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. Specifically, it is used in the modeling of thin structures that react elastically to external forces. Notation. It is written as or or where formula_4, which is the fourth power of the del operator and the square of the Laplacian operator formula_5 (or formula_6), is known as the biharmonic operator or the bilaplacian operator. In Cartesian coordinates, it can be written in formula_7 dimensions as: Because the formula here  contains a summation of indices, many mathematicians prefer the notation formula_9 over formula_4 because the former makes clear which of the indices of the four nabla operators are contracted over. For example, in three dimensional Cartesian coordinates the biharmonic equation has the form As another example, in "n"-dimensional Real coordinate space without the origin formula_12, where which shows, for "n=3 and n=5" only, formula_15 is a solution to 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 where formula_18, formula_19 and formula_20 are harmonic functions and formula_19 is a harmonic conjugate of formula_18. 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 formula_24 and formula_25 are analytic functions.