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

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: 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 ∂...
View Full Document

This document was uploaded on 01/12/2014.

Ask a homework question - tutors are online