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

References

External links

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