Regular Perturbation Approaches

Sometimes a problem is encountered in which the governing equation is almost identical to a simpler problem that one already knows how to solve. Hopefully, some modification of the new problem can resolve the difference and lead to useful results. For example, consider the following model for a species A in a reacting system [12]  

Suppose Equation 6.141 is subject to the following boundary conditions  

except for Equation 6.145, the method of separation of variables is applicable. Observe that the terms of the denominator of the right-hand side of Equation 6.145 are all positive physical quantities. That is, the right-hand side of Equation 6.145 is expected to have a power series expansion. Eirst, the variables are redefined in terms of dimensionless quantities by 

The otherwise nonlinear problem can be reduced to a set of linear problems by substituting Equation 6.151 and Equation 6.152 into Equation 6.146 to Equation 6.150 to get 

Then equate like powers of e in both the differential equation and the given conditions as follows  

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Following Whitehead s approach, we seek a solution of (9-75) and (9-76) by means of a regular perturbation expansion of the form... 

The sixth chapter provides instruction in a number of solution techniques for linear partial differential equations. Also included is a section introducing regular perturbation, a common approach to solving some nonlinear differential equations. Since the material in this chapter will be a new experience for a large segment of the readership, a substantial number of drill-type examples are included. 

One final approach to the development of a regular perturbation series is worth examining, and that is the renormalization perturbation theory of Sadlej et al. (1994, 1995). This approach follows the line of direct perturbation theory, but the formal perturbation is different. Writing the Dirac equation with 1 /c extracted from the small component according to (17.60) and replacing l/c2 in the metric with we have... 

A consistent and regular perturbation theory has, however, been set up recently in the four-component framework. It is known under the name of direct perturbation theory . This approach uses the four-component non-relativistic... 

Instead of estimating -/ 1 ln(exp(-/W(f)) directly using (5.44), one can use cumulant expansion approaches, as in regular free energy perturbation theory (see e.g., [20, 39] for combining cumulant expansions about the initial and final states). Unbiased estimators for cumulants can be used. Probably the most useful relations involve averages and variances of the work ... 

Expansion is considered for finite, regular polyethylene stars perturbed by the excluded volume effect. An RIS model is used for the chain statistics. The number of bonds in each branch ranges up to 10 240, and the functionality of the branch point ranges up to 20. The form of the calculation employed here provides a lower bound for the expansion. If the number, n, of bonds in the polymers is heid constant, expansion is found to decrease with increasing branch point functionality. Two factors dictate the manner in which finite stars approach the limiting behavior expected for very large stars, These two factors are the chain length dependence at small n of the characteristic ratio and of fa -a3) / n1/2. 

Charged point defects on regular lattice positions can also contribute to additional losses the translation invariance, which forbids the interaction of electromagnetic waves with acoustic phonons, is perturbed due to charged defects at random positions. Such single-phonon processes are much more effective than the two- or three phonon processes discussed before, because the energy of the acoustic branches goes to zero at the T point of the Brillouin zone. Until now, only a classical approach to account for these losses exists, which has been... 

Table 3 Optimized regular interatomic distance (in A) and gain in energy per C2H2 unit (in eV) of regular polyacetylene by report to the alternating ground state, with the Neel state based higher order and the RVB ansatze. Results from Local-density-functional approach of Mintmire and White [48], Hartree-Fock and Mpller-Plesset perturbation Theory of Suhai [50], or Ashkenazi [49] are included.

As an alternative approach to conventional uptake measurements, in the frequency response technique [44-48] one follows the response of the sample to a regular periodic perturbation, e.g. a sinusoidal variation of the system volume. Using complex notation, one may write for the time dependence of the system volume,... 

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