Numerical Integration of Differential Equations

For simple rate equations, the integrated form can be obtained analytically. For example, the rate equation. 

Some complex rate laws cannot be integrated analytically (or are too difficult to integrate analytically) and one must appeal to numerical methods. Numerical methods yield [A] at time increments of At, beginning with [A]q  

This algorithm of numerical integration is called the Euler method. The error is proportional to At. A more detailed discussion of the Euler method, as well as methods with better accuracy (and concomitant higher complexity), can be found in any text on numerical methods. A fine text with many examples in the context of chemical engineering is Applied Numerical Methods by B. Carnahan, H. A. Luther, and J. O. Wilkes (Krieger Publ., Melbourne, FL, 1990). 

10 Integrate numerically rate equation (1) given in the previous page, with k = sec Use a spreadsheet (or write a computer program) to produce a table with at least the following six columns  

Column 1. the iteration step. This column should begin with 0 and increase in increments of 1. 

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In this age of powerful computers, it is no longer even necessary to find analytical solution to differential equations. There are many software packages available that cany out numerical integration of differential equations followed by non-linear regression to fit the model and assess its quality by comparing with experimental data. In this study we have used a numerical integration approach to compare kinetic properties of Photinus pyralis and Luciola mingrelica firefly luciferases. 

Error Growth in Numerical Integration of Differential Equations... 

The coupling of fast electron transfer with homogeneous chemical reaction requires the combination of chemical reaction rate term (see Table 1) with material fluxes evoked by the production/consumption of electroactive species and products during the current flow. The mathematical solution (numerical integration of differential equations) results in relations that seem to be rather confused for electrochemical practice. 

Once the differential equations and mass balance have been written down, three approaches can be followed in order to model complex reaction schemes. These are (1) numerical integration of differential equations, (2) steady-state approximations to solve differential equations analytically, and (3) exact analytical solutions of the differential equations without using approximations. 

The formulas for the first- and second-order derivatives, developed in the preceding four sections, together with those of the third- and fourth-order derivative, are summarized in Table 4.1. It can be concluded from these examples that any derivative can be expressed in terms of finite differences with any degree of accuracy desired. These formulas may be used to differentiate the function y(x) given a set of values of this function at equally spaced intervals of X, such as a set of experiment data. Conversely, these same formulas may be used in the numerical integration of differential equations, as shown in Chaps. 5 and 6. 

Starting with an initial value of and knowing c t), Eq. (8-4) can be solved for c t + At). Once c t + At) is known, the solution process can be repeated to calciilate c t + 2At), and so on. This approach is called the Euler integration method while it is simple, it is not necessarily the best approach to numerically integrating nonlinear differential equations. To achieve accurate solutions with an Eiiler approach, one often needs to take small steps in time. At. A number of more sophisticated approaches are available that allow much larger step sizes to be taken but require additional calculations. One widely used approach is the fourth-order Bunge Kutta method, which involves the following calculations ... 

Solution of Sets of Simultaneous Linear Equations 71. Least Squares Curve Fitting 76. Numerical Integration 78. Numerical Solution of Differential Equations 83. 

Kinetic analysis of the data obtained in differential reactors is straightforward. One may assume that rates arc directly measured for average concentrations between the inlet and the outlet composition. Kinetic analysis of the data produced in integral reactors is more difficult, as balance equations can rarely be solved analytically. The kinetic analysis requires numerical integration of balance equations in combination with non-linear regression techniques and thus it requires the use of computers. 

By the method of numerical integration of differential kinetic equations of the Scheme 3 it has been found that this scheme quantitatively describes experimental data on kinetics of accumulation of acetic acid and radicals at CA photolysis (Figure 2.1. - 2.4) at the following set of kinetic parameters ... 

The in-cloud S(IV) oxidation reactions to be considered are those involving 03 and H202. The necessary equilibrium and kinetic data can be obtained from Chapter 7. The dynamic calculations should be done by numerically integrating the differential equations for a and c,1 together with the relevant electroneutrality relation. The results of the computation should be presented in terms of the ratio of c, at t = 25 min to c, at t = 0. Discuss your results. 

Predictor-corrector methods [47,48] are appropriate for the integration because they require only one evaluation of the slope for each integration step. Molecular simulation is unusual in the context of the numerical treatment of differential equations, because an approximation to the slope is available before the simulation is complete. This information can be used to update the state point as the simulation proceeds. An increment in a typical GDI series entails the following steps, which for concreteness we describe for an integration in the P-p plane... 

SE7 Mathematically inexact deconvolution. Numerical procedures such as numerical integration, numerical solution of differential equations, and some matrix-vector formulations of linear systems are numerical approximations and as such contain errors. This type of error is largely eliminated in the direct deconvolution method where the deconvolution is based on a mathematical exact deconvolution formula (see above). Similarly, the prescribed input function method ( deconvolution through convolution ) wiU largely eliminate this numerical type of error if the convolution can be done analytically so that numerical convolution is avoided. 

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