Algebraic equation systems

Tjoa, I.-B., and Biegler, L. T., Simultaneous solution and optimization strategies for parameter estimation of differential-algebraic equations systems, I EC Research, 30, 376 (1991). 

The plug-flow problem may be formulated with a variable cross-sectional area and heterogeneous chemistry on the channel walls. Although the cross-sectional area varies, we make a quasi-one-dimensional assumption in which the flow can still be represented with only one velocity component u. It is implicitly assumed that the area variation is sufficiently small and smooth that the one-dimensional approximation is valid. Otherwise a two- or three-dimensional analysis is needed. Including the surface chemistry causes the system of equations to change from an ordinary-differential equation system to a differential-algebraic equation system. 

The general solution of the algebraic equation system obtained by means of the internal [(14.11a) and (14.11b)] conditions withm= 1,2,..., Mandexternal boundary conditions [(14.3a) and (14.3b)] are given in Appendix B. The mass-transfer rate on the upstream side of the membrane can be given, for that case, as follows ... 

Equation 14.22 takes into account that the membrane and cake layers could have different partition coefficient. Thus, you can get an algebraic equation system with 6 equations, which can be solved relatively easily (as will be discussed elsewhere). Obviously, for prediction of the filtration efficiency, the values of transport parameters have to be known. 

Applying the boundary conditions, defined by Equations 14.3a, 14.3b, 14.11a, 14.11b, an algebraic equation system can be obtained. The form of these equations is the same as are given here by Equation 14.B1 to Equation 14.10. The value of the... 

Differential algebraic equation systems whose algebraic equations explicitly involve manipulated input variables are referred to as nonregular (Kumar and Daoutidis 1999a). See also Definition A.6. 

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

By solving the algebraic equation system given by the constraints of Equation (B.3), the steady-state solutions yJiSS can be obtained for the fast variables. Subsequently, yJiSS can be used for the derivation of an equivalent n-dimensional ODE representation of the slow subsystem ... 

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

Investigation 10 was a study of fixed-bed reactor models and their application to the data of Hettinger et al. (1955) on catalytic reforming of C7 hydrocarbons. The heuristic posterior density function p 6 Y) proposed by Stewart (1987) was used to estimate the rate and equilibrium parameters of various reaction schemes, two of which were reported in the article. The data were analyzed with and without models for the intraparticle and boundary-layer transport. The detailed transport model led to a two-dimensional differential-algebraic equation system, which was solved via finite-element discretization in the reactor radial coordinate and... 

Bain, R. S., Solution of nonlinear algebraic equation systems and single and multiresponse nonlinear parameter estimation problems, Ph. D. thesis, University of Wisconsin-Madison (1993). 

This chapter gives a brief summary of properties of linear algebraic equation systems, in elementary and partitioned form, and of certain elimination methods for their solution. Gauss-Jordan elimination, Gaussian elimination, LU factorization, and their use on partitioned arrays are described. Some software for computational linear algebra is pointed out, and references for further reading are given. 

The DDAPLUS algorithm (Caracotsios and Stewart, 1985), updated here, is an extension of the DDASSL (Petzold, 1982) implicit integrator. DDAPLUS solves differential-algebraic equation systems of the form... 

It must be noticed the cascade interconnection between the algebraic and the differential items. At each time t, the two algebraic equations system admits a unique and robust solution for in any motion in which S (the Jacobian matrix of ( ) is nonsingular for S 0). Feeding the solution into the differential equation, the following differential estimator, driven by the output and the input signals u, is obtained ... 

Solving the ODEs using numerical integration routines, which generally involve another discretization into a nonlinear algebraic equation system and subsequent iterative solution... 

For brevity, further discussion is restricted to the spatial discretization used to obtain ordinary differential equations. Often the choice and parameters selection for this methods is left to the user of commercial process simulators, while the numerical (time) integrators for ODEs have default settings or sophisticated automatic parameter adjustment routines. For example, using finite difference methods for the time domain, an adaptive selection of the time step is performed that is coupled to the iteration needed to solve the resulting nonlinear algebraic equation system. For additional information concerning numerical procedures and algorithms the reader is referred to the literature. 

L. R. Petzold, A description of DASSL A differential algebraic equation system solver. 

Assemble and solve the linearized algebraic equation systems for the velocity components (momentum equations) to obtain v. ... 

Both methods are implicit by their nature because fn usually is a non-linear function of y . Eqns (10.66) and (10.67) imply a solution of a non-linear algebraic equation system of the type... 

Models used in dynamic simulation lead typically to a differential-algebraic equation system (DAE), whose general form may be written as follows ... 

A differential- algebraic equations system is indicated with the acronym DAE. 

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