In this case the nonvanishing coecients for the

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Unformatted text preview: are kept at zero temperature. We expect that there should exist a unique function u(x, t), defined for 0 ≤ x ≤ L and t ≥ 0 such that 1. u(x, t) satisfies the heat equation ∂u ∂2u = c2 2 , ∂t ∂x (4.8) where c is a constant. 2. u(x, t) satisfies the boundary condition u(0, t) = u(L, t) = 0, in other words, the temperature is zero at the endpoints. (This is sometimes called the Dirichlet boundary condition.) 85 3. u(x, t) satisfies the initial condition u(x, 0) = h(x), where h(x) is a given function, the initial temperature of the rod. In more advanced courses, it is proven that this initial value problem does in fact have a unique solution. We will shortly see how to find that solution. Note that the heat equation itself and the boundary condition are homogeneous and linear —this means that if u1 and u2 satisfy these conditions, so does c1 u1 + c2 u2 , for any choice of constants c1 and c2 . Thus homogeneous linear conditions satisfy the principal of superposition. Our method makes use of the dichotomy into homogeneous and nonhomogeneous conditions: Step I. We find all of the solutions to the homogeneous linear conditi...
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This document was uploaded on 01/12/2014.

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