I know this isn't the math-help mailing list, but I'm using maxima for
a problem that's just a little too complicated to complete
symbolically. So I need some help figuring out what to do next.
I'm trying to solve two coupled nonlinear ODEs with boundary
conditions to find two functions h(x) and vx(x). Maxima has been very
helpful so far in formulating the problem. However now I'm having
difficulty solving it and I'm hoping someone has suggestions both for
strategies, and for tools in maxima that can help :-)
----------- detailed description of problem below -----
I know a whole bunch of input parameters, one of them is Q, I know
h(0)*vx(0) = Q, I know that a certain integral involving h(x) from 0
to the first positive root of h(x) = Q also (maxima calculates this
integral easily if I choose an upper bound... see below about the
upper bound)
Ultimately what I am looking for is to find the first x value where
h(x) = 0 under differing conditions. I know that my differential
equation may not work well near where h(x) = 0 (it's got an h(x) in
the denominator but it's removable if vx(x)/h(x) has a finite limit
when h(x) -> 0 which it should)
I tried using two polynomials in x for the two unknown functions h(x)
and vx(x). In order to make things go to zero nicely I multiplied a
general 2nd order polynomial by a factor of (1-x). The idea is if I
solve the problem for the interval [0,1] then I can rescale x and look
for the scale factor that has some minimum potential energy to get the
real value (hopefully, I'm a bit concerned about x essentially being
measured in incompatible units. My parameters are all in SI units).
Now I have 2 equations and 6 unknowns. So following the lead of the
estimable John Boyd who kindly put his book online:
http://www-personal.engin.umich.edu/~jpboyd/BOOK_Spectral2000.html
I tried calculating (x^i,Residfma) and (x^i,Residcont) where the
residual functions are formed by substituting my polynomials into the
ODEs and the inner products are the standard (a(x),b(x)) =
integrate(a(x)*b(x),x,0,len)
However, the integrals require me to answer a long series of unknown
questions about the signs of various extremely complicated
things. Eventually I killed maxima.
Because x^i over-weights the residual error towards the end of the
interval, probably a better choice is some other weight functions
anyway like 1-x^i or the chebyshev polynomials.
Any suggestions as to what method I might try next, and what tools
maxima has to help?
(did anyone get down here??) :-)
--
Daniel Lakeland
dlakelan at street-artists.org
http://www.street-artists.org/~dlakelan