ODE Module is a pro Fit module.
It solves initial value problems of sets of coupled
first-order differential equations (up to 14). The
solutions are plotted over the integration variable and
against each other (phase diagrams). For information on the
scientific analysis package pro Fit, visit the
QuantumSoft pro Fit
site.
ODE Module is
Freeware.
How It
Works
Ordinary
differential equations (ODEs) of any order can be reduced
to a set of first-order differential equations which can be
numerically integrated. The ODEs can be coupled as well.
Consider a second-order ODE of the function
x(t)
like
x'' + q(t)
x' = r(t)
,
where x' denotes the first and
x'' the second derivative
of x(t)
with respect
to t. Rename the first derivative
to
y = x'
and rewrite the ODE like
y' = r(t)
-
q(t)
y
.
Now you have a set of two
first-order differential equations:
x' = y
y' = r(t)
-
q(t)
y
Starting from any initial
point t0,
where x and y (that is, x') are known, one can calculate
the derivatives of x and y and thus integrate to a second
point t1,
using a numerical integration method like Euler's,
Runge-Kutta or the like.
For
example, a third-order ODE results in three first-order
differential equations, two second-order ODEs result in
four equations, respectively. In order to be flexible for
systems of different degrees, we use the following naming
conventions in the ODE Module:
y1
=
x, y2
=
y, y3
=
z, ...
Then we write the differential
equations above in the form
y1'
= y2
y2'
= r(t)
-
q(t)
y2
When using ODE Module to
integrate such systems of differential equations, you have
to supply a pro Fit function that calculates the values of
the derivatives at a given point using this set of
equations. ODE Module uses the Runge-Kutta method (fixed or
variable stepsize) for integration and calls this function.
For futher information on installing and using ODE Module,
see the Readme
file.
History
1.4
- Version for pro Fit 6.1
- Mac OS X Universal (PPC and Intel)
- Default line thickness 0.5 pt instead of 0.25 pt
- Carbon version for pro Fit 5.6 up to 6.0
- Increased number of equations to 14
- Fixed bug with N > 6 (I didn't consider the limitation of the pro Fit InputBox function)
- Increased the number of equations to 12
- Solved problem with phase diagrams when a function is constant
- Added option to save the calculated data in a pro Fit data window. This is useful for subsequent calculations with pro Fit.
- First public release
The Runge-Kutta algorithms were implemented by Press, Teukolsky, Vetterling, Flannery: Numerical Recipes in C. Cambridge University Press. pro Fit is made by QuantumSoft, Switzerland.
Warranties
None. ;-) Please contact me if you experience any problems.