Frobenius method for system of equations
- From: btatting@xxxxxxxxxxx
- Date: 21 Feb 2006 13:30:37 -0800
I am trying to solve the following system for u and w in terms of s:
s^2 u'' + s u' - mu^2 u + mu^2 Cot[alpha] (nu s w' - w) = 0
eta^2 (s^2 w'' - s w') - Cot[alpha] (nu s^3 u' + u + Cot[alpha] s^2 w)
= 0
where all greek variables are known positive constants and ' denotes
differentiation. Application of the Frobenius method, i.e. assuming
u(s) = Summation( U_k s^(sigma+k), k=0,infinity)
w(s) = Summation( W_k s^(sigma+k), k=0,infinity)
leads to an indicial equation with roots 0, 2 and +/- mu. Note also
that mu may equal integral values 1,2... depending on physical
properties of the system, while nu is bounded to be between 0 and 1/2.
For non-integral values of mu, I can use the Frobenius method to find
the recurrence relations for the expansion coefficients. This involves
straightforward solutions for sigma = +/- mu and sigma = 2, with a
slighter more complicated solution for sigma = 0 since it differs by an
integer from the sigma = 2 solution (involving taking the limit as
sigma approaches zero, similar to Bessel function derivations for
integral roots to generate a second independent solution).
My question is, what happens when mu is an integer (it is often equal
to 1, but could possibily be any positive odd or even integer). I am
sure I must take similar limits with regards to sigma approaching +/-
mu, but not sure which solutions have precedence or if that even
matters. For example, if mu = 1, then I have four roots of 2, 1, 0, -1,
so do I need to start with the highest root and then generate
successsive independent solutions for each root in decreasing order?
Will this alter the (previously independent) solution for sigma=0?
Alternatively, if mu = 2 (roots 2 [twice], 0, -2) or mu > 2 (but
integral), must I begin with the highest root and now alter my previous
solutions for sigma=2 and 0?
My predilection is to use the sigma = 2 and sigma = 0 solutions as two
independent solutions, and then alter the +/-mu solutions as they
approach any integral value, but I want to make sure it will always
furnish a suitable solution. I know I can always check for independence
for each case after I generate the four solutions, but after all that
work I do not want to find out that they are NOT linearly independent
and then not know how to proceed.
Any insights or references to point to? I have not found a good
reference for SYSTEMS that require Frobenius method solutions.
Thanks
bt
.
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