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Unformatted text preview: , t), deﬁned for 0 ≤ x ≤ π and t ≥ 0, such
that
∂u
∂2u
,
=
∂t
∂x2 u(0, t) = 10, 90 u(π, t) = 50. (4.13) (Hint: Let v = u − w, where w is the solution to part a. Determine what
conditions v must satisfy.)
c. Find the particular solution to (4.13) which in addition satisﬁes the initial
condition
40
u(x, 0) = 10 + x + 2 sin x − 5 sin 2x.
π
4.2.7.a. Find the eigenvalues and corresponding eigenfunctions for the diﬀerential operator
d2
L = 2 + 3,
dt
which acts on the space V0 of wellbehaved functions f : [0, π ] → R which vanish
at the endpoints 0 and π by
L(f ) = d2 f
+ 3f.
dt2 b. Find the function u(x, t), deﬁned for 0 ≤ x ≤ π and t ≥ 0, which satisﬁes
the following conditions:
∂u
∂2u
+ 3u,
=
∂t
∂x2 u(0, t) = u(π, t) = 0, u(x, 0) = sin x + 3 sin 2x. 4.2.8. The method described in this section can also be used to solve an initial
value problem for the heat equation in which the Dirichlet boundary condition
u(0, t) = u(L, t) = 0 is replaced by the...
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 Winter '14
 Equations

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