Biharmonic Equation

The differential equation obtained by applying the biharmonic operator and setting to zero:

 del ^4phi=0.

In Cartesian coordinates, the biharmonic equation is

del ^4phi=del ^2(del ^2)phi

In polar coordinates (Kaplan 1984, p. 148)

del ^4phi=phi_(rrrr)+2/(r^2)phi_(rrthetatheta)+1/(r^4)phi_(thetathetathetatheta)+2/rphi_(rrr)-2/(r^3)phi_(rthetatheta)-1/(r^2)phi_(rr)+4/(r^4)phi_(thetatheta)+1/(r^3)phi_r=0.

For a radial function phi(r), the biharmonic equation becomes

del ^4phi=1/rd/(dr){rd/(dr)[1/rd/(dr)(r(dphi)/(dr))]}

The solution to the homogeneous equation is


The homogeneous biharmonic equation can be separated and solved in two-dimensional bipolar coordinates.

The solution to the inhomogeneous equation

 del ^4phi=64beta

is given by


See also

Biharmonic Operator, Thin Plate Spline, von Kármán Equations

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Kantorovich, L. V. and Krylov, V. I. Approximate Methods of Higher Analysis. New York: Interscience, 1958.Kaplan, W. Advanced Calculus, 3rd ed. Reading, MA: Addison-Wesley, 1984.Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 417, 1995.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 129, 1997.

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Biharmonic Equation

Cite this as:

Weisstein, Eric W. "Biharmonic Equation." From MathWorld--A Wolfram Web Resource.

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