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?
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?