The heat equation is typically developed using the following

The heat equation is typically developed using the following 5 assumptions:

A1. Rod material is homogeneous. Thus c, k, p are independent of the position, temperature, and time.

A2. Length of rod is independent of the temperature.

A3. Rod is perfectly insulated along its lateral surface (i.e, heat must flow horizontally, and at a fixed time t, the temperature on a vertical corss section is the same).

A4. Heat flows from left to right.

A5. Rod is very thin (1-dimensional)

Compound the heat equation by imposing the following additional assumption:

The cross-sectional area of the rod, A, varies very slowly with x [so that the temperature U remains approximately a function of x and t]

Solution

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