Biharmonic equation

It is written as

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

or

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

or

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

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

${\displaystyle \nabla ^{4}\varphi =\sum _{i=1}^{n}\sum _{j=1}^{n}\partial _{i}\partial _{i}\partial _{j}\partial _{j}\varphi =\left(\sum _{i=1}^{n}\partial _{i}\partial _{i}\right)\left(\sum _{j=1}^{n}\partial _{j}\partial _{j}\right)\varphi .}$

Because the formula here contains a summation of indices, many mathematicians prefer the notation ${\displaystyle \Delta ^{2}}$ over ${\displaystyle \nabla ^{4}}$ because the former makes clear which of the indices of the four nabla operators are contracted over.

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

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

As another example, in n-dimensional Real coordinate space without the origin ${\displaystyle \left(\mathbb {R} ^{n}\setminus \mathbf {0} \right)}$,

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

where

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

which shows, for n=3 and n=5 only, ${\displaystyle {\frac {1}{r}}}$ 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, the biharmonic equation is

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

which can be solved by separation of variables. The result is the Michell solution.

The general solution to the 2-dimensional case is

${\displaystyle xv(x,y)-yu(x,y)+w(x,y)}$

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

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

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

where ${\displaystyle f(z)}$ and ${\displaystyle g(z)}$ are analytic functions.

• Harmonic function

• 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.

• Weisstein, Eric W. "Biharmonic Equation". MathWorld.
• Weisstein, Eric W. "Biharmonic Operator". MathWorld.