Consider the flow of heat in a cylinder 0 < r < 1, 0 < < 2 , < z < of radius 1 and of
Chapter 11, Problem 8(choose chapter or problem)
Consider the flow of heat in a cylinder 0 < r < 1, 0 < < 2 , < z < of radius 1 and of infinite length. Let the surface of the cylinder be held at temperature zero, and let the initial temperature distribution be a function of the radial variable r only. Then the temperature u is a function of r and t only and satisfies the heat conduction equation 2_urr + 1 r ur = ut, 0< r < 1, t > 0, and the following initial and boundary conditions: u(r, 0) = f (r), 0 r 1, u(1, t) = 0, t > 0. Show that u(r, t) = _ n=1 cn J0(nr)e22n t , where J0(n) = 0. Find a formula for cn.
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