T 516 to derive this equation we represent the uid ow

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Unformatted text preview: y . (5.8) 2. u satisfies the “Dirichlet boundary condition” u(x, y, t) = 0, for (x, y ) ∈ ∂D. 3. u satisfies the initial condition u(x, y, 0) = h(x, y ). The first two of these conditions are homogeneous and linear, so our strategy is to treat them first by separation of variables, and then use Fourier analysis to satisfy the last condition. In this section, we consider the special case where D = {(x, y ) ∈ R 2 : 0 ≤ x ≤ a, 0 ≤ y ≤ b}, so that the Dirichlet boundary condition becomes u(0, y, t) = u(a, y, t) = u(x, 0, t) = u(x, b, t) = 0. In this case, separation of variables is done in two stages. First, we write u(x, y, t) = f (x, y )g (t), and substitute into (5.8) to obtain f (x, y )g (t) = c2 ∂2f ∂2f +2 ∂x2 ∂y g (t). Then we divide by c2 f (x, y )g (t), 1 1 g (t) = 2 g (t) c f (x, y ) ∂2f ∂2f +2 ∂x2 ∂y . The left-hand side of this equation does not depend on x or y while the right hand side does not depend on t. Hence neither side can depend on x, y , or...
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This document was uploaded on 01/12/2014.

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