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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.
 Winter '14
 Equations

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