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.

Notation

, which is the fourth power of the del operator and the square of the Laplacian), is known as the biharmonic operator or the bilaplacian operator. In Cartesian coordinates because the former makes clear which of the indices of the four nabla operators are contracted over.

For example, in three dimensional Cartesian coordinates As another example, in n-dimensional real coordinate space which shows, for n=3 and n 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 which can be solved by separation of variables. The result is the Michell solution.

2-dimensional space

are harmonic functions is a harmonic conjugate.

Just as harmonic functions in 2 variables are closely related to complex analytic functions are analytic functions.

See also

• Harmonic function

References

• Eric W Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 2002. .

• S I Hayek, Advanced Mathematical Methods in Science and Engineering, Marcel Dekker, 2000. .

External links