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
It is written as
∇ ∇ -->
4
φ φ -->
=
0
{\displaystyle \nabla ^{4}\varphi =0}
∇ ∇ -->
2
∇ ∇ -->
2
φ φ -->
=
0
{\displaystyle \nabla ^{2}\nabla ^{2}\varphi =0}
Δ Δ -->
2
φ φ -->
=
0
{\displaystyle \Delta ^{2}\varphi =0}
∇ ∇ -->
4
{\displaystyle \nabla ^{4}}
del operator and the square of the Laplacian operator
∇ ∇ -->
2
{\displaystyle \nabla ^{2}}
Δ Δ -->
{\displaystyle \Delta }
biharmonic operator or the bilaplacian operator . In Cartesian coordinates , it can be written in
n
{\displaystyle n}
∇ ∇ -->
4
φ φ -->
=
∑ ∑ -->
i
=
1
n
∑ ∑ -->
j
=
1
n
∂ ∂ -->
i
∂ ∂ -->
i
∂ ∂ -->
j
∂ ∂ -->
j
φ φ -->
=
(
∑ ∑ -->
i
=
1
n
∂ ∂ -->
i
∂ ∂ -->
i
)
(
∑ ∑ -->
j
=
1
n
∂ ∂ -->
j
∂ ∂ -->
j
)
φ 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 .}
Δ Δ -->
2
{\displaystyle \Delta ^{2}}
∇ ∇ -->
4
{\displaystyle \nabla ^{4}}
For example, in three dimensional Cartesian coordinates the biharmonic equation has the form
∂ ∂ -->
4
φ φ -->
∂ ∂ -->
x
4
+
∂ ∂ -->
4
φ φ -->
∂ ∂ -->
y
4
+
∂ ∂ -->
4
φ φ -->
∂ ∂ -->
z
4
+
2
∂ ∂ -->
4
φ φ -->
∂ ∂ -->
x
2
∂ ∂ -->
y
2
+
2
∂ ∂ -->
4
φ φ -->
∂ ∂ -->
y
2
∂ ∂ -->
z
2
+
2
∂ ∂ -->
4
φ 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.}
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}}}
r
=
x
1
2
+
x
2
2
+
⋯ ⋯ -->
+
x
n
2
.
{\displaystyle r={\sqrt {x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}}.}
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 φ -->
∂ ∂ -->
r
)
)
)
+
2
r
2
∂ ∂ -->
4
φ φ -->
∂ ∂ -->
θ θ -->
2
∂ ∂ -->
r
2
+
1
r
4
∂ ∂ -->
4
φ φ -->
∂ ∂ -->
θ θ -->
4
− − -->
2
r
3
∂ ∂ -->
3
φ φ -->
∂ ∂ -->
θ θ -->
2
∂ ∂ -->
r
+
4
r
4
∂ ∂ -->
2
φ φ -->
∂ ∂ -->
θ 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}
Michell solution .
2-dimensional space
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)}
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))}
f
(
z
)
{\displaystyle f(z)}
g
(
z
)
{\displaystyle g(z)}
analytic functions .
See also
References
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
