# 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:

au+u+u=u

2

rr

1

r

r

1

r

2

θθ

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 , .

u

a

The initial conditions are

u(r,0)=sin(4πr)

u(r,0)=0

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

au+u=u

2

rr

1

r

r

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)

-λ

2

r(R''+rR'+λrR)=0

2

2

2

T''+λaT=0

2

2

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=cJ(λr)+cY(λr)

1

0

2

0

T=ksin(λat)+kcos(λat)

1

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, , can be expressed by the series

c

1

c

2

k

1

k

2

u(1,t)=0

r=1

J(z)

0

J(z)=(-1)

0

∞

∑

k=0

k

z

2k

2(k !)

2k

2

Then with , , equal to the zeros of , the solution satisfying the boundary conditions is given by

λ

1

λ

2

λ

3…

J(λ)

0

u(r,t)=cJ(rλ)cos(aλt)

∞

∑

n=1

n

0

n

n

with

c=∫rsin(4πr)J(λr)dr/∫rJ(λr)dr

n

1

0

0

n

1

0

0

n

2