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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

${\mathrm{\nabla }}^4\phi =0$

or

${\mathrm{\nabla }}^2{\mathrm{\nabla }}^2\phi =0$

or

${\mathrm{\Delta }}^2\phi =0$

where ${\mathrm{\nabla }}^4$ is the fourth power of the del operator and the square of the laplacian operator ${\mathrm{\nabla }}^2$ (or $\mathrm{\Delta }$), and it is known as the biharmonic operator or the bilaplacian operator.

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

$\frac{{\partial }^4\phi }{\partial x^4}+\frac{{\partial }^4\phi }{\partial y^4}+\frac{{\partial }^4\phi }{\partial z^4}+2\frac{{\partial }^4\phi }{\partial x^2\partial y^2}+2\frac{{\partial }^4\phi }{\partial y^2\partial z^2}+2\frac{{\partial }^4\phi }{\partial x^2\partial z^2}=0.$

As another example, in n-dimensional Euclidean space,

${\mathrm{\nabla }}^4\left(\frac{1}{r}\right)=\frac{3(15-8n+n^2)}{r^5}$

where

$r=\sqrt{x_1^2+x_2^2+\cdots +x_n^2}.$

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

$\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial }{\partial r}\left(\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial \phi }{\partial r}\right)\right)\right)+\frac{2}{r^2}\frac{{\partial }^4\phi }{\partial {\theta }^2\partial r^2}+\frac{1}{r^4}\frac{{\partial }^4\phi }{\partial {\theta }^4}-\frac{2}{r^3}\frac{{\partial }^3\phi }{\partial {\theta }^2\partial r}+\frac{4}{r^4}\frac{{\partial }^2\phi }{\partial {\theta }^2}=0$

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

$xv(x,y)-yu(x,y)+w(x,y)$

where $u(x,y)$, $v(x,y)$ and $w(x,y)$ are harmonic functions and $v(x,y)$ is a harmonic conjugate of $u(x,y)$.

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

$\mathrm{\Im }(\overline{z}f(z)+g(z))$

where $f(z)$ and $g(z)$ are analytic functions.

See also

• Harmonic function

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.
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