Boundary Value Problem Using Series of Bessel Functions

parameter a

0.1

time t

0.

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

(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

.