Show that the general solution to the heat equation u

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Unformatted text preview: 2x sin y, 5.5.2. Solve the following initial value problem for a vibrating square membrane: Find u(x, y, t), 0 ≤ x ≤ π, 0 ≤ y ≤ π , such that ∂2u =4 ∂t2 ∂2u ∂2u +2 ∂x2 ∂y , u(x, 0, t) = u(x, π, t) = u(0, y, t) = u(π, y, t) = 0, u(x, y, 0) = 0, ∂u (x, y, 0) = 2 sin x sin y + 13 sin 2x sin y. ∂t 5.5.3. Solve the following initial value problem for a vibrating square membrane: Find u(x, y, t), 0 ≤ x ≤ π, 0 ≤ y ≤ π , such that ∂2u ∂2u ∂2u = + 2, ∂t2 ∂x2 ∂y u(x, 0, t) = u(x, π, t) = u(0, y, t) = u(π, y, t) = 0, ∂u (x, y, 0) = 0, ∂t u(x, y, 0) = p(x)q (y ), where p(x) = 5.6 x, for 0 ≤ x ≤ π/2, π − x, for π/2 ≤ x ≤ π , q (y ) = y, π − y, for 0 ≤ y ≤ π/2, for π/2 ≤ y ≤ π . The Laplace operator in polar coordinates In order to solve the heat equation over a circular plate, or to solve the wave equation for a vibrating circular drum, we need to express the Laplace operator in polar coordinates (r, θ). These coo...
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This document was uploaded on 01/12/2014.

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