428 Chapter 7 Numerical Methods
11.Approximate the solution of the wave equation in a semi-infinite strip 3
units wide. Assumeu=0 on all boundaries, zero initial velocity, and an
initial value foruthat is 1 in a corner and 0 elsewhere. Take x= y= 1
andρ^2 = 1 /2.7.6 Comments and References
Our objective in this chapter has been to survey some elementary numerical
methods for problems like those we attacked analytically in earlier chapters.
We have only had enough space to touch on the central topics: obtaining re-
placement equations, solving linear systems of equations by direct and iterative
methods, numerical stability, and order of error.
The methods we have introduced are satisfactory for a first introduction and
for learning something about partial differential equations, but they are not
adequate for any serious problem solving. New techniques for these problems
are superior in speed, accuracy and stability but are also more complicated. Of
the many texts available, two excellent ones areNumerical Analysisby Burden
and Faires, for general methods, andNumerical Solution of Partial Differential
Equationsby Smith. (See the Bibliography.)
Almost all numerical methods for linearpartial differential equations rely
on the symbolism and theory of matrices. Two outstanding texts on matrix
theory areApplied Linear Algebra, 3rd ed., by Noble and Daniel, andMatrices
by Barnett.
Miscellaneous Exercises
1.Set up and solve replacement equations for this boundary value problem.
Use x= 1 /3.
d^2 u
dx^2 −√
24 xu= 0 , 0 <x< 1 ,
du
dx( 0 )= 1 , u( 1 )= 1.2.Use the change of variablesx=(r−a)/(b−a)andv(r)=u(x)to con-
vert the equation
1
rd
dr(
rdv
dr)
−q(r)v=f(r), a<r<b,to an equation inuon the interval 0<x<1.