Fourth-order PDE in continuum mechanics
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 flows . Specifically, it is used in the modeling of thin structures that react elastically to external forces.
Notation [ edit ] It is written as
∇ 4 φ = 0 {\displaystyle \nabla ^{4}\varphi =0} or
∇ 2 ∇ 2 φ = 0 {\displaystyle \nabla ^{2}\nabla ^{2}\varphi =0} or
Δ 2 φ = 0 {\displaystyle \Delta ^{2}\varphi =0} where ∇ 4 {\displaystyle \nabla ^{4}} del operator and the square of the Laplacian operator ∇ 2 {\displaystyle \nabla ^{2}} Δ {\displaystyle \Delta } biharmonic operator or the bilaplacian operator . In Cartesian coordinates , it can be written in n {\displaystyle n}
∇ 4 φ = ∑ i = 1 n ∑ j = 1 n ∂ i ∂ i ∂ j ∂ j φ = ( ∑ i = 1 n ∂ i ∂ i ) ( ∑ j = 1 n ∂ j ∂ j ) φ . {\displaystyle \nabla ^{4}\varphi =\sum _{i=1}^{n}\sum _{j=1}^{n}\partial _{i}\partial _{i}\partial _{j}\partial _{j}\varphi =\left(\sum _{i=1}^{n}\partial _{i}\partial _{i}\right)\left(\sum _{j=1}^{n}\partial _{j}\partial _{j}\right)\varphi .} Because the formula here contains a summation of indices, many mathematicians prefer the notation Δ 2 {\displaystyle \Delta ^{2}} ∇ 4 {\displaystyle \nabla ^{4}}
For example, in three dimensional Cartesian coordinates the biharmonic equation has the form
∂ 4 φ ∂ x 4 + ∂ 4 φ ∂ y 4 + ∂ 4 φ ∂ z 4 + 2 ∂ 4 φ ∂ x 2 ∂ y 2 + 2 ∂ 4 φ ∂ y 2 ∂ z 2 + 2 ∂ 4 φ ∂ x 2 ∂ z 2 = 0. {\displaystyle {\partial ^{4}\varphi \over \partial x^{4}}+{\partial ^{4}\varphi \over \partial y^{4}}+{\partial ^{4}\varphi \over \partial z^{4}}+2{\partial ^{4}\varphi \over \partial x^{2}\partial y^{2}}+2{\partial ^{4}\varphi \over \partial y^{2}\partial z^{2}}+2{\partial ^{4}\varphi \over \partial x^{2}\partial z^{2}}=0.} As another example, in n -dimensional Real coordinate space without the origin ( R n ∖ 0 ) {\displaystyle \left(\mathbb {R} ^{n}\setminus \mathbf {0} \right)}
∇ 4 ( 1 r ) = 3 ( 15 − 8 n + n 2 ) r 5 {\displaystyle \nabla ^{4}\left({1 \over r}\right)={3(15-8n+n^{2}) \over r^{5}}} where
r = x 1 2 + x 2 2 + ⋯ + x n 2 . {\displaystyle r={\sqrt {x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}}.} which shows, for n=3 and n=5 only, 1 r {\displaystyle {\frac {1}{r}}}
A solution to the biharmonic equation is called a biharmonic function . Any harmonic function is biharmonic, but the converse is not always true.
In two-dimensional polar coordinates , the biharmonic equation is
1 r ∂ ∂ r ( r ∂ ∂ r ( 1 r ∂ ∂ r ( r ∂ φ ∂ r ) ) ) + 2 r 2 ∂ 4 φ ∂ θ 2 ∂ r 2 + 1 r 4 ∂ 4 φ ∂ θ 4 − 2 r 3 ∂ 3 φ ∂ θ 2 ∂ r + 4 r 4 ∂ 2 φ ∂ θ 2 = 0 {\displaystyle {\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)+{\frac {2}{r^{2}}}{\frac {\partial ^{4}\varphi }{\partial \theta ^{2}\partial r^{2}}}+{\frac {1}{r^{4}}}{\frac {\partial ^{4}\varphi }{\partial \theta ^{4}}}-{\frac {2}{r^{3}}}{\frac {\partial ^{3}\varphi }{\partial \theta ^{2}\partial r}}+{\frac {4}{r^{4}}}{\frac {\partial ^{2}\varphi }{\partial \theta ^{2}}}=0} which can be solved by separation of variables. The result is the Michell solution .
2-dimensional space [ edit ] The general solution to the 2-dimensional case is
x v ( x , y ) − y u ( x , y ) + w ( x , y ) {\displaystyle xv(x,y)-yu(x,y)+w(x,y)} where u ( x , y ) {\displaystyle u(x,y)} v ( x , y ) {\displaystyle v(x,y)} w ( x , y ) {\displaystyle w(x,y)} harmonic functions and v ( x , y ) {\displaystyle v(x,y)} harmonic conjugate of u ( x , y ) {\displaystyle u(x,y)}
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
Im ( z ¯ f ( z ) + g ( z ) ) {\displaystyle \operatorname {Im} ({\bar {z}}f(z)+g(z))} where f ( z ) {\displaystyle f(z)} g ( z ) {\displaystyle g(z)} analytic functions .
See also [ edit ]
References [ edit ]
Eric W Weisstein, CRC Concise Encyclopedia of Mathematics , CRC Press, 2002. ISBN 1-58488-347-2 . S I Hayek, Advanced Mathematical Methods in Science and Engineering , Marcel Dekker, 2000. ISBN 0-8247-0466-5 . J P Den Hartog (Jul 1, 1987). Advanced Strength of Materials . Courier Dover Publications. ISBN 0-486-65407-9 External links [ edit ] 
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