The Euler method provides the simplest approach for solving ODEs
and most of the more sophisticated techniques derive from it. In
the Euler approach, the first order differential equation
can be approximated by making small increments in t
and assuming that the slope, f(x,t),
is nearly constant over those increments. So, beginning at point
x0, the
next point x1
becomes
for small dx, which
relates to the increment in dt
as
then
Repeating for the next point
This can be summarized in the finite difference equation:
The Euler method assumes that the slope f(xn,tn)
evaluated at the point (xn,tn)
remains approximately constant over the dt
interval to x(n+1).
See the demonstration of the Euler Method
for the case of an object falling in a constant gravitational field.
Latest update: Dec.12.2003
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