# Biharmonic equation

Jump to navigation Jump to search

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 $\nabla ^{4}$ , which is the fourth power of the del operator and the square of the Laplacian operator $\nabla ^{2}$ (or $\Delta$ ), 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.