Information Technology Reference
In-Depth Information
Exercise 6.8.
Suppose we have a differential equation on the form
u
0
.t / D f.
u
;t/;
u
.0/ D U
0
;
that is,
f
can be an explicit function of
t
.Asimpleexampleis
f.
u
;t/D2t
u
,
with the corresponding solution
u
D e
t
2
.
Understand the idea behind the derivation of Heun's method, and use this under-
standing to generalize Algorithm
6.12
such that it works for the right-hand side
functions
f.
u
;t/
in the above differential equation.
˘
6.7
Projects
6.7.1
Computing the Volume of a Cylindrical Container
The purpose of this project is to compute the volume of water in a cylindrical
container. An example of such a container is given in Fig.
6.3
. The radius of the
container can change along the center axis, and is given by a function
R.x/
. Water
is stored in the container up to a height
H
. The goal is to develop a program that
can compute the water volume in such a container, given the function
R.x/
and the
parameter
H
as input.
10
x
6
5
H
3
R(x)
Fig. 6.3
Sketch of a
cylindrical container with
circular cross section and
radius
R.x/
. The container
contains water up to a
height
H
0
1
3
0
