Question

Differential algebraic equations matlab example

Answer 1

A differential-algebraic equation (DAE) is an equation involving an unknown function and its derivatives. A (first order) DAE in its most general form is given by 

F(t,x,x′)=0,t0≤t≤tf,(1)

where x=x(t) , the unknown function, and F=F(t,u,v) have Ncomponents, denoted by xi and Fi, i=1,2,...,N , respectively. Every DAE can be written as a first order DAE. 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′ . The Jacobian ∂F/∂v along a particular solution of the DAE may be singular. Systems of equations like (1) are also called implicit systems, generalized systems, or descriptor systems. The DAE may be an initial value problem where x is specified at the initial time, x(t0)=x0 , or a boundary value problem, where the solution is subject to N two-point boundary conditions g(x(t0),x(tf))=0 .

The method of solution of a DAE will depend on its structure. A special but important class of DAEs of the form (1) is the semi-explicit DAE or ordinary differential equation (ODE) with constraints 

y′0==f(t,y,z)g(t,y,z),

which appear frequently in applications. Here x=(y,z) and g(t,y,z)=0 are the explicit constraints.