Boundary Value Problem Using Series of Bessel Functions
Boundary Value Problem Using Series of Bessel Functions
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 , a function of the coordinates and time, is the vertical displacement and , a constant independent of coordinates and time, which is determined by the density and tension in the membrane. The initial conditions are and (r,0)=0, .
u
a
u(r,0)=sin(4πr)
u
t
0≤r≤1
In this example we assume circular symmetry. Thus the term can be removed from the equation, yielding the traditional form of Bessel's equation:
θ
2
a
u
rr
1
r
u
r
u
tt
Using separation of variables with and the separation constant reduces the problem to two ordinary differential equations:
u(r,t)=R(r)T(t)
-
2
λ
2
r
2
λ
2
r
T''+T=0
2
λ
2
a
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=(λr)+(λr)
c
1
J
0
c
2
Y
0
T=sin(λat)+cos(λat)
k
1
k
2
The boundary conditions that determine the constants , , , and are that , meaning that the function vanishes on the perimeter . The Bessel function of the first kind, (z), can be expressed by the series
c
1
c
2
k
1
k
2
u(1,t)=0
r=1
J
0
J
0
∞
∑
k=0
k
(-1)
2k
z
2k
2
2
(k!)
Then with , , equal to the zeros of (λ), the solution satisfying the boundary conditions is given by
λ
1
λ
2
λ
3…
J
0
u(r,t)=(r)cos(at)
∞
∑
n=1
c
n
J
0
λ
n
λ
n
with
c
n
1
∫
0
J
0
λ
n
1
∫
0
2
r[(r)]
J
0
λ
n