# Biharmonic equation

In mathematics, the **biharmonic equation** is a fourth-order partial differential equation which arises in areas of continuum mechanics, including linear elasticity theory and the solution of Stokes flows. Specifically, it is used in the modeling of thin structures that react elastically to external forces.

## NotationEdit

It is written as

or

or

where , which is the fourth power of the del operator and the square of the Laplacian operator (or ), is known as the **biharmonic operator** or the **bilaplacian operator**. In Cartesian coordinates, it can be written in dimensions as:

Because the formula here contains a summation of indices, many mathematicians prefer the notation over because the former makes clear which of the indices of the four nabla operators are contracted over.

For example, in three dimensional Cartesian coordinates the biharmonic equation has the form

As another example, in *n*-dimensional Real coordinate space without the origin ,

where

which shows, for *n=3 and n=5* only, is a solution to the biharmonic equation.

A solution to the biharmonic equation is called a **biharmonic function**. Any harmonic function is biharmonic, but the converse is not always true.

In two-dimensional polar coordinates, the biharmonic equation is

which can be solved by separation of variables. The result is the Michell solution.

## 2-dimensional spaceEdit

The general solution to the 2-dimensional case is

where , and are harmonic functions and is a harmonic conjugate of .

Just as harmonic functions in 2 variables are closely related to complex analytic functions, so are biharmonic functions in 2 variables. The general form of a biharmonic function in 2 variables can also be written as

where and are analytic functions.

## See alsoEdit

## ReferencesEdit

- Eric W Weisstein,
*CRC Concise Encyclopedia of Mathematics*, CRC Press, 2002. ISBN 1-58488-347-2. - S I Hayek,
*Advanced Mathematical Methods in Science and Engineering*, Marcel Dekker, 2000. ISBN 0-8247-0466-5. - J P Den Hartog (Jul 1, 1987).
*Advanced Strength of Materials*. Courier Dover Publications. ISBN 0-486-65407-9.