We write ux y t f x y g t and substitute into 521

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Unformatted text preview: nd u(x, y, t), where 0 ≤ x ≤ π, 0 ≤ y ≤ π and t ≥ 0, such that ∂u ∂2u ∂2u + 2, = ∂t ∂x2 ∂y u(x, 0, t) = u(x, π, t) = u(0, y, t) = u(π, y, t) = 0, u(x, y, 0) = 2 sin x sin y + 5 sin 2x sin y. You may assume that the nontrivial solutions to the eigenvalue problem ∂2f ∂2f (x, y ) + 2 (x, y ) = λf (x, y ), 2 ∂x ∂y f (x, 0) = f (x, π ) = f (0, y ) = f (π, y ) = 0, are of the form λ = −m2 − n2 , f (x, y ) = bmn sin mx sin ny, for m = 1, 2, 3, . . . and n = 1, 2, 3, . . . , where bmn is a constant. 5.3.2. Solve the following initial value problem for the heat equation in a square region: Find u(x, y, t), where 0 ≤ x ≤ π, 0 ≤ y ≤ π and t ≥ 0, such that ∂u ∂2u ∂2u + 2, = ∂t ∂x2 ∂y u(x, 0, t) = u(x, π, t) = u(0, y, t) = u(π, y, t) = 0, u(x, y, 0) = 2(sin x)y (π − y ). 5.3.3. Solve the following initial value problem in a square region: Find u(x, y, t), where 0 ≤ x ≤ π, 0 ≤ y ≤ π and t ≥ 0, such that 1 ∂u ∂2u ∂2u = +...
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This document was uploaded on 01/12/2014.

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