Next: 7.38 U, §4 Up: 7.38 Unsteady Previous: 7.38 U, §2

7.38 U, §3 Eigenfunctions

Try separation of variables:

The first form won't separate, so use the second.

Substitute into the homogeneous P.D.E. :

Sturm-Liouville problem for which was already solved in 7.38. For any function

For brevity we will write this as

with

, and

. Also

,and any Fourier coefficient Fⁱ_n can be found as

Written in terms of these eigenfunctions, the equation for u will become a separate equation for each Fourier coefficient C_n(r,t) of u as before. This will still be a P.D.E. since it still involves derivatives with respect to both r and t. Now use separation of variables on each of these individual P.D.E.s. In other words, put into the homogeneous P.D.E. :

where

drops out. Separate:

Sturm-Liouville problem for Rⁱ_n:

For the case , this is the same equation as solved in 7.38: there the solutions that are regular at r=0 were found to be rⁿ. The only one that satisfies the boundary condition at r=a is the case n=0, giving an eigenfunction 1.

For the case , define

Solutions are the Bessel functions of the first kind J_n and of the second kind Y_n:

Regularity at r=0 requires B_n=0. Boundary condition

requires:

There will be an infinite set of

values and corresponding eigenvalues and eigenfunctions:

There are no nontrivial solutions for negative , since . To include the special case in the general formula, we can simply add to the list.

Conclusion: we can write every Fourier coefficient as a sum of Bessel functions. For any arbitrary function , with

we can now expand further as

For the formula for the coefficients Fⁱ_nm, we use orthogonality. Multiplying the O.D.E. by ,

which is standard S.L. form if

. (This equation is not quite acceptable at r=0, but you can imagine a tiny solid cylinder right in the center and then let the radius of this solid cylinder go to zero. Or see Mathematical Handbook 24.109/27.109 ff.)

According to the S.L. theorem: