Partial Differential Equations

Derivation of the Wave Equation. (pdf file) These notes show that the equation of motion for small amplitude transverse oscillations of an elastic string is the wave equation.

The Telegraph Equation. (pdf file) A model for signal transmission along wires.

Numerical Solution of Partial Differential Equations. (pdf file) These notes introduce discretization as a method for generating approximate solutions for partial differential equations. This method is the analog, for PDE's, of Euler's method.

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). This demonstration also illustrates the behaviour of solutions of the wave equation. It also plots the solution given by separation of variables that you have found in class. The demonstration animates the solution by successively plotting u(x,0), followed by u(x,dt), followed by u(x,2dt) and so on.

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.