Approximate analytic solutions for

Boyd and Gordon (1961) showed that explicit analytical expressions for the electric field distribution in the transverse modes may be obtained by allowing the limits of integration in equation (12.4) to tend to infinity. This is a valid approximation for stable resonators having Fresnel numbers which satisfy tlue condition F 1, i.e. the size of the mirror aperture is large compared with the size of the laser cavity mode. The condition that stable solutions exist for radiation which is propagated back and forth within the resonator is then equivalent to requiring that the field 

Frequently the symmetry of the transverse modes is determined by the circular cross-section of the laser discharge tube. In this case the two-dimensional Fourier transform is expressed in polar coordinates (r,0) as 

Hevmite polynomials and generalized Laguerre polynomials of low order 

The mode patterns of a laser oscillating in some of the lowest-order modes of rectangular symmetry are shown in Fig.12.4. These should be compared with the diagrammatic representation of electric fields shown in Fig.12.2. The higher-order modes clearly have more extended field distributions and consequently higher diffraction losses than the lower-order modes, as we have already remarked. The wavefronts of these modes contain one or more phase reversals of Ti arising from the particular form of the Hermite or Laguerre polynomial which determines the field distribution. 

The lowest-order mode has identical form in both rectangular and cylindrical symmetry  

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In a recent paper, an approximate calculation was made of effects (b) to (d) above (19), using an approximate analytical solution for the diffusion problem, for the case where the reaction occurs readily over a short range of separation distances of the reactants. In the present report, we summarize the results of our recent calculations on a numerical solution of the same problem. A more complete description is given elsewhere (28). One additional modification made here to (19) is to ensure that the current available rate constant data at AG° = 0 (Appendix) are satisfied. 

Thus even approximate analytical solutions are often more instructive than the more accurate numerical solutions. However considerable caution must be used in this approach, since some of the approximations, employed to make the equations tractable, can lead to erroneous answers. A number of approximate solution for the hot spot system (Eq 1) are reviewed by Merzhanov and their shortcomings are pointed out (Ref 14). More recently, Friedman (Ref 15) has developed approximate analytical solutions for a planar (semi-infinite slab) hot spot. These were discussed in Sec 4 of Heat Effects on p H39-R of this Vol. To compare Friedman s approximate solutions with the exact numerical solution of Merzhanov we computed r, the hot spot halfwidth, of a planar hot spot by both methods using the same thermal kinetic parameters in both calculations. Over a wide range of input variables, the numerical solution gives values of r which are 33 to 43% greater than the r s of the approximate solution. Thus it appears that the approximate solution, from which the effect of the process variables are much easier to discern than from the numerical solution, gives answers that differ from the exact numerical solution by a nearly constant factor... 

We have so far been able to obtain exact explicit analytic solutions for (a) the case where only processes (i) and (ii) are significant, and (b) the case where only processes (ii) and (iii) are significant. We have also obtained an approximate analytic solution for the case where all three processes (i), (ii) and (iii) occur, but where the loss of radicals occurs predominantly by process (ii) rather than by prodess (iii). As a generalisation of case (a), we have obtained a general solution which covers the case where the parameters which characterise the processes (i) and (ii) are themselves time-dependent. The general solution to case (b) requires modification if processes of type (ii) do not occur. Complete solutions have been obtained for three special cases of (b), namely, decay from a Stockmayer-01Toole distribution of locus populations, decay from a Poisson distribution of locus populations, and decay from a homogeneous distribution of locus populations. 

Approximate Analytical Solutions for Models of Three-Dimensional Electrodes by Adomian s Decomposition Method... 

In this equation A is the molecular diffusion coefficient expressed as m2/sec, and is the number of liquid moles of component i. This equation can be simplified using approximate analytical solutions for the transient diffusion equation in the vertical direction to ... 

The only approximate analytical solution for the RSA of a binary mixture of hard disks was proposed by Talbot and Schaaf [27], Their theory is exact in the limit of vanishing small disks radius rs — 0, but fails when the ratio y = r Jrs of the two kinds of disk radii is less than 3.3 its accuracy for intermediate values is not known. Later, Talbot et al. [28] observed that an approximate expression for the available area derived from the equilibrium Scaled Panicle Theory (SPT) [19] provided a reasonable approximation for the available area for a non-equilibrium RSA model, up to the vicinity of the jamming coverage. While this expression can be used to calculate accurately the initial kinetics of adsorption, it invariably predicts that the abundant particles will be adsorbed on the surface until 6=1, because the Scaled Particle Theory cannot predict jamming. 

The simulation of catalyst deactivation by coke formation using a 3-dimensional site-bond-site network model is highly attractive, especially for zeolites, as the processes occurring in cavities (also referred to as voids or intersections) and in channels (also referred as necks, capillaries or arcs) can be readily distinguished. This model is flexible and the cormectivity of pores as well as the local homogeneity of the catalyst can be readily altered. Further, a percolation theory is available for site-bond-site models. In the particular case of Bethe lattices, approximated analytical solutions for the percolation probabilities have been derived[7]. 

Establishing the bifurcation point and finding out the post-bifurcation trends are the main and most valuable results of the theoretical study of non-equilibrium physico-chemical systems. After that, the investigation of mathematical model is essentially completed. Sometimes, in simplest cases, one can derive an approximate analytical solution for evolution of the system in time, or obtain the numerical solution by computer simulations, and compare these with real experimental dependencies. Though, such solutions could never be rigorous. Because of complexity of real processes and unavoidable simplifications adopted in the model, they would reflect only qualitative trends and, thus, could not be used for quantitative calculations. 

The approximate analytical solutions for wave functions and energy levels had shown the following. P-type contribution is dominant at the defect distance zo from the surface 0 10 ag. Here is effective Bohr radius that depends strongly on effective mass p, and dielectric permittivity 82 S2/p- (see Fig. 4.13). It is seen from the Fig. 4.13... 

The Xi are the solutions of a fourth degree equation the coefficients of which are given explicitly in the literature [34, 70]. In the next paragraph we will present exact numerical simulations for g x) for a single three level system. Later we will use approximate analytical solutions for specific time regimes of g (t) which render the physical interpretation of the observed effects more intelligible. 

Gorius, A. Bailly, M., and Tondeur, D.. Perturbative solutions for non-linear fixed-bed adsorption Approximate analytical solutions for asymptotic fronts, Chem. Eng. Sci., 46(2), 677-684 (1991). 

Scott K, Sun Y-P (2007) Approximate analytical solutions for models of three-dimensional electrodes by Adomian s decomposition method. In Vayenas C, White R, Gamboa-Aldeco ME (eds) Modem aspects of electrochemistry 41. Springer, New York, pp 221-304... 

Mimson-McGee, S.H., 2002. An approximate analytical solution for the fluid dynamics of laminar flow in a porous tube. J. Membr. Sci. 197, 223-230. 

The standard approaches usually use the quasiclassical approximation, that is, the initial conditions are selected in accordance with quantum mechanics and then the evolution of the phase space points is treated purely classically. The quasiclassical approximation thus requires that the trajectories are initiated in the semiclassical eigenstates of molecules. This is easily accomplished for diatomic molecules. For example, approximate analytical solutions for the rotating Morse oscillator have been derived which allow for straightforward selections of initial conditions. If the molecular vibrations are such that the harmonic approximation can be made, then the required analytical relationship is even simpler. ... 

FIG. 3.20. Asymptotic and approximate analytical solutions for the maximum temperature rise. — Exact solution,-Asymptotic solution (a < 0.1), - - O - - Approxi-... 

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