Boundary Value Problem Using Series of Bessel Functions

​
parameter a
0.1
time t
0.
0.2
0.4
0.6
0.8
1.0
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
This Demonstration solves a Bessel equation problem of the first kind. The equation is for a thin elastic circular membrane and is governed by the partial differential equation in polar coordinates:
2
a
u
rr
+
1
r
u
r
+
1
2
r
u
θθ
=
u
tt
.
Here
u
, a function of the coordinates and time, is the vertical displacement and
a
, a constant independent of coordinates and time, which is determined by the density and tension in the membrane. ​
​The initial conditions are
u(r,0)=sin(4πr)
and
u
t
(r,0)=0
,
0≤r≤1
.
In this example we assume circular symmetry. Thus the
θ
term can be removed from the equation, yielding the traditional form of Bessel's equation:
2
a
u
rr
+
1
r
u
r
=
u
tt
.
Using separation of variables with
u(r,t)=R(r)T(t)
and the separation constant
-
2
λ
reduces the problem to two ordinary differential equations:
2
r
(R''+rR'+
2
λ
2
r
R)=0
,
T''+
2
λ
2
a
T=0
.
The solution of these ODE equations is done using the techniques outlined in[1] for series solutions of ordinary differential equations. The general solution has the form:
R=
c
1
J
0
(λr)+
c
2
Y
0
(λr)
,
T=
k
1
sin(λat)+
k
2
cos(λat)
.
The boundary conditions that determine the constants
c
1
,
c
2
,
k
1
, and
k
2
are that
u(1,t)=0
, meaning that the function vanishes on the perimeter
r=1
. The Bessel function of the first kind,
J
0
(z)
, can be expressed by the series
J
0
(z)=
∞
∑
k=0
k
(-1)
2k
z
2k
2
2
(k!)
.
Then with
λ
1
,
λ
2
,
λ
3…
equal to the zeros of
J
0
(λ)
, the solution satisfying the boundary conditions is given by
u(r,t)=
∞
∑
n=1
c
n
J
0
(r
λ
n
)cos(a
λ
n
t)
with
c
n
=
1
∫
0
rsin(4πr)
J
0
(
λ
n
r)dr
1
∫
0
2
r[
J
0
(
λ
n
r)]
dr
.

Details

This example comes from[1], and the discussions given in Chapter 8.7 on series solutions and Bessel's equation. Also see Chapter 10.5.

References

[1] J. R. Brannan and W. E. Boyce, Differential Equations with Boundary Value Problems: An Introduction to Modern Methods and Applications, New York: John Wiley and Sons, 2010.
​

External Links

Differential Equation (Wolfram MathWorld)
Bessel Differential Equation (Wolfram MathWorld)
Bessel Function of the First Kind (Wolfram MathWorld)

Permanent Citation

Stephen Wilkerson
​
​"Boundary Value Problem Using Series of Bessel Functions"​
​http://demonstrations.wolfram.com/BoundaryValueProblemUsingSeriesOfBesselFunctions/​
​Wolfram Demonstrations Project​
​Published: December 22, 2010