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