Asymptotic methods

Brujin, N. G. de. Asymptotic Methods in Analysis, Dover, New York (1981). 

BenE74a Bender, E. A. Asymptotic methods in enumeration. SIAM Revue 16 (1974) 485-515. 

In 1958 N. N. Bogoliubov and Y. A. Mitropolsky (B.M.) published a treatise entitled Asymptotic Methods in the Theory of Nonlinear Oscillations,18 which presents a considerable generalization of the early K.B. theory. Since a detailed account of this work is beyond the scope of this book, we give only a few of its salient points. 

In line with the general asymptotic method, Mitropolsky introduced a further generalization18 by means of an additional independent variable r (the slow time ), whereby it becomes possible to investigate not only the stationary state, but also the transient one. 

This enlarges the scope of problems that can be treated by these asymptotic methods. For example, the important problem of nonlinear resonance could otherwise be solved only in the stationary state. With this extension it is possible to determine what happens when the zone of resonance is passed at a certain rate. Likewise, with the additional extension for the slow time it is possible to attack the problem of modulated oscillations, which has previously remained outside the scope of the general theory. 

The asymptotic method For most systems, Vij ikiA VgIQg- Then at long times. 

REFERENCES de Brujin, N. G. Asymptotic Methods in Analysis, Dover, New York (1981) Folland, G. B., Advanced Calculus, Prentice-Hall, Saddle River, N.J. (2002) Gradshteyn, I. S., and I. M. Ryzhik, Tables of Integrals, Series, and Troducts, Academic, New York (2000) Kaplan, W., Advanced Calculus, 5th ed., Addison-Wesley, Redwood City, Calif. (2003). 

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I - Asymptotic Methods and Perturbation Theory, Springer, Berlin, 1999. 

Williams, F. A. 1988. Asymptotic methods for flames with detailed chemistry. In Mathematical modeling in combustion science. Eds. J. D. Buckmaster and T. Takeno. New York Springer-Verlag. 44-51. 

Williams, F. A. 1996. Use of asymptotic methods for predicting pollutant production in turbulent combustion. In Dynamics of exothermicity in honor of Antoni Kazimierz Oppenheim. Ed. J. R. Bowen. Combustion science and technology ser. Amsterdam Gordon and Breach 2 75-95. 

The procedure outlined thus allows a simple construction of a uniformly valid formal asymptotic solution, free of the difficulties invoked by the use of the standard asymptotic methods when applied to a situation similar... 

M. J. Ward, Asymptotic Methods in Semiconductor Device Modeling, Ph.D. thesis, Caltech, Pasadena, CA, 1988. 

B.M. Smirnov, Asymptotic Methods in the Theory of Atomic Collisions, Atomizdat, Moscow, 1973 (in Russian). 

In the three-dimensional case, the computation of the exchange matrix element can be carried out by the asymptotic method [15]. Without dwelling upon the details of the calculation, we quote the final result... 

Despite the diversity of the studies being carried out, they had a single ideological and methodological platform at their foundation was the strong dependence of the chemical reaction rate on temperature, and various related threshold phenomena. To obtain the basic laws of combustion, asymptotic methods were used, complemented by an explicitly physical interpretation. 

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... 

Yu.A. Mitropolskii and N. Van Dao Applied Asymptotic Methods in Nonlinear Oscillations. 

ASYMPTOTIC METHODS IN ANALYSIS, N.G. de Bruijn. An inexpensive, comprehensive guide to asymptotic methods—the pioneering work that teaches by explaining worked examples in detail. Index. 224pp. 5)4 x 8H. 64221-6 Pa. 5.95... 

We shall consider mainly publications in which asymptotic methods are used, but we also mention numerical methods, since we use numerical results for comparison with our phase-integral results. For a general review of the field we refer to Bethe and Salpeter (1957), Ryde (1976), Bayfield (1979), Koch (1981), Gallas, Leuchs, Walther and Figger (1985), Lisitsa (1987) and Gallagher (1988, 1994). 

There are also papers on calculations of M M systems NaLi [121] and Li2, Na2, K2, LiNa, LiK, NaK [122]. Comparison of the asymptotic method of calculation with results of ab initio approach [123] helps to estimate the accuracy of the former. The asymptotic method was found to give a different term pattern for different M M pairs, significant, e.g., for the possibility of crossing of and terms of M M. New information on potential curves will probably emerge from studies of radiation by M excited in collisions with atoms X [124] and M [125]. 

This tutorial paper begins with a short introduction to multicomponent mass transport in porous media. A theoretical development for application to single and multiple reaction systems is presented. Two example problems are solved. The first example is an effectiveness factor calculation for the water-gas shift reaction over a chromia-promoted iron oxide catalyst. The methods applicable to multiple reaction problems are illustrated by solving a steam reformer problem. The need to develop asymptotic methods for application to multiple reaction problems is apparent in this example. 

In particular, the receptivity to vortical disturbances was investigated theoretically by Rogler Reshotko (1975) who modeled the free stream disturbance as a convected array of harmonic vortices. Highly damped near-wall disturbances were calculated from this model. Kerschen (1991) used asymptotic method to calculate vortical receptivity and showed that to vary with the convection speed of vortices. Other Orr-Sommerfeld based models in the literature also could not reveal the physical picture seen in the experiments of Kendall (1987) and Dietz (1999), except in Sengupta et al. (2002) - that is discussed in details here. 

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