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:
.
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 , .
In this example we assume circular symmetry. Thus the term can be removed from the equation, yielding the traditional form of Bessel's equation:
.
Using separation of variables with and the separation constant reduces the problem to two ordinary differential equations:
,
.
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:
,
.
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
.
Then with , , equal to the zeros of , the solution satisfying the boundary conditions is given by
with
.
  • Contributed by: Stephen Wilkerson
  • (United States Military Academy West Point, Department of Mathematics)

THINGS TO TRY

SNAPSHOTS

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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.
Reference
[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.
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