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:

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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:

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Using separation of variables with and the separation constant reduces the problem to two ordinary differential equations:

,

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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:

,

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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

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

with

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