Partial differential equation
An equation with a derivative isolating one of the independent variables. The variables themselves can
be functions. There are various examples of these from Wolfram for physics and engineering.
The above has second partial derivatives. The above involves partial derivatives of a function w. This is a
partial differential equation (PDE). These are important in modelling because they unlock certain
phenoman. The above is a wave equation measuing water, light, and sound waves. c could be a
constant to the wave speed.
Show that a solution is given by w(t,x) :=cos(2x+2ct).
The partial derivative needs to satify the above solution. We need to calculate the partial derivatives of
the above function w(t,x) :=cos(2x+2ct) that appear in the wave formula. |e need to show that they
balance. We formulate the PDE and extract the solution somehow.
So:
if w=cos(2x+2ct) then: (you need to calculate first partial derivative of the wave function first)
refer to 3:34 in video for rational of -2csin(2x+2ct) and 4:25 for -4c^2*cos(2x+2ct) and 5:05 for 2sin(2x+2ct) and 5:25
Note * is the wave equation.
Show that the wave equation has solution of a more general form:
w(t,x)= f(x+ct) where f is unknown. This becomes the chain rule.
http://www.youtube.com/watch?v=ezH88VekVWU&feature=related