s11_mthsc208_hw21

# s11_mthsc208_hw21 - MthSc 208, Fall 2011 (Differential...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MthSc 208, Fall 2011 (Differential Equations) Dr. Macauley HW 21 Due Friday April 29th, 2011 (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 ) = 4sin x- 3sin2 x + 2sin3 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) together (superposition), 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 = 0 , u (0 ,y ) = 0 , u ( π,y ) = y ( π- y ) , u ( x, 0) = 0 , u ( x,π ) = 4sin x- 3sin2 x + 2sin3 x. (d) Sketch the solutions to (a), (b), and (c). Hint: it is enough to sketch the boundaries, and then use the fact that the solutions are harmonic functions ....
View Full Document

## This note was uploaded on 03/11/2012 for the course MTHSC 208 taught by Professor Staufeneger during the Spring '09 term at Clemson.

### Page1 / 3

s11_mthsc208_hw21 - MthSc 208, Fall 2011 (Differential...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online