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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

4φ=0

or

22φ=0

or

Δ2φ=0

where 4 is the fourth power of the del operator and the square of the laplacian operator 2 (or Δ), and it is known as the biharmonic operator or the bilaplacian operator.

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

4φx4+4φy4+4φz4+24φx2y2+24φy2z2+24φx2z2=0.

As another example, in n-dimensional Euclidean space,

4(1r)=3(158n+n2)r5

where

r=x12+x22++xn2.

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

1rr(rr(1rr(rφr)))+2r24φθ2r2+1r44φθ42r33φθ2r+4r42φθ2=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

(z¯f(z)+g(z))

where f(z) and g(z) are analytic functions.

See also

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.
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