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
=
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}
, which is the fourth power of the del operator and the square of the Laplacian operator
∇
2
{\displaystyle \nabla ^{2}
(or
Δ
{\displaystyle \Delta }
), is known as the biharmonic operator or the bilaplacian operator . In Cartesian coordinates , it can be written in
n
{\displaystyle n}
dimensions as:
∇
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}
over
∇
4
{\displaystyle \nabla ^{4}
because the former makes clear which of the indices of the four nabla operators are contracted over.
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}
is a solution to the biharmonic equation.
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
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)}
and
w
(
x
,
y
)
{\displaystyle w(x,y)}
are harmonic functions and
v
(
x
,
y
)
{\displaystyle v(x,y)}
is a 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)}
and
g
(
z
)
{\displaystyle g(z)}
are 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