T2 514 on the other hand the potential energy stored

Info iconThis preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: + B2 sin(2πx/a) sinh(2πb/a)+ . . . + Bk sin(2πk/a) sinh(kπb/a) + . . . . We see that Bk sinh(kπb/a) is the k -th coefficient in the Fourier sine series for f (x). For example, if a = b = π , and f (x) = 3 sin x + 7 sin 2x, then we must have f (x) = B1 sin(x) sinh(π ) + B2 sin(2x) sinh(2π ), and hence B1 = 3 , sinh(π ) B2 = 7 , sinh(2π ) Bk = 0 for k = 3, . . . . Thus the solution to Dirichlet’s problem in this case is u(x, y ) = 3 7 sin x sinh y + sin 2x sinh 2y. sinh(π ) sinh(2π ) Exercises: 5.2.1. Which of the following functions are harmonic? a. f (x, y ) = x2 + y 2 . b. f (x, y ) = x2 − y 2 . c. f (x, y ) = ex cos y . d. f (x, y ) = x3 − 3xy 2 . 5.2.2. 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 ∂2u ∂2u + 2 = 0, ∂x2 ∂y u(x, 0) = 0, u(0, y ) = u(π, y ) = 0, u(x, π ) = sin x − 2 sin 2x + 3 sin 3x. 122 10 5 3 0 -5 -10 0 2 1 1 2 3 Figure 5.1: Graph of u(x, y ) = 3 sinh(π ) 0 sin...
View Full Document

This document was uploaded on 01/12/2014.

Ask a homework question - tutors are online