s09_mthsc208_hw24

s09_mthsc208_hw24 - MTHSC 208 (Differential Equations) Dr....

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Unformatted text preview: MTHSC 208 (Differential Equations) Dr. Matthew Macauley HW 24 Due Friday April 24th, 2009 (1) Which of the following functions are harmonic? (a) f ( x ) = 10- 3 x . (b) f ( x, y ) = x 2 + y 2 . (c) f ( x, y ) = x 2- y 2 . (d) f ( x, y ) = e x cos y . (e) f ( x, y ) = x 3- 3 xy 2 . (2) (a) Solve the following Dirichlet problem for Laplace’s equation in a square region: Find u ( x, y ), 0 ≤ x ≤ π , 0 ≤ y ≤ π such that ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 = 0 , u (0 , y ) = u ( π, y ) = 0 , u ( x, 0) = 0 , u ( x, π ) = x ( π- x ) . (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 ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 = 0 , u (0 , y ) = 0 , u ( π, y ) = y ( π- y ) , u ( x, 0) = u ( x, π ) = 0 (c) By adding the solutions to parts (a) and (b) togeter, find the solution to the Dirichlet problem: Find u ( x, y ), 0 ≤ x ≤ π , 0 ≤ y ≤ π such that ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2...
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This note was uploaded on 03/12/2012 for the course MTHSC 208 taught by Professor Staufeneger during the Spring '09 term at Clemson.

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s09_mthsc208_hw24 - MTHSC 208 (Differential Equations) Dr....

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