Page 305 - Excel for Scientists and Engineers: Numerical Methods

282 EXCEL: NUMERICAL METHODS

Vapor Diffusion in a Tube
Solved by Using a Custom Function
This example, using the same data as the preceding one, illustrates the use of
the custom function. The spreadsheet, not shown here, can be examined on the
accompanying CD-ROM. Unlike the preceding spreadsheets, tables of
coefficients and constants are not required. The x-increment is 2 cm, thus
creating a table of values that is 11 columns wide, including the boundary values.
The function returns values identical to those shown in Figure 12-1 1.

Hyperbolic Partial Differential Equations

Hyperbolic second-order differential equations result from problems
involving vibration processes, and are of the form
d2F - d2y
-
p g q&T (12-30)
For example, the wave equation in one dimension
d2y Tg d2y
-=-- (12-31)
at2 w at2
describes the vibration (i.e., the lateral displacement y) of a string of length L,
weight W, tension T and weightlunit length w = WIL, as a function of distance x
along the length of the string.

Solving Hyperbolic Partial Differential Equations:
Replacing Derivatives with Finite Differences
Once again, we can solve the problem by replacing derivatives by finite
differences.

which, when rearranged, yields

If we set Tg(A,t)2/w(hX)2 = 1, equation 12-33 is simplified to equation 12-34.
Interestingly, this simplified expression also yields the most accurate results.
(1 2-34)
F.x,t+l = Fx+l,t + Fx-l,l - F*,I-l