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. It is written as

or

or

where ${\displaystyle \nabla ^{4}}$, which is the fourth power of the del operator and the square of the Laplacian operator ${\displaystyle \nabla ^{2}}$ (or ${\displaystyle \Delta }$), is known as the biharmonic operator or the bilaplacian operator. In summation notation, it can be written in ${\displaystyle n}$ dimensions as:

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