Biharmonic equation - Alchetron, The Free Social Encyclopedia

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

∇ 4 φ = 0

or

∇ 2 ∇ 2 φ = 0

or

Δ 2 φ = 0

where ∇ 4 , which is the fourth power of the del operator and the square of the Laplacian operator ∇ 2 (or Δ ), is known as the biharmonic operator or the bilaplacian operator. In summation notation, it can be written in n dimensions as:

∇ 4 φ = ∑ i = 1 n ∑ j = 1 n ∂ i ∂ i ∂ j ∂ j φ .

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

∂ 4 φ ∂ x 4 + ∂ 4 φ ∂ y 4 + ∂ 4 φ ∂ z 4 + 2 ∂ 4 φ ∂ x 2 ∂ y 2 + 2 ∂ 4 φ ∂ y 2 ∂ z 2 + 2 ∂ 4 φ ∂ x 2 ∂ z 2 = 0.

As another example, in n-dimensional Euclidean space,

∇ 4 ( 1 r ) = 3 ( 15 − 8 n + n 2 ) r 5

where

r = x 1 2 + x 2 2 + ⋯ + 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

1 r ∂ ∂ r ( r ∂ ∂ r ( 1 r ∂ ∂ r ( r ∂ φ ∂ r ) ) ) + 2 r 2 ∂ 4 φ ∂ θ 2 ∂ r 2 + 1 r 4 ∂ 4 φ ∂ θ 4 − 2 r 3 ∂ 3 φ ∂ θ 2 ∂ r + 4 r 4 ∂ 2 φ ∂ θ 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

Im ⁡ ( z ¯ f ( z ) + g ( z ) )

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

References

Biharmonic equation Wikipedia

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