# 14 and 515 that x y d 2u x y dxdy t2 x y t

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Unformatted text preview: x sinh y + 7 sinh(2π ) sin 2x sinh 2y . b. Solve the following Dirichlet problem for Laplace’s equation in the same square region: Find u(x, y ), 0 ≤ x ≤ π, 0 ≤ y ≤ π , such that ∂2u ∂2u + 2 = 0, ∂x2 ∂y u(π, y ) = sin 2y + 3 sin 4y, u(0, y ) = 0, u(x, 0) = 0 = u(x, π ). c. By adding the solutions to parts a and c together, ﬁnd the solution to the Dirichlet problem: Find u(x, y ), 0 ≤ x ≤ π, 0 ≤ y ≤ π , such that ∂2u ∂2u + 2 = 0, ∂x2 ∂y u(π, y ) = sin 2y +3 sin 4y, 5.3 u(x, 0) = 0, u(0, y ) = 0, u(x, π ) = sin x−2 sin 2x+3 sin 3x. Initial value problems for heat equations The physical interpretation behind the heat equation suggests that the following initial value problem should have a unique solution: Let D be a bounded region in the (x, y )-plane which is bounded by a piecewise smooth curve ∂D, and let h : D → R be a continuous function which vanishes on ∂D. Find a function u(x, y, t) such that 123 1. u satisﬁes the heat equation ∂u = c2 ∂t ∂2u ∂2u +2 ∂x2 ∂...
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## This document was uploaded on 01/12/2014.

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