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

$x v(x,y) - y u(x,y) + w(x,y)$

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.