Solving Differential Equations in Excel

This appendix provides a detailed description of how to build Excel spreadsheet solutions for several of the problems that were presented and solved in Chapter 4. Generically, these include an ordinary-differential-equation boundary-value problem, a one-dimensional parabolic partial differential equation, and a two-dimensional elliptic partial differential equation. 

Like computer programming generally, the analyst has great freedom to create spreadsheets to meet particular needs and to suit his or her own style preferences. The spreadsheets in this appendix are certainly not intended to specify any particular programming style, but simply to highlight the essential elements that are needed to solve particular classes of problems. 

The examples herein are done using Excel98, on a Macintosh personal computer. It is easy to anticipate some changes as Excel progresses to newer releases, and there well may be some small differences between IBM-compatible PCs and the Macintosh implementations. While the syntax and communication interfaces will likely change in the future, the functional requirements for solving certain problems will not. The descriptions provided in this appendix are quite detailed nevertheless, some knowledge of spreadsheet operations is presumed. 

Once a solution has been found, there are a variety of charting options available to create graphical representations of the results. This is a valuable capability and very easy to use. There are no special considerations in using the charting capabilities, so we do not provide further documentation in this appendix. 

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The correction to the relaxing density matrix can be obtained without coupling it to the differential equations for the Hamiltonian equations, and therefore does not require solving coupled equations for slow and fast functions. This procedure has been successfully applied to several collisional phenomena involving both one and several active electrons, where a single TDHF state was suitable, and was observed to show excellent numerical behavior. A simple and yet useful procedure employs the first order correction F (f) = A (f) and an adaptive step size for the quadrature and propagation. The density matrix is then approximated in each interval by... 

When the data are known for each of the contributing reactions it is easy, as shown in the table, to calculate what the sum total effect will be, but in laboratory practise the situation is reversed and we try to find out what reaction steps and what rate constants are operating to give us our observed facts. This is a much more difficult task and frequently the differential equation can not be solved by ordinary methods. Furthermore, if a set of reaction steps is found to reproduce the facts we can not be sure that it is the only set of reactions which will account for the over-all observed rate. Excellent examples of these consecutive reactions are found among the disintegrations of the radioactive elements. Frequently the kineticist has to work out intermediate steps in this way for ordinary chemical reactions. It is always to be hoped that one... 

The differential equations (69) and (70) were then solved numerically for different values of Cj Cj = 2 is clearly preferred (Fig. 25a,b). A similar calculation was carried out for the (T3) flow (Fig. 26a), where Cj = 2 also gives excellent agreement. Note that the predicted length-scale variations do model the integral-scale changes as measured (C4) (Fig. 25b). It therefore appears that a satisfactory model equation for the dissipation history in homogeneous flows is... 

The differential equation, (1.2), is an ordinary differential equation because there is only one independent variable, x In this case, equations in one space dimension are boundary value problems, because the conditions are provided at two different locations. While it is also possible to solve this problem using Excel and MATLAB, it is much simpler to use FEMLAB. Transient heat transfer in one space dimension is governed by... 

It is clear from these values and Figure 7.14 that p 1 / is an excellent approximation for this reactor. Substituting this equation for p into the mass balance and solving the differential equation produces the results shown in Figure 7.2 7. The concentration of O2 is nearly constant, which Justifies the pseudo-first-order rate expression. Reactor volume... 

Fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes [37-50]. This is the main advantage of fractional derivatives when compared with the classical integer-order models, in which such effects are in fact neglected. The mathematical modeling and simulation of systems and processes, based on the description of their properties in terms of fractional derivatives, naturally leads to differential equations of fractional order and to the necessity for solving such equations. 

Iserles, Arieh. A First Course in the Numerical Analysis of Differential Equations. 2d ed. New York Cambridge University Press, 2009. This text gives an excellent overview of the various techniques used to solve ordinary differential equations. 

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