## Biharmonic Equation

The differential equation obtained by applying the Biharmonic Operator and setting to zero.

 (1)

In Cartesian Coordinates, the biharmonic equation is

 (2)

In Polar Coordinates (Kaplan 1984, p. 148)
 (3)

For a radial function , the biharmonic equation becomes
 (4)

Writing the inhomogeneous equation as
 (5)

we have
 (6)

 (7)

 (8)

 (9)

 (10)

Now use
 (11)

to obtain
 (12)

 (13)

 (14)

The homogeneous biharmonic equation can be separated and solved in 2-D Bipolar Coordinates.

References