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Lecture 7.1: Harmonic functions and Laplace’s equation

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

• M. Macauley (Clemson)

Lecture 7.1: Harmonic functions & Laplace’s equation Advanced Engineering Mathematics 1 / 8

Higher dimensional PDEs

Recall the del operator ∇ from vector calculus: ∇ = ∂ ∂x1 , . . . , ∂ ∂xn

• ,

∆ := ∇ · ∇ = ∂2 ∂x2

1

+ · · · + ∂2 ∂x2

n

.

In Rn

Heat equation: ut = c2∆u Wave equation: utt = c2∆u

Remark

Steady state solutions:

• ccur for the heat equation (heat dissipates)

do not occur for the wave equation (waves propagate)

Definition

A steady-state solution means ut = 0. Thus, all steady-state solutions satisfy ut = c2∆u = 0, i.e., ∆u = 0 = ⇒ ∂2u ∂x2

1

+ ∂2u ∂x2

2

+ · · · + ∂2u ∂x2

n

= 0. A function u is harmonic if ∆u = 0.

• M. Macauley (Clemson)

Lecture 7.1: Harmonic functions & Laplace’s equation Advanced Engineering Mathematics 2 / 8

Properties of harmonic functions

Key properties

The graphs of harmonic functions (∆f = 0) are as flat as possible. If f is harmonic, then for any closed bounded region R, the function f achieves its minimum and maximum values on the boundary, ∂R.

• M. Macauley (Clemson)

Lecture 7.1: Harmonic functions & Laplace’s equation Advanced Engineering Mathematics 3 / 8

Examples of harmonic functions

• M. Macauley (Clemson)

Lecture 7.1: Harmonic functions & Laplace’s equation Advanced Engineering Mathematics 4 / 8

Solving Laplace’s equation on a bounded domain

Example 1a

Solve the following BVP for Laplace’s equation: uxx + uyy = 0, u(0, y) = u(x, 0) = u(π, y) = 0, u(x, π) = x(π − x) .

• M. Macauley (Clemson)

Lecture 7.1: Harmonic functions & Laplace’s equation Advanced Engineering Mathematics 5 / 8

Solving Laplace’s equation on a bounded domain

Example 1b

Solve the following BVP for Laplace’s equation: uxx + uyy = 0, u(0, y) = u(x, 0) = u(x, π) = 0, u(π, y) = y(π − y) .

• M. Macauley (Clemson)

Lecture 7.1: Harmonic functions & Laplace’s equation Advanced Engineering Mathematics 6 / 8

Solving Laplace’s equation on a bounded domain

Example 1c

Solve the following BVP for Laplace’s equation: uxx + uyy = 0, u(0, y) = u(x, 0) = 0, u(x, π) = x(π − x), u(π, y) = y(π − y) .

• M. Macauley (Clemson)

Lecture 7.1: Harmonic functions & Laplace’s equation Advanced Engineering Mathematics 7 / 8

Unbounded domains and Fourier transforms

Example 2

Solve the following BVP for Laplace’s equation, where x ∈ R and y > 0, and the solution u is bounded as y → ∞: uxx + uyy = 0, u(x, 0) = f (x).

• M. Macauley (Clemson)

Lecture 7.1: Harmonic functions & Laplace’s equation Advanced Engineering Mathematics 8 / 8