Asymptotic expansion methods

For Gr = 0, the Sherwood or Nusselt number is given by Eq. (3-44). For Gr 0, neither perturbation nor asymptotic expansion methods have proved capable of yielding solutions for Sh comparable to Eq. (3-55). At larger Gr... 

This paper gives a brief survey of the variational and asymptotic expansion methods used to solve the nonrelativistic problem, and the principal effects that must be taken into account in order to estimate the higher-order QED corrections. The lowest order QED shift can now be calculated to high precision, and there has been much recent progress on the next-to-lowest order terms. 

Fig. 3. Illustration of the physical basis for the asymptotic expansion method in which the Rydberg electron moves in the field generated by the polarized core...

Also as in the case of helium, asymptotic expansion methods can be applied to the Rydberg states of lithium and compared with high precision measurements [73,74]. This case is more difficult because the Li+ core is a nonhydrogenic two-electron ion for which the multipole moments cannot be calculated analytically, and variational basis set methods must be used instead. However, the method is in principle capable of the same high accuracy as for helium. 

Using asymptotic expansion methods Shraiman (1987) and Rosen-bluth et al. (1987) have shown that for large Pe the effective diffu-sivity is of the form... 

Bhatia and Drachman have also made important progress in applying the asymptotic expansion methods to the Rydberg states of lithium [27-29]. The calculations in this case are more difficult because the polarizable core now consists of the nucleus and two Is electrons, and so its multipole moments cannot be calculated analytically. 

Klaibring, A. (1991) Derivation of a model of adhesively bonded joints by the asymptotic expansion method. International Journal of Engineering Science, 29, 493-512. 

Two of the major techniques of homogenization are the two-scale asymptotic expansion method and the notion of two-scale convergence. When using two-scale asymptotic expansions one makes the ansatz that, e.g., the variable (the superscript e indicates the parametrization) can be written as a formal powers series... 

Matched-Asymptotic Expansions Sometimes the coefficient in front of the highest derivative is a small number. Special perturbation techniques can then be used, provided the proper scaling laws are found. See Kevorkian, J., and J. D. Cole, Perturbation Methods in Applied Mathematics, Springer-Verlag, New York (1981) and Lager-strom, P. A., Matched Asymptotic Expansions Ideas and Techniques, Springer-Verlag, New York (1988). 

Earlier modeling studies were aimed at predicting the current and temperature distributions, as the nonuniform distributions contribute to stress formation, a major technical challenge associated with the SOFC system. Flow and multicomponent transport were typically simplified in these models that focused on SOFC electrochemistry. Recently, fundamental characteristics of flow and reaction in SOFCs were analyzed using the method of matched asymptotic expansions. " ... 

Since momentum densities are unfamiliar to many. Section II outlines the connection between the position and momentum space representations of wavefunctions and reduced-density matrices, and the connections among one-electron density matrices, densities, and other functions such as the reciprocal form factor. General properties of momentum densities, including symmetry, expansion methods, asymptotic behavior, and moments, are described in... 

These results have been obtained with a time-independent perturbation method, described in Landau and Lifshitz (1977). They are actually the leading terms of the exact asymptotic expansion of the potential curves (Damburg and Propin, 1968 Cizek et al., 1986). Up to the third term, the exact result of the coupling energies for the Ictj and lo- states is... 

Jones, D. S., and M. Kline, 1958. Asymptotic expansions of multiple integrals and the method of stationary phase, J. Math. Phys., 37, 1-28. 

Another extension of the perturbation method. In the preceding chapter we have shown that p.m. interpreted in the sense of asymptotic expansion is valid... 

Perturbation method as asymptotic expansion. In the preceding section we have shown that the solution power series of k, which is absolutely convergent for every k and for every initial state bounded operator. But this is the only case in which we have succeeded in proving the convergence of the series (16. 9). 

This paper is one of the first applications of the asymptotic method in world scientific literature, a method which twenty years later has received widespread use. Now it is called the method of matched asymptotic expansions. Without introducing the terminology which later appeared, the author essentially made use of the full arsenal of this method, which today makes the problem studied in this article a textbook example of its application. An exposition of the general technique of the method of matched asymptotic expansions and numerous examples of its use may be found in monographs.3,4... 

Using field theoretic methods this divergence has been analyzed to some extent. The analysis strongly suggests that the expansion is in fact an asymptotic expansion in the mathematical sense. Using the powerful method of Borel transformation one can resum the expansion to get a unique finite result. We can thus state that the theory is well defined and all the qualitative... 

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