# 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. 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 summation notation, it can be written in dimensions as:

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

As another example, in *n*-dimensional Euclidean space,

where

which, for *n=3 and n=5* only, becomes 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 space

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 also

## References

- 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.

## External links

- Weisstein, Eric W. "Biharmonic Equation".
*MathWorld*. - Weisstein, Eric W. "Biharmonic Operator".
*MathWorld*.