Laboratory for Advanced Computing

Annotated Publications in Exponential Asymptotics

32. F. B. Hanson and A. M. Wazwaz,
*Singular and turning point resonance: Small perturbations - big
effects*, **Appl. Math. Letters**, vol. 1, no. 2, pp. 137-140, 1988.

- This paper, it is shown that subtle problem differences can lead to drastic
solution differences in singular perturbation problems for 2nd order ordinary
differential equations having singular boundary and an interior turning point.
As a consequence, the problem of the authors and the intended comparison
problem of de Groen in a recent {\it Appl. Math. Letter} are quite different,
so that de Groen's model in not a reasonable approximation of ours and his
comparison is not correct.

38. F. B. Hanson,
*Singular Point and Exponential Asymptotics *,
in **Asymptotic and Computational Analysis: Conference in Honor
of Frank W. J. Olver's 65th Birthday**,
R. Wong, Editor, Marcel Dekker, NY, pp. 211-240, 1990.

- This invited publication established the seminal result that exponential
precision asymptotic expansions are valid using Poincar\'{e} algebraic
asymptotics when viewed in an extended space of multiple scaled variables, thus
removing the Poincar\'{e} degeneracy due to ignoring exponentially small terms.
This is an very important area of asymptotics because exponentially small terms
may have a globally cumulative effect, or exponentially large terms may suffer
catastrophic cancellation in some applications, such as fiber optics, even
though exponentially small terms are usually neglected. This paper was an
invited presentation at the International Symposium on Asymptotics and
Computational Analysis in honor of Frank W. J. Olver, Winnepeg, Manitoba,
Canada, June 1989.

43. A. M. Wazwaz and F. B. Hanson,
*Moments of Extinction for a Singular Perturbation Problem in the
Diffusion Process *, **SIAM J. Appl. Math.**, vol. 51(1), pp. 233-265,
February 1991.

- This paper, presented by Wazwaz at the International Conference on
Industrial and Applied Mathematics, Paris, July 1987, is a lengthy exponential
precision asymptotics application to the stochastic diffusion moments problem.
A remarkable result in this paper is that the leading order exit time
distribution can be computed asymptotically through use of uniform
approximations and exponential precision asymptotics, while not by standard
algebraic asymptotics using a similar, finite number of terms.

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