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[MUG] Coupled, nonlinear ODEs.
| [MUG] Coupled, nonlinear ODEs. |
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Author: Anders Ballestad
Posted: Tue, 14 May 2002 13:48:18 -0700
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>> From: Anders Ballestad "anders"
I am trying to solve a set of coupled nonlinear ODEs, and have
tried mostly a numerical approach so far (LSODA from LLNL); however, I'd
like the analytical solutions, if possible. A simple version of the
equations at hand are:
dx/dt = a - b*x^2 - (c*x-d)*z
dy/dt = b*x^2*(1-y) + (c*x-d)*z
where
z(t) = y*(1-y) and
x = x(t), y=y(t), and the constants (a, b, c, d) are all constant.
Another numerical snag here is the stiffness of this problem: a~1,
b~c~10^10 and d~10^5.
Does Maple solve nonlinear systems like these?
Thanks!
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| [MUG] Re: Coupled, nonlinear ODEs. |
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Author: Robert Israel
Posted: Thu, 16 May 2002 13:28:07 -0700
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>> From: Robert Israel "israel"
Some nonlinear systems can be solved, but most will have no closed form
solutions. Maple 7 has so far spent 4450 CPU seconds on yours (on a Sun)
with no result yet, so I think I'll give up on it.
This autonomous 2 x 2 system can be reduced to a single DE for, say, y as
a function of x:
2
d b x (1 - y(x)) + (c x - d) y(x) (1 - y(x))
ode := -- y(x) = -------------------------------------------
dx 2
a - b x - (c x - d) y(x) (1 - y(x))
But it seems that either generating or analyzing the symmetries of this DE
may be a difficult task (or else I've encountered a bug in Maple).
Robert Israel "israel"
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
On Tue, 14 May 2002, Anders Ballestad wrote:
> I am trying to solve a set of coupled nonlinear ODEs, and have
> tried mostly a numerical approach so far (LSODA from LLNL); however, I'd
> like the analytical solutions, if possible. A simple version of the
> equations at hand are:
>
> dx/dt = a - b*x^2 - (c*x-d)*z
> dy/dt = b*x^2*(1-y) + (c*x-d)*z
>
> where
>
> z(t) = y*(1-y) and
> x = x(t), y=y(t), and the constants (a, b, c, d) are all constant.
>
> Another numerical snag here is the stiffness of this problem: a~1,
> b~c~10^10 and d~10^5.
>
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| [MUG] Re: Coupled, nonlinear ODEs. |
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Author: Edgardo S Cheb-Terrab
Posted: Tue, 21 May 2002 21:32:54 -0700
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>> From: "Edgardo S. Cheb-Terrab" "ecterrab"
> Some nonlinear systems can be solved, but most will have no closed form
> solutions. Maple 7 has so far spent 4450 CPU seconds on yours (on a Sun)
> with no result yet, so I think I'll give up on it.
The system,
2
x' = a - b x - (c x - d) y (1 - y)
2
y' = b x (1 - y) + (c x - d) y (1 - y)
if tackled "as is", it cannot be solved using current Maple - either 7 or 8.
But installing the latest ODEtools - which in any case I recommend since it
contains fixes to all the known Maple exact solution DE bugs - dsolve does not
hang and returns a reduction of the system to a single first order ODE in
around 1 minute (fast PC). The latest ODEtools is available at
http://lie.uwaterloo.ca/odetools.htm
Regarding possible approaches for tackling this system, I put a Maple
worksheet with some input/output at
http://lie.uwaterloo.ca/odetools/cm_system.mws. This worksheet shows the
results of some experimentation, by taking one or two of the parameters a,b
equal to zero. The purpose was to find a particular case which could
eventually be taken as "generating problem" - i.e. such that the original
problem could be obtained from this particular case by performing some change
of variables. Also it is sometimes of use.. (?) to know that the problem is
solvable for certain extreme values of the parameters.
I tried, in order if simplicity: a=b=0, then b=0, then a=0. At the end it is
shown the result of dsolve directly without taking any parameter equal to
zero.
In the first case, a=b=0, there is a closed form solution. For the other three
cases there is a (expected) reduction of the problem to a single first order
ODE in the original variables (could be either y(t) or x(t)). If this reduced
ODE could be solved by any mean, then by using DEtools[buildsol] it is
possible to use the returned reduction of order to compute a closed form
solution.
In the first three of the four cases considered in the worksheet there are
also singular solutions which - although too simple - are explicit closed form
solutions. More comments are in the worksheet.
Edgardo
___________________________________________________________________________
Edgardo S. Cheb-Terrab http://lie.uwaterloo.ca/ecterrab
Centre for Experimental and Constructive Mathematics SFU, Canada
Theoretical Physics Department UERJ,Brazil
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