# For 0 i n let xi in and ui t uxi t the temperature

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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: ∂u ∂2u , = ∂t ∂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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