Differential algebraic equation
In a differential- algebraic equation (also differential-algebraic equation or the descriptor system) are ordinary differential equations and algebraic (ie here: derivative-free ) constraints coupled and as an equation or system of equations considered. In some cases this structure is already created in the form of the equation system, such as
This form is usually achieved with problems of the mechanics of bodies under constraint conditions than instructive example of the pendulum is often chosen.
The general form of a differential algebraic equation is implicit differential equation of the form
For a vector-valued function with. An equation in this implicit form is ( locally ) to be resolved if the partial derivative is regular. This follows from the classical implicit function theorem. In this particular case, one can rewrite the implicit equation in the form
And will thus have an explicit ordinary differential equation.
A true differential-algebraic equation exists when the partial derivative is singular. Then the implicit differential equation decomposes locally into an inherent differential equation and an algebraic constraint. This practically corresponds to a differential equation, which is considered on a manifold. The practical problem with the implied differential equations, however, is that this manifold is not explicitly known initially.
In contrast to ordinary differential equations whose solution is determined by integration, there are parts of the solution of a differential- algebraic equation by differentiation. This places further demands on the system function Q. Do these just be steadily and continuously differentiable for ordinary differential equations in order to guarantee the solvability, so now also higher derivatives of the solution required. The exact order of the necessary derivatives depends on the selected approach and is commonly referred to as an index of differential- algebraic equation.
By hinzuzuziehenden in the solution process derivatives of the components of the system of equations created an overdetermined system. One consequence is that the solutions must satisfy also a number of explicit or implicit algebraic constraints. In particular, this applies to initial values of initial value problems. The search for consistent initial values, for example, near a given inconsistent initial values , is a non-trivial first problem in the practical solution of differential- algebraic equations.
- 2.1 differentiation index 2.1.1 Examples
Types of differential- algebraic equations
Semi - explicit differential-algebraic equation
A special case of a differential-algebraic equations is a system in the form of
By differentiating the second differential equation and inserting the first is obtained, as a further condition for a solution
If the factor in front is different from zero, we obtain an explicit system of ordinary differential equations. But initial values for this system must also meet the undifferentiated second equation so that only one parameter can be chosen freely.
Linear differential-algebraic equation
Very often occur differential-algebraic equations in the form
With constant matrix coefficients
And a function
A true differential-algebraic equation is present here if the matrix function on a non-trivial core has. A particularly simple case occurs when the dies are square with constant entries.
Linear differential-algebraic equation with clearly formulated leading term
Another notation for linear differential-algebraic equations is the form
With (at least) continuous matrix coefficients
And a function
In this notation it is taken into account that, for a differential- algebraic equation only a part of the variable vector is differentiated. In fact only the component is differentiated and not the entire vector variables. As a classical solutions of this equation are functions of the space
Considered, so the area of continuous functions for which the component is continuously differentiable.
The two matrix functions and form the main term of the equation and this means proper formulated when two properties are satisfied:
1 It is
2 There exists a continuously differentiable projector function
With the feature
Here, the first condition ensures that " nothing is lost " between the two matrix functions and. At the core of the matrix nothing can disappear from the image of the matrix. The projector function accurately realized by the matrix functions and given decomposition of the space and is useful for the analysis of the equation.
A simple special case for a proper formulated leading term is given by matrix functions and with the property
Then the identity matrix can be selected for the projector function.
Index terms for DAEs
Differentiation index
Often, the solution of a differential-algebraic system of equations can be represented by (special ) solution curves of ordinary differential equations, although it is singular. A key role is taken up by the differentiation index of differential-algebraic system of equations.
Numerical methods for solving differential-algebraic equation systems can usually integrate only systems whose differentiation index does not exceed a certain maximum value. Thus, the differentiation index of the system as implicit Euler method, for example, must not be greater than one.
The differentiation index of differential-algebraic equation system
Is the number of time derivatives that are necessary from the resulting system of equations
By algebraic manipulations an ordinary differential equation system
Be able to extract.
Examples
A differential-algebraic system of equations with a regular matrix, which can be converted by algebraically Thus, the differentiation index is zero.
A purely algebraic equation
With regular Jacobian matrix, which is interpreted as a differential-algebraic equation, differentiation index has one: simply differentiating we obtain the equation
Which is resolvable by:
This fact is sometimes used for the construction of homotopy.
Are ( with one on normalized acceleration due to gravity and pendulum length ) The Euler -Lagrange equations for the mathematical pendulum
This differential-algebraic system of equations has the three differentiation index: twice the time derivative of the constraint condition (third equation) with respect to time supplies
With the help of two differential equations in the Euler -Lagrange equations can be the second time derivatives and replace what
Supplies. With one obtains the equation
Through time derivation of this equation ( this is the third time-derivative ) then to get the missing differential equation for
Again using the equations from the Euler -Lagrange equations were used and to replace and in addition takes into account that was valid.
Geometric Index
A mathematically clear concise and well geometrically interpretable term is the geometric index of a differential-algebraic system of equations. The basic idea is that you determined using the iterative method presented below the maximum constraint manifold on which the differential-algebraic equation a vector field ( as a vector field on a manifold ) describes. The geometric index of the algebraic - differential equations is then the minimum number of iteration steps required in this process.
The geometric index is equal to the differentiation index.
Consider an autonomous differential-algebraic equation
With sufficiently differentiable function.
As part of the algorithm as multiplicity is interpreted with the tangent bundle. The pairs are also referred to as the tangent vectors.
By function, the amount is fixed, associated with each point for all admissible solutions of algebraic differential equation system velocity vectors at that point.
It is possible that for a point no pair, exactly one such pair or more such pairs exist in.
The items, possibly solutions can go through, you recognized in the amount
( with the projection on the first component, that is ). At this point, it should be assumed that a differentiable submanifold represents the.
Each tangent vector to a solution of the differential-algebraic equation must also tangent
Of lie (this means that a defined on an interval, once continuously differentiable curve lying entirely in ).
The tangent vectors to solutions of differential-algebraic equation must also be in the crowd and therefore the solutions are themselves in the crowd.
This process can be continued (under certain conditions) and from the forced manifold the forced manifold
Form. It is possible that after a any point in exactly one tangent vector is assigned. Then describes a vector field on the manifold.
The geometric index of the differential-algebraic equation is precisely describes the minimum number for which a vector field on the manifold.
By Equation
Defined function and the corresponding differential-algebraic equation used in the following text as co-running example.
In the example, there are all points that do not lie in the plane defined by, no couples. So run in this example outside this plane no solutions of differential-algebraic equation.
It arises and, thus
As you can see, is the tangent vector given by ( the ) for values because not in the tangent space, so it can not correspond to a solution of the differential-algebraic system of equations. This results in
We obtain
And the amount
Assigns to each point of the set to (which here is just equal to) exactly one tangent vector. The amount that is not the case, since tangent vectors from this set, the component is not restricted.
The geometric index of the differential-algebraic equation system in this example is thus equal to two.
Is a manifold, then this can help with a function in the form
Are shown. The limiting equations in this representation are called constraints of the differential-algebraic equation.
In addition, can be singled out for the manifold with the help of a function from the manifold: The equations are also called hidden constraints of the differential-algebraic equation ( engl.: hidden constraints).
Remarks:
- Only autonomous differential-algebraic equations are considered that in this section, facilitates the geometric interpretation and is not really a restriction, since any time-dependent differential-algebraic equation can be rewritten by introducing an additional variable and an additional differential equation into an autonomous differential-algebraic equation.
- In this section, it was assumed that a submanifold is the. If this is not the case, the geometric index is not explained for the respective differential-algebraic equations.
- There are also differential-algebraic equations in which the geometric index is infinite.
Consistent initial values
Given is again a differential-algebraic equation
With sufficiently often differentiable.
A point is called consistent initial value at the time, if there's one in an open interval is defined by solving the differential-algebraic equation applies.
The calculation is to be noted that consistent initial values except the constraints, including hidden constraints are met (see Section Geometric Index).