Approximate analytical methods

Optimal control theory, as discussed in Sections II-IV, involves the algorithmic design of laser pulses to achieve a specified control objective. However, through the application of certain approximations, analytic methods can be formulated and then utilized within the optimal control theory framework to predict and interpret the laser fields required. These analytic approaches will be discussed in Section VI. 

Example 5.3 The Semi-infinite Solid with Variable Thermophysical Properties and a Step Change in Surface Temperature Approximate Analytical Solution We have stated before that the thermophysical properties (k, p, Cp) of polymers are generally temperature dependent. Hence, the governing differential equation (Eq. 5.3-1) is nonlinear. Unfortunately, few analytical solutions for nonlinear heat conduction exist (5) therefore, numerical solutions (finite difference and finite element) are frequently applied. There are, however, a number of useful approximate analytical methods available, including the integral method reported by Goodman (6). We present the results of Goodman s approximate treatment for the problem posed in Example 5.2, for comparison purposes. 

Another approach involves the analysis of equations corresponding to the discrete steps of combustion. The ignition parameters are calculated from the coupling condition of the steps by computerized numerical, or approximate analytical methods. In Ref. where the heterogeneous ignition of a condensed system was analyzed, a sharp transition from activation to diffusion control was assumed as ignition criterion ... 

When analytical solutions are not known and the approximate analytical methods give results of limited applicability, the numerical methods may be a solution. Let us first discuss a method based on the diagonalization of the second-harmonic Hamiltonian [48,49]. As we have already said, the two parts of the Hamiltonian Ho and Hi given by (55), commute with each other, so they are both constants of motion. The //0 determines the total energy stored in both modes, which is conserved by the interaction ///. This means that we can factor the quantum evolution operator... 

The boundary layer equations are valid only in the region between the front stagnation point and the separation point. Behind the separation point there is a wake region with absolutely different hydrodynamic laws. The position of the separation point can be determined either experimentally or by using numerical or approximate analytical methods. 

As we learned in this chapter, the formulation of unsteady distributed problems leads to partial differential equations. The solution of these equations is much more involved than that of ordinary differential equations. Among the techniques available, the analytical and computational methods are most frequently referred to. Exact analytical methods such as separation of variables and transform calculus are beyond the scope of the text. However, the method of complex temperature and the use of charts based on exact analytical solutions, being useful for some practical problems, are respectively discussed in Sections 3.4 and 3.6. Among approximate analytical methods, the integral method, already introduced in Sections 2.4 and 3.1, is further discussed in Section 3.5. The analog solution technique is also briefly treated in Section 3.7. 

The coupling of the transport of momentum with the mass transport practically excludes any analytical solution in the field of physico-chemical hydrodynamics of bubbles and drops. However, a large number of effective approximate analytical methods have been developed which make solutions possible. Most important is the fact, that the calculus of these methods allows to characterise different states of dynamic adsorption layers quantitatively weak retardation of the motion of bubble surfaces, almost complete retardation of bubble surface motion, transient state at a bubble surface between an almost completely retarded and an almost completely free bubble area. 

Approximate analytical methods, based on a second quantization approach, have been developed by HOPAGKER and LEVINE /93/, making use of curvilinear coordinates, to describe the non-adiabatic transitions leading to a population inversion of the products vibrational states. 

Methods to calculate the pressure distribution in the contact zone are presented by Johannesson (4) and Johannesson and Kassfeldt (7). In (4) a semi-empirical method for the calculation of the pressure distribution in an 0-ring seal contact for arbitrary sealed pressures is presented, cind in (7) an approximate analytical method, for calculation of the pressure distribution in an arbitrary elastomeric seal contact, is suggested. In both these papers measurements verifying the calculated pressure distributions are also presented. 

The function g(X), arrived at by Smith for a crosslinked styrene-butadiene rubber is related to the strain measure in the sense of the present approach by the equation S-g(X) - 3(X - X"l)/g(X). SgCX) calculated from Smith s data is also shown in Fig. 1. Using the approximate analytical method to analyze constant-stretching-rate data, Hong et al. obtained a function F(X), which equals Se(X)/3, also for a styrene-butadiene rubber vulcanlzate. Figure 1 also shows 3F(X). 

The error in doing this involves the neglect of coupling entropy and consequently the free energy and enthalpy of interaction become equal. Cvdw calculation is a difficult task, and is usually done by numerical integration of potential functions of the Lennard-Jones type as in equation (48). An approximate analytical method for Cvdw evaluation is described in Ref. 35. 

Calculated with an approximate analytical method adapted for nonspherical solute. ... 

We would also like to record our observation of the favorable trend toward improved accuracy in rate constant determinations that was evident in recent work and deserves comment. These advances may be attributed primarily to improved experimental techniques, particularly the development of new, sensitive detection schemes and data recording methods, and to the expanding role of the computer in both the design of experiments and in data reduction. The growing use of computers to handle extensive reaction mechanisms, and thereby account for the effects of secondary reactions more accurately than by approximate analytical methods, allows the experimentalist to optimize the... 

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