Although it yields only approximate solutions it can

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Unformatted text preview: b = 0 or sin(ωL) = 0, Now, however, f (L) = 0 ⇒ and hence either b = 0 and we obtain only the trivial solution or sin(ωL) = 0. The latter possibility will occur if ωL = nπ , or ω = (nπ/L), where n is an integer. In this case, we obtain f (x) = b sin(nπx/L). Therefore, we conclude that the only nontrivial solutions to (4.11) are constant multiples of f (x) = sin(nπx/L), with λ = −(nπ/L)2 , n = 1, 2, 3, . . . . For each of these solutions, we need to find a corresponding g (t) solving equation (4.9), g (t) = λc2 g (t), where λ = −(nπ/L)2 . This is just the equation of exponential decay, and has the general solution 2 g (t) = be−(ncπ/L) t , where a is a constant of integration. Thus we find that the nontrivial product solutions to the heat equation together with the homogeneous boundary condition u(0, t) = 0 = u(L, t) are constant multiples of un (x, t) = sin(nπx/L)e−(ncπ/L) t . 2 88 It follows from the principal of superposition that u(x, t) = b1 sin(πx/L)e−(cπ/L) t + b2 sin(2πx/L...
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This document was uploaded on 01/12/2014.

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