In two dimensions, the heat equation is

In the limit as *t* → ∞, and in the absence of sources or sinks of energy, the effects of the initial conditions are no longer felt in the solution and we think of *u*_{t} as zero to obtain *Laplace’s equation*

Laplace’s equation is also known as the *potential equation* and also as the *steady state heat equation.*

In three space variables, Laplace’s equation is

In vector calculus, the del symbol denotes the gradient operator, where

and, in three dimensions,

Now Laplace’s equation can be written

^{2} is read del squared. The del symbol is also called *nabla* after the Greek word ...

Start Free Trial

No credit card required