Similarity Solution Method and Initial Condition

[Metronome]

In this video, the professor solves a heat equation with two boundary conditions and one initial condition, [imath]u(x, 0) = 0[/imath], via the similarity solution method.

The PDE is used to construct a second order ODE, and the two boundary conditions determine the two arbitrary constants that arise out of the ODE. However, the initial condition is never used.

How is it possible to find the particular solution to the heat equation without using the initial condition? In other contexts this generally requires a Fourier Series. Does the similarity solution method only work for that particular initial condition, [imath]u(x, 0) = 0[/imath], and apply it implicitly behind the scenes?

 

[nasi112]

When you use the Similarity Solution Method on the heat equation, the main idea is to look for a solution in this form:

\(\displaystyle u = t^{c/b}y\left(\frac{x}{t^{a/b}}\right)\)

which means the initial condition \(\displaystyle u(x,0) = 0\) is satisfied automatically.

Also, the Similarity Solution Method can work on a lot of nonlinear equations. One such an equation is the Burger's Equation. Use this method when the domain is infinite. You can always try to use other techniques that you have already learnt and when they fail, you can try this method as an extra resource.