Differential-algebraic Equations: Analysis and Numerical SolutionDifferential-algebraic equations are a widely accepted tool for the modeling and simulation of constrained dynamical systems in numerous applications, such as mechanical multibody systems, electrical circuit simulation, chemical engineering, control theory, fluid dynamics and many others. This is the first comprehensive textbook that provides a systematic and detailed analysis of initial and boundary value problems for differential-algebraic equations. The analysis is developed from the theory of linear constant coefficient systems via linear variable coefficient systems to general nonlinear systems. Further sections on control problems, generalized inverses of differential-algebraic operators, generalized solutions, and differential equations on manifolds complement the theoretical treatment of initial value problems. Two major classes of numerical methods for differential-algebraic equations (Runge-Kutta and BDF methods) are discussed and analyzed with respect to convergence and order. A chapter is devoted to index reduction methods that allow the numerical treatment of general differential-algebraic equations. The analysis and numerical solution of boundary value problems for differential-algebraic equations is presented, including multiple shooting and collocation methods. A survey of current software packages for differential-algebraic equations completes the text. The book is addressed to graduate students and researchers in mathematics, engineering and sciences, as well as practitioners in industry. A prerequisite is a standard course on the numerical solution of ordinary differential equations. Numerous examples and exercises make the book suitable as a course textbook or for self-study. |
Contents
Introduction | 3 |
Linear differentialalgebraic equations with constant coefficients | 13 |
Linear differentialalgebraic equations with variable coefficients 556 | 56 |
Nonlinear differentialalgebraic equations | 151 |
Numerical methods for strangenessfree problems | 217 |
Numerical methods for index reduction | 273 |
Boundary value problems | 298 |
Software for the numerical solution of differentialalgebraic | 352 |
359 | |
373 | |
Common terms and phrases
algebraic equations analysis apply assumptions BDF methods block row boundary value problem C¹(I canonical form characteristic values Cm,n collocation columns computation consider consistent initial values constant coefficients constraints control problem convergence corank Corollary corresponding defined Definition derivative array differential-algebraic equation differentiation index Drazin inverse equivalent F(ti feedback follows Fréchet derivative full row rank given global Hence Hypothesis 3.48 implicit function theorem implies index reduction inhomogeneity initial condition initial value problem inverse iord Jacobian kernel Lemma linear differential-algebraic equations manifold matrix pair multi-step method multibody systems nilpotent nodal analysis nonsingular matrix numerical solution O(hk obtain ordinary differential equation pointwise nonsingular polynomial Proof pseudoinverse reduced problem Runge-Kutta methods satisfies Hypothesis 4.2 Section solve strangeness index sufficiently small sufficiently smooth Theorem ti+1 transformations unique solution uniquely solvable variables vector x(to Xi,j Xi+1 yields