of numbers k and I2 unambiguously characterizes chemical size (l = h + l2) and composition ( 1 = l] //, 2 = h/l) of a macromolecule. The essence of this approach consists of invoking the Superposition Principle [81] that enables the problem of finding the Laplace transform G(pi,p2) of distribution f(li,k) to be reduced to the solution of two subsidiary problems. The first implies the derivation of the expression for the generating function [/(z1",z 2n ZjX,z ) of distribution P(ti, M2 mt, m2), and the second is concerned with finding the Laplace transforms g (pi,p2) and (pi,p2) of distributions (Eq. 91). With these two problems solved, it is possible to obtain the characteristic function G(pi,p2) of distribution f(li,h) using the Superposition Principle formula... 

The pair of differential equations are solved either directly or with Laplace transform, with a table of inverses. 

A study of electron scavenging in multipair clusters [21] has shown that the total scavenging probability decreases with increasing number of ion pairs in the cluster. However, the Laplace transform relationship [Eq. (28)] between the scavenging probability and the recombination kinetics was found to work reasonably well also in the multipair case. 

Before discussing the solution of these equations, it is prudent to known where we are aiming to go In Chaps. 6 and 7, the quantities of interest were shown to be the survival and recombination probabilities. From the diffusion equation, the former was found to be the volume integral of the probability density, p(r, and was discussed in Chap, 6, Sect. 2. Recombination probabilities are similarly the time and volume integrals of the rate of reaction of the pair. In a similar manner, the survival probability in Laplace transform space is... 

Equation (345) shows that the average scavenging probability of an ion-pair initially separated by a distance r0 is the Laplace transform (into k%c space) of the lifetime distribution function. 

Equation (366) is a formal solution for the Laplace transform of the pair density [103]. This can be inverted to give a time-dependent form of P(r, t) t... 

This same pair of reactions was also studied by Ishida,10 who used a Laplace transform technique. Although superficially quite different results were obtained, they are in fact identical to those presented above. 

The distribution function f(R) can be also found from experimental data on the kinetics of recombination luminescence [19,20] either by applying the inverse Laplace transform or by using eqn. (16). An example of the determination of the form of the distribution of reacting pairs over the distances from the data on the kinetics of the tunneling recombination luminescence will be given below in Chap. 6, Sect 2.3. 

In the case of the non-pair reagent distribution, the distribution function f(i ) can also be found from the experimental kinetic curve using the inverse Laplace transform. Actually, taking into account that i(R) = N(R,Q)/N, it is easy to obtain [16] from eqns. (21) and (22) that... 

We shall now determine the inverse Laplace transform of this function to find the concentration of B at time t. In tables of Laplace transforms the following pair has been found ... 

Table F.l Some Laplace transform properties and pairs of functions used as probability density functions for semi-Markov modeling.

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