Page 268 - Excel for Scientists and Engineers: Numerical Methods

Chapter 11

Numerical Integration of

Ordinary DHerential Equations

Part 11: Boundary Conditions

In the preceding chapter, we saw that a differential equation of order n could
be converted into a set of n first-order differential equations. For example, if the
problem to be solved is a second-order differential equation, it is converted into
two first-order differential equations; two "known" values of the function or its
derivative will be needed in order to solve the problem. In the second-order
differential equation example illustrated in Figure 10-16, the value of the
function and its first derivative were both known at x = 0. The problem was then
solved using the standard methods described in Chapter 10.
If information about a second-order differential equation is known at two or
more different values of the independent variable, then the problem is known as a
boundary-value problem (BVP). The points where the function is known are
usually (but not always) the limits of the domain of interest - hence the term
boundary-value problem. Problems of this type must be solved by different
methods than those we applied to initial-value problems.
Two approaches are commonly used to solve boundary-value problems: the
"shooting'' method and the finite-difference method. This chapter shows how to
apply these methods to differential equations of order two; fortunately, most
important physical systems are described by differential equations of order no
higher than two.

The Shooting Method

The shooting method is a trial-and-error method. To solve a problem where
the values of y are known at xo and x,, the boundaries of the interval of interest,
we set up the problem as though it were an initial-value problem, with two
llknowns" given at the same boundary - for example, at xo. (See Figure 10-17
for an example of an initial-value problem of this type: the two knowns, shown in
bold, are the value of y at xo and a trial value of y' at xo.) Using the trial value of

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