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Unformatted text preview: SECTION 10.2 FOURIER SERIES – BUT FIRST, THE HEAT EQUATION First, some motivation. Suppose we have a rod of length L , insulated along its sides so heat flows only along the rod. Suppose further that the rod is made of a homogeneous material and has uniform cross section and a diameter so small that we can ignore any variation in directions perpendicular to the rod’s length. The temperature at a given point along the rod will vary according to both the location along the rod and the time, so it is given by a function u ( x, t ). Empirical laws governing conduction of heat then imply that this function satisfies the partial differential equation α 2 u xx ( x, t ) = u t ( x, t ) for some positive constant α . A derivation begins on page 649. We suppose that the initial temperature distribution along the rod is given, so that u ( x, 0) = f ( x ) , ≤ x ≤ L. For now we’ll also assume that the temperature is always 0 at the ends of the rod, that is, u (0 , t ) = u ( L, t ) = 0 , t >...
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- Spring '10
- Fourier Series, Partial differential equation, Joseph Fourier, 2L, 2 L, nπx