Page 276 - Excel for Scientists and Engineers: Numerical Methods

CHAPTER 11 ORDINARY DIFFERENTIAL EOUATIONS. PART I1 253

RK formulas to calculate z, columns H through M a similar series to calculate y.
The RK formulas in cells C7 through G7 are, respectively
=B6*( A7-A6)
=( B6+C7/2)*(A7-A6)

=( B6+ D7/2)*( A7-A6)
=( B6+E7)*(A7-A6)

=G6+(C7+2*D7+2*E7+F7)/6
As expected, application of the RK method reduces the errors significantly.
The results from the more precise calculation are shown in Figure 11-9. Every
fifth data point has been plotted.
Even better accuracy can be obtained by using the RK method with a smaller
kc. When a kc value of 0.0 1 is used instead of 0.1, the maximum error is 0.25%

0 1 2 3
x

Figure 11-9. Solution of the differential equationy" -y = 0 by the shooting method,
using the RK method to calculate y' and y. Maximum error is ca. 1%.
(folder 'Chapter 1 1 Examples', workbook 'ODE-BVP', worksheet 'y"-y=O (RK)')