## 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.

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