We give an overview of the numerical solution of the initial value differential-algebraic equation (DAB) from the underlying theory, through both the development of numerical techniques and software and a survey of the major areas of application, to the implementation of available software codes in the solution of DABs arising in applications. The experimental part serves to demonstrate the need for specific DAB, rather than simply ordinary differential equation (ODE), methods and the special considerations requited for the successful numerical solution of DAEs, as well.as verifying predictions made in the theory. It is hoped that '.his dissertation can be used as a reference by those working in areas of application, or at least as a pointer to the relevant literature. / GR 2016

For any given complex algebraic ordinary differential equation (ODE), one major task of both pure and applied mathematicians is to find explicit meromorphic solutions due to their extensive applications in science. In 2010, Conte and Ng in [12] proposed a new technique for solving complex algebraic ODEs. The method consists of an idea due to Eremenko in [20] and the subequation method of Conte and Musette, which was first proposed in [9]. Eremenko’s idea is to make use of the Nevanlinna theory to analyze the value distribution and growth rate of the solutions, from which one would be able to show that in some cases, all the meromorphic solutions of the studied differential equation are in a class of functions called “class W”, which consists of elliptic functions and their degenerates. The establishment of solutions is then achieved by the subequation method. The main idea is to build subequations which have solutions that also satisfy the original differential equation, hoping that the subequations will be easier to solve. As in [12], the technique has been proven to be very successful in obtaining explicit particular meromorphic solutions as well as giving complete classification of meromorphic solutions. In this thesis, the necessary theoretical background, including the Nevanlinna theory and the subequation method, will be developed. The technique will then be applied to obtain all meromorphic stationary wave solutions of the real cubic Swift-Hohenberg equation (RCSH). This last part is joint work with Conte and Ng and will appear in Studies in Applied Mathematics [13]. RCSH is important in several studies in physics and engineering problems. For instance, RCSH is used as modeling equation for Rayleigh- B?nard convection in hydrodynamics [43] as well as in pattern formation [16]. Among the explicit stationary wave solutions obtained by the technique used in this thesis, one of them appears to be new and could be written down as a rational function composite with Weierstrass elliptic function. / published_or_final_version / Mathematics / Master / Master of Philosophy

Thesis (Ph. D.)--University of Washington, 1999. / Vita. Includes bibliographical references (p. 124-129).

Systems of differential-algebraic equations (DAEs) arise in many areas including electrical circuit simulation, chemical engineering, and robotics. The difficulty of solving a DAE is characterized by its index. For index-1 DAEs, there are several solvers, while a high-index DAE is numerically much more difficult to solve. The DAETS solver by Nedialkov and Pryce integrates numerically high-index DAEs. This solver is based on explicit Taylor series method and is efficient on non-stiff to mildly stiff problems, but can have severe stepsize restrictions on highly stiff problems. Hermite-Obreschkoff (HO) methods can be viewed as a generalization of Taylor series methods. The former have smaller error than the latter and can be A- or L- stable. In this thesis, we develop an implicit HO method for numerical solution of stiff high-index DAEs. Our method reduces a given DAE to a system of generally nonlinear equations and a constrained optimization problem. We employ Pryce’s structural analysis to determine the constraints of the problem and to organize the computations of higher-order Taylor coefficients (TCs) and their gradients. Then, we use automatic differentiation to compute these TCs and gradients, which are needed for evaluating the resulting system and its Jacobian. We design an adaptive variable-stepsize and variable-order algorithm and implement it in C++ using literate programming. The theory and implementation are interwoven in this thesis, which can be verified for correctness by a human expert. We report numerical results on stiff DAEs illustrating the accuracy and performance of our method, and in particular, its ability to take large steps on stiff problems. / Thesis / Doctor of Philosophy (PhD)

Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references. Available also from UMI/Dissertation Abstracts International.

Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references (leaves 104-112). Available also in a digital version from Dissertation Abstracts.

Thesis (Ph. D.)--University of Tennessee, Knoxville, 2005. / Title from title page screen (viewed on Feb. 14, 2006). Vita. Includes bibliographical references.

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Thesis (Ph. D.)--University of Texas at Austin, 2005. / Supervisor: Thomas F. Edgar. Vita. Includes bibliographical references.