Applied Mathematics Graduate Theses & Dissertations

Spring 1-1-2013

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Applied Mathematics

Bengt Fornberg

Mark Ablowitz

Thomas Manteuffel

Barbara Prinari

Harvey Segur

Abstract

This thesis explores two unrelated research topics. The first is a numerical study of the fourth Painleve equation, while the second is a characterization of the stability domains of Adams predictor-corrector methods. First, the six Painleve equations were introduced over a century ago, motivated by theoretical considerations. Over the last several decades these equations and their solutions have been found to play an increasingly central role in numerous areas of mathematical physics. Due to extensive dense pole fields in the complex plane, their numerical evaluation remained challenging until the recent introduction of a fast `pole field solver' (Fornberg and Weideman, J. Comp. Phys. 230 (2011), 5957-5973). This study adapts this numerical method to allow for either extended precision or faster numerical solutions to explore the solution space of the fourth Painleve (PIV) equation. This equation has two free parameters in its coefficients, as well as two free initial conditions. After summarizing key analytical results for PIV , the present study applies this new computational tool to the fundamental domain and a surrounding region of the parameter space. We confirm existing analytic and asymptotic knowledge about the equation, and also explore solution regimes which have not been described in the previous literature. In particular, solutions with the special characteristic of having adjacent pole-free sectors, but with no closed form, are identified. Second, the extent that the stability domain of a numerical method reaches along the imaginary axis indicates the utility of the method for approximating solutions to certain differential equations. This maximum value is called the imaginary stability boundary (ISB). It has previously been shown that exactly half of Adams-Bashforth (AB), Adams-Moulton (AM), and staggered Adams-Bashforth methods have nonzero stability ordinates. In the last chapter of this thesis, two categories of Adams predictor-corrector methods are considered, and it is shown that they have a nonzero ISB when (for a method of order p) p = 1,2, 5,6, 9,10,... for ABp-AMp and p = 3,4, 7,8, 11,12,... in the case of and AB(p-1)-AMp.

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