Biharmonic function

A mathematical function is called biharmonic in a domain if it has the biharmonic equation ${\displaystyle u(x,y)}$ ${\displaystyle D}$

${\displaystyle \Delta \Delta u(x,y)=0}$

fulfilled for all points ; is the Laplace operator . ${\displaystyle (x,y)\in D}$ ${\displaystyle \Delta }$

The biharmonic equation is thus a partial differential equation of the fourth order of . ${\displaystyle u(x,y)}$

In practice this equation occurs e.g. B. in continuum mechanics with plates . The deformation of a plate at a point obeys the inhomogeneous biharmonic equation as a first approximation : ${\displaystyle u(x,y)}$ ${\displaystyle (x,y)}$

${\displaystyle \Delta \Delta u(x,y)=f(x,y)}$

Here is the force (density) that is being exerted on the plate. ${\displaystyle f(x,y)}$

Harmonic functions are always biharmonic functions; however, the reverse does not have to apply.

Web links

• Eric W. Weisstein : Biharmonic Equation . In: MathWorld (English).