NUMERICAL TREATMENT OF DIFFERENTIAL ALGEBRAIC EQUATIONS

Abstract

Fatma Mohamed Yousry Mohamed. "Numerical Treatment of Differential Algebraic Equations." Doctor of Philosophy of Science dissertation (Pure Mathematics) .University College of Women for Art, Science and Education , Ain Shams University The main purpose of this thesis is to study; the proposed numerical methods for solving ordinary differential equations and differential algebraic equations. This thesis is divided into six chapters: In chapter 1, the definition of differential equation and it’s sources are presented. Some fundamentals are mentioned such as, index, Index Reduction and Consistent initial values. Types of the differential algebraic equation are presented. Hessenberg forms are discussed. In chapter 2, numerical methods for solving ODEs and DAEs are discussed such as Runge-Kutta method, linear multistep method, Backward Differentiation Formulae, Extended BDF, Modified Extended BDF, Parametric class and its extended and Hybrid method. Order of MEBDF applied to DAEs. In chapter 3, three classes of hybrid methods to solve systems of differential algebraic equations (DAEs) and its stability analysis are introduced. These classes are based on a free parameter class of linear multistep method (LMM). Two classes contain one step point and one stage point (off-step point) of the first derivative of the solution. The third one contains two step points and one stage point of the first derivative of the solution. ii In chapter 4, the one-leg twin of the first two hybrid classes in chapter 3 are studied for step k=2 and k=3. The order of convergence of these methods are determined according to the value of the parameters and compared to the order of convergence of their twin hybrid multistep methods. The G-stability of these methods are studied. Finally, the methods are tested by solving DAEs. Chapter 5 focuses on the implemented of the three hybrid classes and its twin one-leg methods on the implicit mixed differential algebraic equations. The orders of convergence for the above methods are discussed. Numerical tests are introduced. In chapter 6 some practical problem are solved by the proposed classes which introduced in chapters 3 and 4. Keywords: Stiff ODEs; DAEs; Multistep Methods; BDF; Hybrid Methods; Stability Aspects ; One-leg Methods; G-Stability; Order of convergence.