Page 661 - Advanced_Engineering_Mathematics o'neil

LAPLACE’S EQUATION DIRICHLET
PROBLEM FOR A RECTANGLE
CHAPTER 18 DIRICHLET PROBLEM FOR A DISK
POISSON’S INTEGRAL FORMULA
The Potential

Equation

18.1 Laplace’s Equation

The partial differential equation
2
2
∂ u ∂ u
+ = 0
∂x 2 ∂y 2
is called Laplace’s equation in two dimensions. In three dimensions Laplace’s equation is
2
2
2
∂ u ∂ u ∂ u
+ + = 0.
∂x 2 ∂y 2 ∂z 2
2
2
These equations are often written ∇ u = 0, in which the symbol ∇ is read “del” and ∇ is read
“del squared”. We saw the del operator previously with the gradient vector field.
Laplace’s equation arises in several contexts. It is the steady-state heat equation, occurring
when ∂u/∂t = 0. It is also called the potential equation. If a vector field F has a potential ϕ, then
ϕ must satisfy Laplace’s equation.

A function satisfying Laplace’s equation in a region of the plane (or 3-space) is said to be
harmonic on that region. For example, x − y and xy are both harmonic over the entire
2
2
plane.
A Dirichlet problem for a region D consists of finding a function that is harmonic on
D and assumes specified values on the boundary of D. We will be primarily concerned
with Dirichlet problems in the plane, in which D is a region that is bounded by one or
more piecewise smooth curves. Denote the boundary by ∂ D. The Dirichlet problem for D
is to solve

641

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 14, 2010 15:27 THM/NEIL Page-641 27410_18_ch18_p641-666