Introduction to numerical integration

Numerical integration of sets of differential equations is a well developed field of numerical analysis, e.g. most engineering problems involve differential equations. Here, we only give a very brief introduction to numerical integration. We start with the Euler method, proceed to the much more useful Runge-Kutta algorithm and finally demonstrate the use of the routines that are part of the Matlab package. 

Without solving differential Eqs. (6.95)-(6.98) we have predicted the qualitative concentrations of the reactant, intermediate, and undesired product in our batch reactor. Quantitative estimates using time steps done with computers is called numerical integration. An introduction to numerical integration is given in the exercises at the end of this chapter. 

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. 

This section introduces some of the basic concepts of system theory in relation to modeling. Our presentation is rather brief since our aim is to integrate known models for chemical/biological processes with numerical techniques to solve these models for simulation and design purposes, rather than to give a broad introduction to either system theory or modeling itself. For references on modeling, see the Resources appendix. 

The Monte Carlo method is a very powerful numerical technique used to evaluate multidimensional integrals in statistical mechanics and other branches of physics and chemistry. It is also used when initial conditions are chosen in classical reaction dynamics calculations, as we have discussed in Chapter 4. It will therefore be appropriate here to give a brief introduction to the method and to the ideas behind the method. 

INTRODUCT ION TO NUMERICAL ANALYSIS (2nd Edition). F.B. Hildebrand. Classic, fundamental treatment covers computation, approximation, interpolation, numerical differentiation and integration, other topics. 150 new problems. 669pp. 55 x 85. 65363-3 Pa. 13.95... 

A brief introduction to IBM s CSMP (Continuous System Modeling Program) is provided. This program is a powerful, easily used tool for numerically integrating complex systems of differential equations, such as are often encountered in considerations of dynamic processes involving polymers. 

Hence, knowledge of the boundary velocity u(xs) and the form of the undisturbed flow evaluated at the body surface Uoo(xj) allows a direct calculation of the surface-force vector T n by means of a solution of the integral equation, (8-198). It is emphasized that we do not actually address the numerical problem of solving (8-198). We note, however, that it is an integral equation of the first kind, and it is known that there can be numerical difficulties with the solution of this class of integral equations. The reader who wishes to learn more about the details of numerical solution should consult one of the general reference books that were listed in the introduction to this section. 

Schiesser, W. DSS/2 - An Introduction to the Numerical Methods of Lines Integration of Partial Differential Equations, Lehigh Univ. Bethlehem, PA, 1976. 

A straightforward method bypasses the introduction of the auxiliary exchange-correlation fitting basis and evaluates those matrix elements directly by three dimensional numerical integration. This not only spares a computational step, it also avoids the limitations of the fitting basis set. There still arises a truncation error due to the numerical integration scheme. 

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