 # Boundary Value Problem Using Series of Bessel Functions

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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 with .

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Contributed by: Stephen Wilkerson (March 2011)
(United States Military Academy West Point, Department of Mathematics)
Open content licensed under CC BY-NC-SA

## Snapshots   ## Details

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

Reference

 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.

## Permanent Citation

Stephen Wilkerson

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