Biharmonic equation

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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



where , which is the fourth power of the del operator and the square of the Laplacian operator (or ), is known as the biharmonic operator or the bilaplacian operator. In summation notation, it can be written in dimensions as:

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

As another example, in n-dimensional Euclidean space,


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

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

2-dimensional space[edit]

The general solution to the 2-dimensional case is

where , and are harmonic functions and is a harmonic conjugate of .

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

where and are analytic functions.

See also[edit]


  • 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[edit]