Euler's method.
This demo contains an annotated implementation
of Euler's method. You can run
Euler's method one step at a time. Each step is accompanied by a
commentary which shows you the computation done during that step.
Math Demonstration Programs |
Euler's method.
This demo contains an annotated implementation
of Euler's method. You can run
Euler's method one step at a time. Each step is accompanied by a
commentary which shows you the computation done during that step.
Simple ODE solvers.
This demo implements
three simple fixed step size methods
- Euler's method, the improved Euler method and the Runge-Kutta algorithm.
You can simultaneously display the results of all three
methods with various step sizes to get some first impressions as to how
well the methods work.
A Simple ODE Solver with Automatic Step Size Adjustment.
This demo contains an annotated implementation of a very naive
variable step size method. You can run it one step at a time.
Each step is accompanied by a commentary which shows you the computation
done during that step.
The RLC Circuit.
This demo illustrates resonance using an RLC circuit.
Phase Plane Plots.
This demo exhibits a phase plane plotter that can be used to
determine the qualitative behaviour of solutions of systems
of first order ordinary differential equations. The particular
system illustrated here arises when Newton's Law of gravitation
is modified to incorporate small
corrections due to general relativity. This is needed to explain
the precession of the perihelion of Mercury.
The Damped Nonlinear Pendulum.
This demo applies the phase plane plotter to the
damped nonlinear pendulum.
An animated pendulum shows the state of the pendulum corresponding
to the current point of the trajectory in the phase plane plotter.
Fourier Series.
This demo shows a
Fourier series
adding up to the expected answer.
The Wave Equation.
This demonstration illustrates the behaviour of solutions of the
wave equation. The demonstration plots the solution
given by separation of variables that you have found in class.
Separation of variables expresses the solution as a sum
b_1(t) sin(pi x/l) + b_2(t) sin(2 pi x/l) + ...
of modes. When the demonstration starts, the initial amplitude is plotted. By clicking the "Advance time" button, you instruct the computer to increase the time by an amount specified in the "time interval window".
The Wave Equation (animated) [
one bump |
two bumps ].
These two demonstrations also illustrate the behaviour of solutions of the
wave equation. They also plot the solution
given by separation of variables that you have found in class.
The demonstrations animate the solution by successively plotting
u(x,0), followed by u(x,dt), followed by u(x,2dt) and so on.
The two demonstrations use different initial conditions.
The Heat Equation.
This demonstration illustrates the behaviour of solutions of the
heat equation. The demonstration plots the solution
given by separation of variables.
When the demonstration starts, the initial amplitude is plotted.
It is the same as that used for the wave equation demonstrations.
By clicking the "Advance time" button, you instruct the computer
to increase the time by an amount specified in the "time interval window".
You should contrast the behaviour of solutions to the heat equation
with that of solutions to the wave equation.
The Telegraph Equation.
This program gives an animated demonstration of the solution to the
telegraph equation. The viewer
may adjust the equation parameters to give signal transmission with and without
distortion due to dispersion.
Parametric Resonance.
This demonstration uses a variable
length pendulum to illustrate parametric resonance, which is a resonance
phenomenon that arises because some parameter of the system is varying
periodically in time.
The Upside Down Pendulum.
This demonstration illustrates that one can stabilize
an upside down pendulum
by shaking it vertically.