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| In [[mathematics]], the '''biharmonic equation''' is a fourth-order [[partial differential equation]] which arises in areas of [[continuum mechanics]], including [[linear elasticity]] theory and the solution of [[Stokes flow]]s. It is written as
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| :<math>\nabla^4\varphi=0</math>
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| or
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| :<math>\nabla^2\nabla^2\varphi=0</math>
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| or
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| :<math>\Delta^2\varphi=0</math>
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| where <math>\nabla^4</math> is the fourth power of the [[del]] operator and the square of the [[laplacian]] operator <math>\nabla^2</math> (or <math>\Delta</math>), and it is known as the '''biharmonic operator''' or the '''bilaplacian operator'''.
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| For example, in three dimensional [[cartesian coordinates]] the biharmonic equation has the form
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| : <math>
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| {\partial^4 \varphi\over \partial x^4 } +
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| {\partial^4 \varphi\over \partial y^4 } +
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| {\partial^4 \varphi\over \partial z^4 }+
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| 2{\partial^4 \varphi\over \partial x^2\partial y^2}+
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| 2{\partial^4 \varphi\over \partial y^2\partial z^2}+
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| 2{\partial^4 \varphi\over \partial x^2\partial z^2} = 0.
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| </math>
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| As another example, in ''n''-dimensional [[Euclidean space]],
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| :<math>\nabla^4 \left({1\over r}\right)= {3(15-8n+n^2)\over r^5}</math> | |
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| where
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| :<math>r=\sqrt{x_1^2+x_2^2+\cdots+x_n^2}.</math>
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| which, for ''n=3 and n=5'' only, becomes the biharmonic equation.
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| A solution to the biharmonic equation is called a '''biharmonic function'''. Any [[harmonic function]] is biharmonic, but the converse is not always true.
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| In two-dimensional [[polar coordinates]], the biharmonic equation is
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| :<math>
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| \frac{1}{r} \frac{\partial}{\partial r} \left(r \frac{\partial}{\partial r} \left(\frac{1}{r} \frac{\partial}{\partial r} \left(r \frac{\partial \varphi}{\partial r}\right)\right)\right)
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| + \frac{2}{r^2} \frac{\partial^4 \varphi}{\partial \theta^2 \partial r^2}
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| + \frac{1}{r^4} \frac{\partial^4 \varphi}{\partial \theta^4}
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| - \frac{2}{r^3} \frac{\partial^3 \varphi}{\partial \theta^2 \partial r}
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| + \frac{4}{r^4} \frac{\partial^2 \varphi}{\partial \theta^2} = 0
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| </math>
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| which can be solved by separation of variables. The result is the [[Michell solution]].
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| ==2 dimensional space==
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| The general solution to the 2 dimensional case is
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| :<math>
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| x v(x,y) - y u(x,y) + w(x,y)
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| </math>
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| where <math>u(x,y)</math>, <math>v(x,y)</math> and <math>w(x,y)</math> are [[harmonic functions]] and <math>v(x,y)</math> is a [[harmonic conjugate]] of <math>u(x,y)</math>.
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| Just as [[harmonic functions]] in 2 variables are closely related to complex [[analytic functions]], so are biharmonic functions in 2 variables. The general form of a biharmonic function in 2 variables can also be written as
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| :<math> | |
| \operatorname{Im}(\bar{z}f(z) + g(z))
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| </math>
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| where <math>f(z)</math> and <math>g(z)</math> are [[analytic functions]].
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| ==See also==
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| * [[Harmonic function]]
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| == References ==
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| * Eric W Weisstein, ''CRC Concise Encyclopedia of Mathematics'', CRC Press, 2002. ISBN 1-58488-347-2.
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| * S I Hayek, ''Advanced Mathematical Methods in Science and Engineering'', Marcel Dekker, 2000. ISBN 0-8247-0466-5.
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| * {{cite book | author=J P Den Hartog | title=Advanced Strength of Materials | publisher=Courier Dover Publications | year=Jul 1, 1987 | isbn= 0-486-65407-9}}
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| ==External links==
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| * {{MathWorld | urlname=BiharmonicEquation | title=Biharmonic Equation}}
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| * {{MathWorld | urlname=BiharmonicOperator | title=Biharmonic Operator}}
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| [[Category:Elliptic partial differential equations]]
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