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Unformatted text preview: )e−(2cπ/L) t + . . .
2 2 (4.12) is a solution to the heat equation together with its homogeneous boundary
conditions, for arbitrary choice of the constants b1 , b2 , . . . .
Step II consists of determining the constants bn in (4.12) so that the initial
condition u(x, 0) = h(x) is satisﬁed. Setting t = 0 in (4.12) yields
h(x) = u(x, 0) = b1 sin(πx/L) + b2 sin(2πx/L) + . . . .
It follows from the theory of the Fourier sine series that h can indeed be represented as a superposition of sine functions, and we can determine the bn ’s as
the coeﬃcients in the Fourier sine series of h. Using the techniques described in
Section 3.3, we ﬁnd that
bn = 2
L L h(x) sin(nπx/L)dx.
0 Example 1. Suppose that we want to ﬁnd the function u(x, t), deﬁned for
0 ≤ x ≤ π and t ≥ 0, which satisﬁes the initial-value problem:
∂x2 u(0, t) = u(π, t) = 0, u(x, 0) = h(x) = 4 sin x +2 sin 2x +7 sin 3x. In this case, the nonvanishing coeﬃcients for the Fourier sine series of h are
b1 = 4, b2 = 2, b3 = 7, so the solution must be
u(x, t) = 4 sin xe−t...
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- Winter '14