Differential Algebraic Equation

Differential Algebraic Equation

In mathematics, differential algebraic equations (DAEs) are a general form of (systems of) differential equations for vector–valued functions x in one independent variable t,

where is a vector of dependent variables and the system has as many equations, . They are distinct from ordinary differential equation (ODE) in that a DAE is not completely solvable for the derivatives of all components of the function x.

This difference is more clearly visible if the system may be rewritten so that instead of x we consider a pair of vectors of dependent variables and the DAE has the form

where, and

Every solution of the second half g of the equation defines a unique direction for x via the first half f of the equations, while the direction for y is arbitrary. But not every point (x,y,t) is a solution of g. The variables in x and the first half f of the equations get the attribute differential. The components of y and the second half g of the equations are called the algebraic variables or equations of the system. The term algebraic in the context of DAEs only means free of derivatives and is not related to (abstract) algebra.

The solution of a DAE consists of two parts, first the search for consistent initial values and second the computation of a trajectory. To find consistent initial values it is often necessary to consider the derivatives of some of the component functions of the DAE. The highest order of a derivative that is necessary in this process is called the differentiation index. The equations derived in computing the index and consistent initial values may also be of use in the computation of the trajectory.

Read more about Differential Algebraic Equation:  Other Forms of DAEs, Examples, Semi-explicit DAE of Index 1, Numerical Treatment of DAE and Applications, Numerical Solution of DAEs

Famous quotes containing the words differential, algebraic and/or equation:

    But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes.
    Antonin Artaud (1896–1948)

    I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?
    Henry David Thoreau (1817–1862)

    A nation fights well in proportion to the amount of men and materials it has. And the other equation is that the individual soldier in that army is a more effective soldier the poorer his standard of living has been in the past.
    Norman Mailer (b. 1923)