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Unformatted text preview: dition u(x, 0) = h(x), where h(x) is a given function, representing the initial
temperature.
For any ﬁxed choice of t0 the function u(x, t0 ) is an element of V0 , the space of
piecewise smooth functions deﬁned for 0 ≤ x ≤ 1 which vanish at the endpoints.
92 Our idea is to replace the “inﬁnitedimensional” space V0 by a ﬁnitedimensional
Euclidean space R n−1 and reduce the partial diﬀerential equation to a system
of ordinary diﬀerential equations. This corresponds to utilizing a discrete model
for heat ﬂow rather than a continuous one.
For 0 ≤ i ≤ n, let xi = i/n and
ui (t) = u(xi , t) = the temperature at xi at time t.
Since u0 (t) = 0 = un (t) by the boundary conditions, the temperature at time t
is speciﬁed by u1 (t) u (t) ,
u(t) = 2 ·
un−1 (t)
a vectorvalued function of one variable. The initial condition becomes h(x1 ) h(x2 ) .
u(0) = h,
where
h= ·
h(xn−1 )
We can approximate the ﬁrstorder partial derivative by a diﬀerence quotient:
∂u xi + xi+1
[ui+1 (t) − ui (t)]
. ui+1 (t) − ui (t)
=
,t =
= n[ui+1 (t) − ui (t)].
∂x
2
xi+1 − xi
1/n
Similarly, we can approximate the sec...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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