Biharmonic equation

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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

$\nabla^4\varphi=0$

where $\nabla^4$ is the fourth power of the del operator and the square of the laplacian operator, 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

${\partial^4 \varphi\over \partial x^4 } + {\partial^4 \varphi\over \partial y^4 } + {\partial^4 \varphi\over \partial z^4 }+ 2{\partial^4 \varphi\over \partial x^2\partial y^2}+ 2{\partial^4 \varphi\over \partial y^2\partial z^2}+ 2{\partial^4 \varphi\over \partial x^2\partial z^2} = 0.$

As another example, in n-dimensional Euclidean space, the following is true:

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

where

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

which, for n=3 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 polar coordinates, the biharmonic equation will read:

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

[edit] See also

Harmonic function

[edit] 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.