A case of heat flow which is virtually one-dimensional arises when the conducting medium is a circular cylinder and the temperature function u depends only on the time t and the distance r from the axis of the cylinder, . For example, imagine a cylindrical pipe filled with a hot fluid and suppose that one wishes to study the loss of heat through the sides of the pipe.

Let c, p, k, K denote the thermal constants of the cylinder. By considering the heat energy contained in a section of pipe of length H and lying between the radiir = a, r = b(Why are the radii different? Shouldn't they be the same?)show that

Hence,obtain the equation for the source-free radial heat flow in a cylinder:

.

How do I start obtaining the equation?