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

$\nabla ^{4}\varphi =0$

or

$\nabla ^{2}\nabla ^{2}\varphi =0$

or

$\Delta ^{2}\varphi =0$

where $\nabla ^{4}$ is the fourth power of the del operator and the square of the laplacian operator $\nabla ^{2}$ (or $\Delta$), and it is known as the biharmonic operator or the bilaplacian operator. In summation notation, it can be written in $n$ dimensions as:

$\nabla ^{4}\varphi =\sum _{{i=1}}^{n}\sum _{{j=1}}^{n}\partial _{i}\partial _{i}\partial _{j}\partial _{j}\varphi .$

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,

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

where

$r={\sqrt {x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}}.$

which, for n=3 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

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

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

$\operatorname {Im}({\bar {z}}f(z)+g(z))$

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.
• 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.