Another difference is that even if consistent initial values are given, the existence and uniqueness theory is more complicated and involves additional technical assumptions besides just sufficient smoothness as in the ODE case. Furthermore, it can be seen that the solution depends on the derivative of the inhomogeneous part (or the derivative of the input, if function \(q\) plays the role of input) which cannot happen in the ODE case. That is, if an arbitrary initial condition is imposed, it may well be inconsistent with the DAE. Clearly, the only solution is \(y=q(t),\ z=q'(t)\ ,\) and no initial or boundary conditions are needed. Where a sufficiently smooth function \(q\) is given. To get an idea of the difference between DAEs and ODEs, consider the very simple example It is easier to derive a complex DAE model so it is highly desirable to be able to work with the DAE model if possible. Models are formulated in a more natural way than explicit ones, as the above examples demonstrate. The class of DAEs includes problems exhibiting fundamental mathematical properties that are different from those of ODEs, and also pose additional challenges for their numerical solution. While the standard-form ODE can be written as a DAE, the more general DAE form admits problems that can be quite different from a standard-form ODE. in the pendulum example, is physical, while the constraint in other problems such as a prescribed path problem is not physical but rather part of the performance specifications.ĭAEs are a generalization of an ordinary differential equations (ODEs)įor which there is a very rich literature for both mathematical theory and numerical solution.
It should be noted that the constraint in mechanics, e.g.
\) However, such a simple elimination procedure is usually impossible in more general situations.Īdditional examples of real-life DAE systems, including multibody mechanical systems, an electrical circuit, and a prescribed path control problem can be found in Brenan et al. In this very simple case of a multibody mechanical system, the change of variables \(x_1=\sin \theta, x_2=\cos \theta \)įollowed by some algebra gives the well-known ODE for a pendulum \(\theta''=-g \sin \theta\. \) After rewriting the two second order equations as four first order ODEs, a DAE system of the form ( 2) with four differential and one algebraic equations results: Which expresses the condition that the rod has fixed length \( 1 \. The \(\lambda x_i\) terms represent the force which holds the solution onto the constraint Where \( g \) is the force of gravity and \( \lambda \) is a Lagrange multiplier. Suppose the pendulum has length 1 and let the coordinates of the tiny ball of mass 1 at the end of the rod be \( (x_1,x_2) \. Here \(x=(y,z)\) and \(g(t,y,z)=0\) are the explicit constraints. The method of solution of a DAE will depend on its structure.Ī special but important class of DAEs of the form ( 1) is the semi-explicit DAE or ordinary differential equation (ODE) with constraints The DAE may be an initial value problem where \(x\) is specified at the initial time, \( x(t_0)=x_0\ ,\) or a boundary value problem, where the solution is subject to \(N\) two-point boundary conditions \(g(x(t_0),x(t_f))=0\. Systems of equations like ( 1) are also called implicit systems, generalized systems, or descriptor systems. \) The Jacobian \(\partial F/\partial v\) along a particular solution of the DAE may be singular. The term DAE is usually reserved for the case when the highest derivative \(x'\) cannot be solved for in terms of the other terms \(t, x,\) when ( 1) is viewed as an algebraic relationship between three variables \(t, x, x'\. Every DAE can be written as a first order DAE. Petzold, Department of Mechanical Engineering, and Department of Computer Science, University of California Santa Barbara, CAĪ differential-algebraic equation ( DAE) is an equation involving an unknown function and its derivatives.Ī (first order) DAE in its most general form is given by Vu Hoang Linh, Faculty of Mathematics, Mechanics and Informatics, Vietnam National University, Hanoi, Vietnam Campbell, North Carolina State University, Raleigh, NC, USA.
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