Re: system of differential equations with boundary conditions in spacetime !


In article <1140112918.518239.249270@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"ventu" <ventu@xxxxxxxxxxxxxxx> writes:
Hi again!

thanks to Peter's suggestions and due to the difficulty of DAEs pointed
out by everybody I thought to handle the problem from a different point
of view, which is basically the first ideas of using directly the
solutions x(t) y(t) z(t) in function of a parameter, such that x(t(a))
y(t(a)) z((t(a)).

So the solving algorithm that peter suggest (midpoint method) can work
fine. Even if... since I am completely new to numerical analysis...

1) how can I solve the sistem of three quadratic equations:

ax + by + cz + d(x^2 + y^2 + z^2) + s1 = 0
ex + fy + gz + h(x^2 + y^2 + z^2) +s2 = 0
ix + ly + mz + n(x^2 + y^2 + z^2) + s3 = 0

(x,y,z are A^{{i}} for i=1,2,3)

in order to find the possible x,y,z that can satisfy my equation?

if it is really only in this form (I had that computed and there
appeared all the mixed products with different coefficients!)
it is too simple: at least one of d,h,n will be not zero. eliminate the sum
x^2+y^2+z^2 from this equation and insert in the others. this gives two
linear equations in the three unknows x,y,z . eliminate two unknowns
with the help of the third, reinsert in the other equation and solve this
for the third unknown (ordinary quadratic equation). select the correct root.
in general such a system can be solved either by special methods for
polynomial systems like the code in
http://www.netlib.org/toms/795

or simply by Newtons method for systems of nonlinear equations
(lots of software, for example
http://plato.la.asu.edu/topics/problems/zero.html

2) is there some better way to solve the equation in function of a
parameter apart from that of imposing directly different x(t), y(t),
z(t) ?

for example one set imposed should be

x(t) = Rcos(at) - R
y(t) = sin(at)
z(t) = 0

is there some way, do you think, to find out u(a), other than solving
the system for each wanted a?
no. your system is nonlinear and there is no chance for a "closed"
solution.
and how would yuo be able to parametrize your side constraints?


Thank you again, you all are a big help!

Davide

PS: I will probably study a more sophisticate method like runge-kutta
soon in order to see if I can make the solution as accurate as
possible. suggestions? Once I know precisely that the problem is
numerically solvable I will try to understand how to minimize the
error...

hth
peter
.