Numerical Evaluation of Integrals

In this section we discuss techniques for numerically evaluating integrals for solving first-order differential equations. 

These formulas are useful in illustrating how the reaction engineering integrals and coupled DDEs (ordinary differential equation(s)) can be solved and also when there is an ODE solver power failure or some other malfunction. 

For the ordinary differential equation solver (ODE solver), contact  

10 Canal Park Cambridge, Massachusetts 02141-2201 USA E-mail info aspentech.com Website http //www.aspentech.com 

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A.l Useful Integrals in Reactor Design 921 A.2 Equal-Area Graphical Differentiation 922 A.3 Solutions to Differential Equations 924 A.4 Numerical Evaluation of Integrals 924 A.5 Software Packages 926 ... 

First, we use a variational integration mesh [49] that allows us to find a set of mesh points [rj for the precise numerical evaluation of integrals required for solution of the density-functional equations. Each matrix element or integral can be rewritten as ... 

S1MPS0N Numerical evaluation of integral by Simpson s 1/3 rule. 

The numerical evaluation of integrals is a much studied subject of numerical methods and many diverse algorithms have been developed for such problems. The general mathematical statement is given flie integral ... 

For the case where t8 = rp — tp, Equation 13 can be integrated directly to give ITto1al, the total amount of tertiary ion formed. For the other two cases, integration cannot be performed directly, and values of ITtotal were evaluated numerically on a KDF 9 computer, using a procedure for Simpson s rule. (Numerical evaluation of the directly integrable case provided a check on this procedure.) Ip and I8 are then given by... 

The numerical evaluation of definite integrals can be carried out in several ways. However, in all cases it must be assumed that the function, as represented by a table of numerical values, or perhaps a known function, is well behaved. While this criterion is not specific, it suggests that the functions haying pathological problems, e.g. singularities, discontinuities,..may not survive under the treatment in question. 

Relationship 23 provides a method for evaluating the parameter "a" that is defined by Equation 2A. The cumulative molar concentration of polymeric species PT0T was numerically evaluated via integration of population density distributions. The contribution of network molecules to the zeroth moment of the distribution is negligible. Results are presented by Figure A and show that... 

The latent-heat terms (3.112) become necessary whenever the integrand ACP undergoes discontinuous change at a phase transition, with accompanying release of hidden AH. [The latent heat contribution is automatically included if one understands J(ACV) dT as Lehesgue integration.] For numerical evaluation of the integral in (3.111), power series... 

When Eq. (6.44) is compared to the result obtained by a numerical evaluation of the integral in Eq. (6.40) (see Table 6.1), one will find good agreement at high temperatures. At low temperatures, the analytical formula can, however, deviate substantially from the exact result. 

I will discuss the current status of theoretical work on the magnetic moment anomalies of the electron and muon, with a particular emphasis on the on-going effort to reduce substantially the statistical and non-statistical uncertainties generated by the adaptive-iterative Monte-Carlo integration routine VEGAS [2] in the numerical evaluation of the QED contribution. 

The numerical evaluation of the Bohr-Sommerfeld integral in the equation,... 

Numerical evaluation of the Kassel integral permitted a comparison between theoretical and experimental fall-off behaviour . With an average molecular diameter of 5.5 A the calculated rate coefficient-azoethane pressure curve showed the best agreement with experiment at an effective number of oscillators of 18, somewhat less than half of the maximum 2N— 6. Because of the complexity of the reaction the experimental curve is probably in error, rendering comparison unreliable. Similar calculations for azomethane using the earlier uninhibited kinetic data showed best agreement with experiments at a molecular diameter of 4.7 A and an effective number of oscillators of 12, one half of the total normal modes of vibrations. 

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