If the charge density is zero, then Laplace's equation results.

Hence both u and v are solutions of Laplace's equation.

The general theory of solutions to Laplace's equation is known as potential theory.

Example initial-boundary value problems using Laplace's equation from exampleproblems.com.

In two or three dimensions, steady state seepage is described by Laplace's equation.

In this method, a time derivative of the dependent variable is added to Laplace's equation.

Thus it cannot be used directly on purely elliptic equations, such as Laplace's equation.

Some important properties of harmonic functions can be deduced from Laplace's equation.

Laplace's equation is a partial differential equation, of the second order.

There are other weak formulations of Laplace's equation that are often useful.

Therefore, the potential of a Laplacian field satisfies Laplace's equation.

The spherical harmonics turn out to be critical to practical solutions of Laplace's equation.

In 1812 Poisson discovered that Laplace's equation is valid only outside of a solid.

For small m, G is a solution to Laplace's equation with a point source:

This equation is known after another Frenchman as Laplace's equation.

Solutions of Laplace's equation are called harmonic functions.

Laplace's equation is linear, and is one of the most elementary partial differential equations.

The combination will reduce the number of constants and the general solution to Laplace's equation may be written:

Substituting for , Laplace's equation may now be written:

Following the technique of separation of variables, a solution to Laplace's equation is written:

A typical problem for Laplace's equation is to find a solution that satisfies arbitrary values on the boundary of a domain.

In 1812, he published his extension of Laplace's equation, which allowed its use for electric charge at the surface of solids.

Many equations can be reduced to Laplace's equation or the Helmholtz equation.

It is found in the solution to Laplace's equation in cylindrical coordinates:

By separation of variables, two differential equations result by imposing Laplace's equation: