The following examples should illustrate the procedure of stating an
initial value problem and solving it with
ode. If these
examples are too elementary, see Input Language, for a formal
specification of the
ode input language. There is also a
directory containing examples of
ode input, which is distributed
along with the GNU plotting utilities. On most systems it is
installed as /usr/share/ode or /usr/local/share/ode.
Our first example is a simple one, namely
y'(t) = y(t)
with the initial condition
y(0) = 1
The solution to this differential equation is
y(t) = e^t.
y(1) = e^1 = 2.718282
to seven digits of accuracy.
You may obtain this result with the aid of
ode by typing on the
command line the sequence of commands
ode y' = y y = 1 print t, y step 0, 1
Two columns of numbers will appear. Each line will show the value of the independent variable t, and the value of the variable y, as t is `stepped' from 0 to 1. The last line will be
as expected. You may use the ‘-p’ option to change the precision. If, for example, you type ‘ode -p 10’ rather than ‘ode’, you will get ten digits of accuracy in the output, rather than seven (the default).
After the above output,
ode will wait for further instructions.
Entering for example the line
step 1, 0
should yield two more columns of numbers, containing the values of t and y that are computed when t is stepped back from 1 to 0. You could type instead
step 1, 2
to increase rather than decrease t. To exit
you would type a line containing only ‘.’, i.e. a single period,
and tap `return'.
ode will also exit if it sees an end-of-file
indicator in its input stream, which you may send from your terminal by
Each line of the preceding example should be self-explanatory. A ‘step’ statement sets the beginning and the end of an interval
over which the independent variable (here, t) will range, and
ode to set the numerical scheme in motion. The initial
value appearing in the first ‘step’ statement (i.e., 0) and the
y = 1
are equivalent to the initial condition y(0) = 1. The statements
‘y' = y’ and ‘y = 1’ are very different: ‘y' = y’
defines a way of computing the derivative of y, while ‘y
= 1’ sets the initial value of y. Whenever a ‘step’
statement is encountered,
ode tries to step the independent
variable through the interval it specifies. Which values are to be
printed at each step is specified by the most recent ‘print’
statement. For example,
print t, y, y'
would cause the current value of the independent variable t, the variable y, and its derivative to be printed at each step.
ode's ability to take its input or the initial part
of its input from a file, you could prepare a file containing the
# an ode to Euler y = 1 y' = y print t, y, y'
Call this file euler. (The ‘#’ line is a comment line,
which may appear at any point. Everything from the ‘#’ to the
end of the line on which it appears will be ignored.) To process
this file with
ode, you could type on your terminal
ode -f euler step 0, 1
These two lines cause
ode to read the file euler, and the
stepping to take place. You will now get three quantities (t,
y, and y') printed at each of the values of t
between 0 and 1. At the conclusion of the stepping,
will wait for any further commands to be input from the terminal. This
example illustrates that
ode -f euler
is not equivalent to
ode < euler
The latter would cause
ode to take all its input from the file
euler, while the former allows subsequent input from the
terminal. For the latter to produce output, you would need to include a
‘step’ line at the end of the file. You would not need to include
a ‘.’ line, however. ‘.’ is used to terminate input only
when input is being read from a terminal.
A second simple example involves the numerical solution of a second-order differential equation. Consider the initial value problem
y''(t) = -y(t) y(0) = 0 y'(0) = 1
Its solution would be
y(t) = sin(t)
To solve this problem using
ode, you must express this
second-order equation as two first-order equations. Toward this end you
would introduce a new function, called yp say, of the independent
variable t. The pair of equations
y' = yp yp' = -y
would be equivalent to the single equation above. This sort of reduction of an n'th order problem to n first order problems is a standard technique.
To plot the variable y as a function of the variable t, you could create a file containing the lines
# sine : y''(t) = -y(t), y(0) = 0, y'(0) = 1 sine' = cosine cosine' = -sine sine = 0 cosine = 1 print t, sine
(y and yp have been renamed sine and cosine, since that is what they will be.) Call this file sine. To display the generated data points on an X Window System display as they are generated, you would type
ode -f sine | graph -T X -x 0 10 -y -1 1 step 0, 2*PI .
After you type the
graph -T X will pop up a window, and after you type the ‘step’ line, the generated
dataset will be drawn in it. The ‘-x 0 10’ and ‘-y -1 1’
options, which set the bounds for the two axes, are necessary if you
wish to display points in real time: as they are generated.
If the axis bounds were not specified on the command line,
graph -T X would wait until all points are read from the
input before determining the bounds, and drawing the plot.
A slight modification of this example, showing how
generate several datasets in succession and plot them on the same graph,
would be the following. Suppose that you type on your terminal the
ode -f sine | graph -T X -C -x 0 10 -y -1 1 step 0, PI step PI, 2*PI step 2*PI, 3*PI .
Then the sine curve will be traced out in three stages. Since the
output from each ‘step’ statement ends with a blank line,
graph -T X will treat each section of the sine curve as a
different dataset. If you are using a color display, each of the three
sections will be plotted in a different color. This is a feature
graph, which normally changes its linemode after each
dataset it reads. If you do not like this feature, you may turn it off
graph -T X -B instead of
graph -T X.
In the above examples, you could use any of the other variants of
graph instead of
graph -T X. For example, you could use
graph -T ps to obtain a plot in encapsulated Postscript format,
ode -f sine | graph -T ps > plot.ps step 0, 2*PI .
You should note that of the variants of
graph, the variants
graph -T png,
graph -T pnm,
graph -T gif,
graph -T svg,
graph -T ai,
graph -T ps,
graph -T fig,
graph -T pcl and
hpgl do not produce output in real time, even when the axis bounds are
specified with the ‘-x’ and ‘-y’ options. So if any
of these variants is used, the plot will be produced only when input
ode is terminated, which will occur when you type ‘.’.
In the preceding examples, the derivatives of the dependent variables
were specified by comparatively simple expressions. They are allowed to
be arbitrarily complicated functions of the dependent variables and the
independent variable. They may also involve any of the functions that
are built into
ode has a fair number of functions
built in, including abs, sqrt, exp, log, log10,
sin, cos, tan, asin, acos, atan, sinh,
cosh, tanh, asinh, acosh, and atanh. Less familiar
functions which are built into it are besj0, besj1,
besy0, besy1, erf, erfc, inverf, lgamma,
gamma, norm, invnorm, ibeta, and igamma. These have
the same definitions as in the plotting program
functions take a single argument, except for ibeta, which takes
three, and igamma, which takes two).
ode also knows the
meaning of the constant ‘PI’, as the above examples show. The
names of the preceding functions are reserved, so, e.g., ‘cos’ and
‘sin’ may not be used as names for variables.
Other than the restriction of avoiding reserved names and keywords, the
names of variables may be chosen arbitrarily. Any sequence of
alphanumeric characters starting with an alphabetic character may be
used; the first 32 characters are significant. It is worth noting
ode identifies the independent variable by the fact that it
is (or should be) the only variable that has not appeared on the left
side of a differential equation or an initial value assignment. If there is more than than one such variable then no stepping takes place;
instead, an error message is printed. If there is no such variable,
a dummy independent variable is invented and given the name