Operator Differential-Algebraic Equations Arising in Fluid Dynamics

2013 ◽  
Vol 13 (4) ◽  
pp. 443-470 ◽  
Author(s):  
Etienne Emmrich ◽  
Volker Mehrmann
Keyword(s):  
Existence And Uniqueness ◽  
Differential Algebraic Equations ◽  
Generalized Solutions ◽  
Operator Matrix ◽  
Algebraic Equations ◽  
Infinite Dimensions ◽  
Initial Value ◽  
Block Operator ◽  
Oseen Problem ◽  
Problem Data

Abstract. Existence and uniqueness of generalized solutions to initial value problems for a class of abstract differential-algebraic equations (DAEs) is shown. The class of equations covers, in particular, the Stokes and Oseen problem describing the motion of an incompressible or nearly incompressible Newtonian fluid but also their spatial semi-discretization. The equations are governed by a block operator matrix with entries that fulfill suitable inf-sup conditions. The problem data are required to satisfy appropriate consistency conditions. The results in infinite dimensions are compared in detail with those known for the DAEs that arise after semi-discretization in space. Explicit solution formulas are derived in both cases.

scholarly journals Semigroups and evolutionary equations

2021 ◽  
Author(s):  
Sascha Trostorff
Keyword(s):  
Initial Value Problem ◽  
Differential Algebraic Equations ◽  
Delay Equation ◽  
Algebraic Equations ◽  
Infinite Dimensions ◽  
Initial Value ◽  
Evolutionary Equations ◽  
Strongly Continuous Semigroups ◽  
Initial States

AbstractWe show how strongly continuous semigroups can be associated with evolutionary equations. For doing so, we need to define the space of admissible history functions and initial states. Moreover, the initial value problem has to be formulated within the framework of evolutionary equations, which is done by using the theory of extrapolation spaces. The results are applied to two examples. First, differential-algebraic equations in infinite dimensions are treated and it is shown, how a $$C_{0}$$ C 0 -semigroup can be associated with such problems. In the second example we treat a concrete hyperbolic delay equation.

Well-Posed Formulations for Nonholonomic Mechanical System Dynamics

2020 ◽  
Vol 15 (10) ◽  
Author(s):  
Edward J. Haug
Keyword(s):  
System Dynamics ◽  
Mechanical System ◽  
Lagrange Multiplier ◽  
Variational Formulation ◽  
Direct Consequence ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
First Order ◽  
Well Posed ◽  
Problem Data

Abstract Four formulations of nonholonomic mechanical system dynamics, with both holonomic and differential constraints, are presented and shown to be well posed; i.e., solutions exist, are unique, and depend continuously on problem data. They are (1) the d'Alembert variational formulation, (2) a broadly applicable manifold theoretic extension of Maggi's equations that is a system of first-order ordinary differential equations (ODE), (3) Lagrange multiplier-based index 3 differential-algebraic equations (index 3 DAE), and (4) Lagrange multiplier-based index 0 differential-algebraic equations (index 0 DAE). The ODE formulation is shown to be well posed, as a direct consequence of the theory of ODE. The variational formulation is shown to be equivalent to the ODE formulation, hence also well posed. Finally, the index 3 DAE and index 0 DAE formulations are shown to be equivalent to the variational and ODE formulations, hence also well posed. These results fill a void in the literature and provide a theoretical foundation for nonholonomic mechanical system dynamics that is comparable to the theory of ODE.

scholarly journals Linear differential algebraic equations with constant coefficients

1970 ◽  
Vol 4 (2) ◽  
pp. 28-35 ◽  
Author(s):  
M Sahadet Hossain ◽  
M Mostafizur Rahman
Keyword(s):  
Existence And Uniqueness ◽  
Differential Algebraic Equations ◽  
Canonical Forms ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Constant Coefficients ◽  
Dae Systems ◽  
International University ◽  
Linear Differential ◽  
Modern Mathematics

Differential-algebraic equations (DAEs) arise in a variety of applications. Their analysis and numerical treatment, therefore, plays an important role in modern mathematics. The paper gives an introduction to the topics of DAEs. Examples of DAEs are considered showing their importance for practical problems. Some essential concepts that are really essential for understanding the DAE systems are introduced. The canonical forms of DAEs are discussed widely to make them more efficient and easy for practical use. Also some numerical examples are discussed to clarify the existence and uniqueness of the system's solutions. Keywords: differential-algebraic equations, index concept, canonical forms. DOI: 10.3329/diujst.v4i2.4365 Daffodil International University Journal of Science and Technology Vol.4(2) 2009 pp.28-35

Solution Numerical of Stiff ODEs and DAEs in Mechanical and Other Systems

1997 ◽  
Author(s):  
H. Pasic
Keyword(s):  
Formal Solution ◽  
Differential Algebraic Equations ◽  
Single Step ◽  
Algebraic Equations ◽  
Stiff Differential Equations ◽  
Index Zero ◽  
Initial Value ◽  
Polynomial Coefficients ◽  
Multi Stage ◽  
Index 2

Abstract Presented is a formal solution of the initial-value problem of the system of general implicit differential-algebraic equations (DAEs) F(x, y, y’) = 0 of index zero or higher, based on perturbations of the polynomial coefficients of the vector y(x). The equation is linearized with respect to the coefficients and brought into a form suitable for implementation of the weighted residual methods. The solution is advanced by a single-step multi-stage collocation qadrature formula which is stiffly accurate and suitable for solving stiff differential equations and DAEs that arise in many mechanical and other systems. The algorithm is illustrated by two index-2 and index-3 examples — one of which is the well known pendulum problem.

Multi-Splitting Waveform Relaxation Methods for Solving the Initial Value Problem of Linear Integral-Differential-Algebraic Equations

2011 ◽  
Vol 403-408 ◽  
pp. 1763-1766
Author(s):  
Xiao Lin Lin ◽  
Yuan Sang ◽  
Hong Wei ◽  
Li Ming Liu ◽  
Yu Mei Wang ◽  
...  
Keyword(s):  
Spectral Radius ◽  
Discrete Time ◽  
Initial Value Problem ◽  
Differential Algebraic Equations ◽  
Waveform Relaxation ◽  
Algebraic Equations ◽  
Initial Value ◽  
Relaxation Methods ◽  
Discrete Time Case ◽  
Linear Integral

We present the multi-splitting waveform relaxation (MSWR) methods for solving the initial value problem of linear integral-differential-algebraic equations. Based on the spectral radius of the derived operator by decoupled process, a convergent condition is proposed for the MSWR method. Finally we discussed the convergent condition of discrete-time case of MSWR.

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