Numerical Methods

CHAPTER

7

7.1 Boundary Value Problems

More often than not, significant practical problems in partial — and even or-
dinary — differential equations cannot be solved by analytical methods. Dif-
ficulties may arise from variable coefficients, irregular regions, unsuitable
boundary conditions, interfaces, or just overwhelming detail. Now that ma-
chine computation is cheap and easily accessible, numerical methods provide
reliable answers to formerly difficult problems. In this chapter we examine
a few methods that are simple and equally adaptable to machine or manual
computation. Implementation of some of these methods with a spreadsheet
program is explained and carried out on the CD.
If we cannot find a simple analytic formula for the solution of a boundary
value problem, we may be satisfied with a table of (approximate) values of the
solution. For instance, the solution of the problem

d^2 u

dx^2

− 12 xu=− 1 , 0 <x< 1 , (1)

u( 0 )= 1 , u( 1 )=− 1 , (2)

may be written out in terms of Airy functions, but the values ofushown in
Table 1 are more informative for most of us. One way to obtain such a table
is to replace the original analytical problem by an arithmetical problem, as
described in what follows.

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