Ordinary differential equations implicit equation

The system illustrated in Figure 6 can be aiodelled with a system of partial and ordinary differential equations. Implicit in this model are the following assumptions ... 

Hindmarsh A. C. (1976) Preliminary Documentation of GEARIB. Solution of Implicit Systems of Ordinary Differential Equations with Banded Jacobians, Rep. UCID - 30130, Lawrence Livermore Laboratory, Livermore. 

In what follows one possible example demonstrates for a system of ordinary differential equations that there is an implicit scheme which is rather economical than the explicit ones requiring the additional operations. 

As discussed in the introduction to this chapter, the solution of ordinary differential equations (ODEs) on a digital computer involves numerical integration. We will present several of the simplest and most popular numerical-integration algorithms. In Sec, 4.4.1 we will discuss explicit methods and in Sec. 4.4.2 we will briefly describe implicit algorithms. The differences between the two types and their advantages and disadvantages will be discussed. 

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

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 CNMMR model with laminar flow liquid stream in the annular region consists of three ordinary differential equations for the gas in the tube core and two partial differential equations for the liquid in the annular region. These equations are coupled through the diffusion-reaction equations inside the membrane and boundary conditions. The model can be solved by first discretizing the liquid-phase mass balance equations in the radial direction by the orthogonal collocation technique. The resulting equations are then solved by a semi-implicit integration procedure [Harold etal., 1989]. 

In this section, we compile some results from nonlinear analysis that are used in the text. The implicit function theorem and Sard s theorem are stated. A brief overview of degree theory is given and applied to prove some results stated in Chapters 5 and 6. The section ends with an outline of the construction of a Poincare map for a periodic solution of an autonomous system of ordinary differential equations and the calculation of its Jacobian (Lemma 6.2 of Chapter 3 is proved). 

The state trajectory u t) is computed by the implicit integrator DDASSL (Petzold 1982 Brenan, Campbell, and Petzold 1989). updated here to handle the initial condition of Eq. (B.1-2). The DDASSL integrator is especially designed to handle stiff, coupled systems of ordinary differential and algebraic equations. It employs a variable-order, variable-step predictor-corrector approach initiated by Gear (1971). The derivative vector applicable at t +i. is approximated in the corrector stage by a... 

Bader, G. Deuflhard, P. A semi-implicit midpoint rule for stiff systems of ordinary differential equations. Numer. Math. 1983, 41, 373-398. 

The assumption implicit in the discussion so far is that the system to be modelled consists of lumped-parameter elements and thus may be described adequately using ordinary differential equations in time. This will be true for a large number of process... 

A few examples will be demonstrated in the following section. Each feature was incorporated in the software because it has been found necessary or useful in some modelling project. All the modelling tasks that previously required tailor made solutions in each project can now be solved in a unified manner in the ModEst environment. ModEst is able to deal with explicit algebraic, implicit algebraic (systems of nonlinear equations) and ordinary differential equations. As the standard way to handle PDE systems, the Numerical Method of Lines, which transforms a PDE system to a number of ODE components is used. In addition, any model with a solver provided may be dealt with as an algebraic system. The basic numerical tools are contained in the well tested public domain software (Bias, Linpack, Eispack, LSODE). 

A finite difference scheme for discretization in time is used at this stage. In order to reduce the set of ordinary differential equations to algebraic equations, a time weighting coefficient is introduced, that allows to use several schemes explicit, implicit or the Crank-Nicolson scheme. 

The new set now has N+l coupled ordinary differential equations (Eqs. 7.5 and 7.4). Thus, the standard form of Eq. 7.1 is recovered, and we are not constrained by the time appearing explicitly or implicitly. In this way, numerical algorithms are developed only to deal with autonomous systems. 

Estimation of the kinetic parameters using experimental data and requiring a mass transfer limited, heterogeneous catalytic liquid phase reaction model, which comprises implicit algebraic and ordinary differential equations. 

Initial value problems with ordinary differential equations (ODE) have well-defined conditions (based on Lipschitz continuity of the time derivatives) that guarantee unique solutions. Conditions for unique solutions of DAEs (Equations 14.2 and 14.3) are less well defined. One way to guarantee existence and uniqueness of DAE solutions is to confirm that the DAE can be converted (at least implicitly) to an initial value ODE. A general analysis of these DAE properties can be found in [5] and is beyond the scope of this chapter. On the other hand, for a workable analysis, one needs to ensure a regularity condition on the DAE characterized by its index. 

Whether the simulation is on a direct discretisation of the equations in cylindrical or transformed coordinates, the discretisation process results in a (usually) linear system of ordinary differential equations, that must be solved. In two dimensions, the number of these will often be large and the equation system is banded. One approach is to ignore the sparse nature of the system and simply to solve it, using lower-upper decomposition (LUD) [212]. The method is very simple to apply and has been used [133,213,214]—it is especially appropriate in curvilinear coordinates and multipoint derivative approximations, where the system is of minimal size [214], and can outperform the more obvious method, using a sparse solver such as MA2 8 (see later). However, many simulators tend to prefer other methods, that avoid using implicit solution in two dimensions simultaneously but still are implicit. Of these, two stand out. 

Systems 3.1 and 3.2 contain N number of equations. The algebraic equation system can be solved with the Newton-Raphson method [1] and the ordinary differential equations with the semi-implicit Runge-Kutta method, Michelen s semi-implicit Runge-Kutta method [2], or, alternatively, with the Rosenbrock-Wanner semi-implicit Runge-Kutta method [2],... 

Levykin, A.I., Novikov, E.A. A study of (m, k)-methods of the order 3 for implicit systems of ordinary differential equations. Novosibirsk, Preprint Computing Center SB RAS 882 (1990)... 

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. 

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