عنوان

Singular complex periodic solutions of van der Pol's equation and uniform approximations for the solution of Lagerstrom's model problem

TLpq303866424

انگلیسی

Singular complex periodic solutions of van der Pol's equation and uniform approximations for the solution of Lagerstrom's model problem

[Thesis]

M. S. Tajdari

C. Hunter

The Florida State University

1990

143

Ph.D.

The Florida State University

1990

Two problems are studied. First, the analytic continuation of the real periodic solutions of van der Pol's equation to complex values of the damping parameter usd\varepsilonusd are discussed. This continuation shows the existence of an infinite family of singular complex periodic solutions associated with values of usd\varepsilonusd lying on two curves usd\Gammausd and usd\bar\Gammausd (see Figure 8) located symmetrically in the usd\varepsilonusd-plane. These singular solutions are found to cause the existence of the moving singularities of the Poincare-Lindstedt series for the real limit cycle which were developed at great length by Andersen and Geer (7), and were analyzed, using Pade approximants, by Dadfar et al. (10). A numerical method for the computation of these singular solutions is described. In addition, an asymptotic description of them for large values of usd\vert\varepsilon\vertusd is obtained using the method of matched asymptotic expansions. Our results suggest that the existence of the complex singular solutions may, in general, play an important role in the utility of computer-generated perturbation expansions at moderate or large values of the perturbation parameter. Our second study involves a model, due to Lagerstrom, of the steady flow of a viscous incompressible fluid past an object in (m + 1) dimensions. The model is a nonlinear boundary-value problem in the range usd\varepsilon \leq x < \inftyusd, with usd\varepsilonusd denoting a positive real parameter. We describe, using the method of matched asymptotic expansions, the construction of a three-term inner and outer expansion for usd\varepsilonusd small and for m = 1 and 2. The predictions from these matchings are then compared with exact results obtained by numerical integration. These comparisons show that the asymptotic results have a small range of usefulness in usd\varepsilonusd. Greatly improved approximations are obtained when a single expansion, structurally similar to the outer expansion, is used throughout the entire interval usd\varepsilon \leq x < \inftyusd. The approximations obtained with this approach are found to be useful for moderate and large values of usd\varepsilonusd as well as for usd\varepsilonusd small. The effectiveness of this approach suggests an iterative method of solution of Lagerstrom's model problem. We prove rigorously that this iteration converges to a unique solution for all real usd\varepsilon >usd 0 and usdm >usd 0. Our results suggest that a similar iteration may be an effective method of approximation of viscous flows at moderate Reynolds numbers.

Mathematics

Pure sciences

C. Hunter

M. S. Tajdari

p

[Thesis]

276903

a

Y