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Re: proper way of posing a non-autonomous ode in mathematica?

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  • Subject: [mg100436] Re: proper way of posing a non-autonomous ode in mathematica?
  • From: Alexei Boulbitch <Alexei.Boulbitch at iee.lu>
  • Date: Wed, 3 Jun 2009 05:29:11 -0400 (EDT)

Hi, Sean,

>From your description it is not quite clear to me, what do you want to obtain. But it seams that the problem is 
in Mathematics, rather than in Mathematica.
Let us see:
1. If you want to solve your system with respect to four unknown functions: a(t), b(t), k1(t)
and k2(t), you MUST fix two other equations. No way around. Information contained in two equations is not enough, 
as you know.

2. If you want to get solution with two unknown functions a(t) and b(t), while k1(t)
and k2(t) are some functions that are known, it may be possible. 

3. For some functions k1(t)and k2(t) it may be even easy (as you mentioned it for t^-h),
 and for some functions it may happen that the system admits an analytical solution. For others it may not admit it,
 but numerical solution can be looked for.
In general however,it is whatever you want, but not trivial. This point becomes clear, if one pays attention that 
your equations look pretty like a 1D Schroedinger equation (though it is not exactly Schroedinger in at least 
two aspects and that does not make it more simple). And it is well known that Schroedinger equation is solved 
analytically only in a very limited number of cases. Therefore, it is unreasonable to expect that Mathematica 
will return exact analytical solution of your equations for general functions k1(t) and k2(t). So, it is probably 
not the output you expect. Then what?

Have a success, Alexei




Hello group,

I was trying to solve a simple ODE system that has parameters that are
functions of time. Then I thought about it and I'm not sure if I'm
doing it right.

Let's say your system is

a = -k1 (t^-h) -k2 (t^-h) b
b = k1 (t) a

Then in mathematica,

a'[t] == - k1 (t ^-h) a[t] - k2 (t ^-h) b[t],
b'[t] == k1 (t ^-h) a[t]

which is solvable.

but if I pose the k1 and k2 as a function of t, like k1[t^-h] and k2
[t^-h]

Then the system becomes under-determined with 4 unknowns and 2
eqns...

So my question is how do you pose/solve a non-autonomous ode in
mathematica?

Thanks for any info in advance.

Sean

-- 
Alexei Boulbitch, Dr., habil.
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