The Eigenvalue Method

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 

Denoting the Laplace transforms of n(r, t) and I(r, t) respectively by n and I, where n is the electron density and I is the outward electron current as seen from the positive ion, the authors derive the following equations  

The second boundary condition assures total finite existence probability at any time the first boundary condition implies that the recombination is fully diffusion-controlled, which has been found to be true in various liquid hydrocarbons (Allen and Holroyd, 1974). [The inner boundary condition can be suitably modified for partially diffusion-controlled reactions, which, however, does not seem to have been done.] 

Given ng and 1(R, p), Eqs. (7.32a, b) can be integrated successively from r = R to a large value of r. By definition Z(R, p) = -y where yis the recombination probability in presence of scavenger. Only for the correct value of ydo the solutions of (7.32a, b) smoothly vanish asymptotically as r—-o° otherwise, they diverge. Thus, the mathematics is reduced to a numerical eigenvalue problem of finding the correct value of I(R, p). 

The eigenvalue method was extended by Abell and Funabashi (1973) to investigate the effect of the initial distribution. This only required an integration over that distribution. However, the authors also used the effect of an external field on the free-ion yield as a further probe of the initial distribution. The 

In general notation, for a polynomial of nth-degree, the new coefficients after application of synthetic division are given by 

This procedure is repeated until aU real roots are extracted. When this is accomplished, the remainder polynomial will contain the complex roots. The presence of a pair of complex roots will give a quadratic equation that can be easily solved by quadratic formula. However, two or more pairs of complex roots require the application of more elaborate techniques, such as the eigenvalue method, which is developed in the next section. 

The concept of eigenvalues will be discussed in Chap. 2 of this textbook. As a preview of that topic, we will state that a square matrix has a characteristic polynomial whose roots are called the eigenvalues of the matrix. However, root-finding methods that have been discussed up to now are not efficient techniques for calculating eigenvalues [5]. There are more efficient eigenvalue methods to find the roots of the characteristic polynomial (see Sec. 2.8). 

It can be shown that Eq. (1.11) is the characteristic polynomial of the n x n) companion matrix A, which contains the coefficients of the original polynomial as shown in Eq, (1.56). Therefore, finding the eigenvalues of A is equivalent to locating the roots of the polynomial inEq. (1.11). 

MATLAB has its own function, roots.m, for calculating all the roots of a polynomial equation of the form in Eq. (1.11). This function accomplishes the task of finding the roots of the polynomial equation [Eq. (1.11)] by first converting the polynomial to the companion matrix A shown in Eq. (1.56), It then uses the built-in function eig.m, which calculates the eigenvalues of a matrix, to evaluate the eigenvalues of the companion matrix, which are also the roots of the polynomial Eq. (1.11)  

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Frost and Pearson treated Scheme XV by the eigenvalue method, and we have solved it by the method of Laplace transforms in the preceding subsection. The differential rate equations are... 

In the context of chemical kinetics, the eigenvalue technique and the method of Laplace transforms have similar capabilities, and a choice between them is largely dependent upon the amount of algebraic labor required to reach the final result. Carpenter discusses matrix operations that can reduce the manipulations required to proceed from the eigenvalues to the concentration-time functions. When dealing with complex reactions that include irreversible steps by the eigenvalue method, the system should be treated as an equilibrium system, and then the desired special case derived from the general result. For such problems the Laplace transform method is more efficient. 

The theoretical foundation for this kind of analysis was, as mentioned, originally laid by Taylor and Aris with their dispersion theory in circular tubes. Recent contributions in this area have transferred their approach to micro-reaction technology. Gobby et al. [94] studied, in 1999, a reaction in a catalytic wall micro-reactor, applying the eigenvalue method for a vertically averaged one-dimensional solution under isothermal and non-isothermal conditions. Dispersion in etched microchannels has been examined [95], and a comparison of electro-osmotic flow to pressure-driven flow in micro-channels given by Locascio et al. in 2001 [96]. 

Equation (5.80), Equation (5.81), and Equation (5.82) may be solved by analytical methods similar to those described in the previous sections and by the Laplace transform method, which will be dealt with in Section 5.8. In this section, the eigenvalue method is discussed. Equation (5.84) is a first-order equation, the solution of which is similar to that of the corresponding scalar equation, Equation (5.12) ... 

In chemical degradation kinetics and pharmacokinetics, the methods of eigenvalue and Laplace transform have been employed for complex systems, and a choice between two methods is up to the individual and dependent upon the algebraic steps required to obtain the final solution. The eigenvalue method and the Laplace transform method derive the general solution from various possible cases, and then the specific case is applied to the general solution. When the specific problem is complicated, the Laplace transform method is easy to use. The reversible and consecutive series reactions described in Section 5.6 can be easily solved by the Laplace transform method ... 

In the Monte Carlo method, the bonds are assigned to the different rotational states according to their conditional probabilities. These are calculated, using the eigenvalue method outlined by Flory (1969), from the corresponding statistical weight matrix ... 

Ranking of judgment matrix. By solving the eigenvalue method, the feature vector W and the maximum characteristic value... 

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