Differential-algebraic equations regularization

Definition A.6. A differential algebraic equation system (A. 10) is said to be regular, if... 

Kumar, A. and Daoutidis, P. (1996). Feedback regularization and control of nonlinear differential-algebraic-equation systems. AIChE J., 42, 2175-2198. 

Differential-algebraic equations (DAEs) differ in main aspects from explicit or regularly implicit ordinary differential equations. This concerns theory, e.g. solvability and representation of the solution, as well as numerical aspects, e.g. convergence behavior under discretization. Both aspects depend essentially on the index of the DAE. Thus, we first define the index. 

When (X = P = 1, Eqs. (6.100)-(6.103) become identical to tho.se developed earlier in this chapter for regular Cartesian coordinate systems. Therefore, for objects with irregular boundaries, the partial differential equations would be converted to algebraic equations using Eqs. (6.100)-(6.103). For points adjacent to the boundary, the parameters a and P would assume values that reflect the irregular shape of the boundary, and for internal points away from the boundary, the value of a and P would be unity,... 

Since X + In X is a transcendental function, Eq. (2-67) cannot be solved for [A], Two methods are usually used. The method of initial rates is the more common one, since it converts the differential equation into an algebraic one. Values of v(, determined as a function of [A]o, are fit to the equation given for v. This application to enzyme-catalyzed reactions will be taken up in Chapter 4. The other method regularly used relies on numerical integration these techniques are given in Chapter 5. 

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