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
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
The general solution to the 2 dimensional case is
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.
- 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.
- Weisstein, Eric W., "Biharmonic Equation", MathWorld.
- Weisstein, Eric W., "Biharmonic Operator", MathWorld.
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