# Thus we substitute u t f g t into the heat equation

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Unformatted text preview: string The Fourier sine series can also be used to ﬁnd the solution to an initial value problem for the vibrating string with ﬁxed endpoints at x = 0 and x = L. We formulate this problem as follows: we seek a function u(x, t), deﬁned for 0 ≤ x ≤ L and t ≥ 0 such that 1. u(x, t) satisﬁes the wave equation ∂2u ∂2u = c2 2 , ∂t2 ∂x (4.18) where c is a constant. 2. u(x, t) satisﬁes the boundary condition u(0, t) = u(L, t) = 0, i.e. the displacement of the string is zero at the endpoints. 3. u(x, t) satisﬁes the initial conditions u(x, 0) = h1 (x) and ∂u (x, 0) = h2 (x), ∂t where h1 (x) and h2 (x) are given functions, the initial position and velocity of the string. 98 Note that the wave equation itself and the boundary condition are homogeneous and linear , and therefore satisfy the principal of superposition. Once again, we ﬁnd the solution to our problem in two steps: Step I. We ﬁnd all of the solutions to the homogeneous linear conditions of the special form u(x, t) = f (x)g (t)....
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## This document was uploaded on 01/12/2014.

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