Lecture 5.3: The transport and wave equations Matthew Macauley - - PowerPoint PPT Presentation

Lecture 5.3: The transport and wave equations

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

• M. Macauley (Clemson)

Lecture 5.3: The transport and wave equations Advanced Engineering Mathematics 1 / 8

Motivation

Some common one-dimensional PDEs

We’ve seen the heat equation: ut = c2uxx. In this lecture, we will introduce the transport equation, from which we will derive the wave equation: utt = c2uxx.

• M. Macauley (Clemson)

Lecture 5.3: The transport and wave equations Advanced Engineering Mathematics 2 / 8

Transport left

Example 1

Consider the following PDE involving a function u(x, t): ∂u ∂t − c ∂u ∂x = 0.

• M. Macauley (Clemson)

Lecture 5.3: The transport and wave equations Advanced Engineering Mathematics 3 / 8

Transport right

Example 2

Consider the following PDE involving a function u(x, t): ∂u ∂t + c ∂u ∂x = 0.

• M. Macauley (Clemson)

Lecture 5.3: The transport and wave equations Advanced Engineering Mathematics 4 / 8

The wave equation

Example 3

Consider the following PDE involving a function u(x, t): ∂ ∂t + c ∂ ∂x ∂ ∂t − c ∂ ∂x

• u = ∂2u

∂t2 − c2 ∂2u ∂x2 = 0

• M. Macauley (Clemson)

Lecture 5.3: The transport and wave equations Advanced Engineering Mathematics 5 / 8

The three most common two-variable PDEs

Summary

Let u(x, t) be a function of position x and time t. Then the heat equation is ut = c2uxx, the wave equation is utt = c2uxx.

One more

Let u(x, y) be a function of position (x, y). Then Laplace’s equation is uxx + uyy = 0.

• M. Macauley (Clemson)

Lecture 5.3: The transport and wave equations Advanced Engineering Mathematics 6 / 8

Example 3

Solve the following B/IVP for the wave equation: utt = c2uxx, u(0, t) = u(L, t) = 0, u(x, 0) = x(L − x), ut(x, 0) = 1 .

• M. Macauley (Clemson)

Lecture 5.3: The transport and wave equations Advanced Engineering Mathematics 7 / 8

Example 3 (cont.)

The general solution to the following BVP for the wave equation: utt = c2uxx, u(0, t) = u(L, t) = 0, u(x, 0) = x(L − x), ut(x, 0) = 1 . is u(x, t) =

∞

• n=1
• an cos

cnπt

L

• + bn sin

cnπt

L

• sin

nπx

L

• . Now, we’ll solve the remaining IVP.
• M. Macauley (Clemson)

Lecture 5.3: The transport and wave equations Advanced Engineering Mathematics 8 / 8