Chapter 18 · Partial Differential Equations
18.07
Laplace's Equation
Laplace's equation ∇²u = 0 describes equilibrium: steady temperatures, electric potentials, ideal fluid flow. Its solutions are as smooth and balanced as possible, and boundary values determine them.
About 15 minutes
You'll learn to
- →Recognize harmonic functions
- →Use the mean value property and maximum principle
- →Solve the Dirichlet problem on a rectangle by separation of variables
- →Know Poisson's equation