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Biharmonic Equation

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

del ^4phi=0.
(1)

In Cartesian coordinates, the biharmonic equation is

del ^4phi=del ^2(del ^2)phi
(2)
=((partial^2)/(partialx^2)+(partial^2)/(partialy^2)+(partial^2)/(partialz^2))((partial^2)/(partialx^2)+(partial^2)/(partialy^2)+(partial^2)/(partialz^2))phi
(3)
=(partial^4phi)/(partialx^4)+(partial^4phi)/(partialy^4)+(partial^4phi)/(partialz^4)+2(partial^4phi)/(partialx^2partialy^2)+2(partial^4phi)/(partialy^2partialz^2)+2(partial^4phi)/(partialx^2partialz^2)
(4)
=0.
(5)

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.
(6)

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

del ^4phi=1/rd/(dr){rd/(dr)[1/rd/(dr)(r(dphi)/(dr))]}
(7)
=phi_(rrrr)+2/rphi_(rrr)-1/(r^2)phi_(rr)+1/(r^3)phi_r=0.
(8)

The solution to the homogeneous equation is

phi(r)=1/4r^2(2C_2-C_3)+C_4+(C_1+1/2r^2C_3)lnr.
(9)

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

The solution to the inhomogeneous equation

del ^4phi=64beta
(10)

is given by

phi(r)=betar^4+1/4r^2(2C_2-C_3)+C_4+(C_1+1/2r^2C_3)lnr.
(11)

See also

Biharmonic Operator, Thin Plate Spline, von Kármán Equations

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References

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. https://mathworld.wolfram.com/BiharmonicEquation.html

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