The differential equation obtained by applying the biharmonic
operator and setting to zero:
![del ^4phi=0.](/images/equations/BiharmonicEquation/NumberedEquation1.svg) |
(1)
|
In Cartesian coordinates, the biharmonic
equation is
In polar coordinates (Kaplan 1984, p. 148)
For a radial function
, the biharmonic equation becomes
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.](/images/equations/BiharmonicEquation/NumberedEquation2.svg) |
(9)
|
The homogeneous biharmonic equation can be separated and solved in two-dimensional
bipolar coordinates.
The solution to the inhomogeneous equation
![del ^4phi=64beta](/images/equations/BiharmonicEquation/NumberedEquation3.svg) |
(10)
|
is given by
![phi(r)=betar^4+1/4r^2(2C_2-C_3)+C_4+(C_1+1/2r^2C_3)lnr.](/images/equations/BiharmonicEquation/NumberedEquation4.svg) |
(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.Referenced on Wolfram|Alpha
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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