Lecture 7.2: Eigenvalues and eigenfunctions of the Laplacian Matthew - - PowerPoint PPT Presentation

Lecture 7.2: Eigenvalues and eigenfunctions of the Laplacian

Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4340, Advanced Engineering Mathematics

• M. Macauley (Clemson)

Lecture 7.2: Laplacian eigenvalues & eigenfunctions Advanced Engineering Mathematics 1 / 5

Overview

The Laplacian is the differenital operator ∆ = ∇2:=

n

• i=1

∂2 ∂x2

i

. In the previous lecture, we found the kernel of this operator (for n = 2) under various boundary conditions. This amounted to solving a PDE called Laplace’s equation: ∆u = 0. In this lecture, we will find the eigenvalues and eigenfunctions of ∆. This amounts to solving a PDE called the Helmholtz equation: ∆u = −λu. This equation arises when solving the heat and wave equations in two dimensions.

• M. Macauley (Clemson)

Lecture 7.2: Laplacian eigenvalues & eigenfunctions Advanced Engineering Mathematics 2 / 5

Dirichlet boundary conditions

Example 1

Find the general solution to the following BVP for the Helmholtz equation uxx + uyy = −λu, u(0, y) = u(π, y) = u(x, 0) = u(x, π) = 0.

• M. Macauley (Clemson)

Lecture 7.2: Laplacian eigenvalues & eigenfunctions Advanced Engineering Mathematics 3 / 5

Neumann boundary conditions

Example 2

Find the general solution to the following BVP for the Helmholtz equation uxx + uyy = −λu, ux(0, y) = ux(π, y) = uy(x, 0) = uy(x, π) = 0.

• M. Macauley (Clemson)

Lecture 7.2: Laplacian eigenvalues & eigenfunctions Advanced Engineering Mathematics 4 / 5

Mixed boundary conditions

Example 3

Find the general solution to the following BVP for the Helmholtz equation uxx + uyy = −λu, u(0, y) = ux(π, y) = uy(x, 0) = uy(x, π) = 0.

• M. Macauley (Clemson)

Lecture 7.2: Laplacian eigenvalues & eigenfunctions Advanced Engineering Mathematics 5 / 5