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Beginning Partial Differential Equations, 3rd Edition by Peter V. O'Neil

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Chapter 4

Dirichlet and Neumann Problems

4.1 Laplace’s Equation and Harmonic Functions

In two dimensions, the heat equation is

equation

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 ut as zero to obtain Laplace’s equation

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

equation

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

equation

and, in three dimensions,

equation

Now Laplace’s equation can be written

equation

2 is read del squared. The del symbol is also called nabla after the Greek word ...

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