Re: solving odes

[Thomas Shores]

On 06/15/2011 06:44 AM, Piotr wrote:
> Hello. I want to solve numerically the system of ODEs:
>
> $y_0(t)=some\_given\_function,$
>
> $y_n'(t)=y_n(t)a(t)+y_{n-1}(t)b(t)+y_n(t)x(t),  y_n(0)=y_n, n>0,$
>
> where $a, b$ are given functions and $x$ is the solution of the 
> auxiliary problem:
>
> $x'(t)=f(t,x(t)), x(0)=x_0,$
>
> and $n>0$ is some natural number, which can be sometimes huge.
>
> I approximated solution of the auxiliary problem by lsode. Then I 
> interpolated it by a piecewise linear function and solve the main 
> equation for $n=1$ by lsode. For $n=2$ I tried to apply the same 
> procedure with interpolation of $x$ and $y_1,$ but it failed. Could 
> somebody suggest me a better (simpler) approach to this problem? I'm a 
> newbie in Octave, so my idea is not sophisticated. I would be grateful 
> for any help. Piotr.
>
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Looks like you're trying to do some sort of method of lines.  There are 
any number of reasons for which lsode might fail, one of the leading 
being that your system might not have a solution over the interval under 
consideration (consider the innocuous looking y'(t) = 1/(1+y^2), y(0) = 
0, over the interval [0,2]).  Without further information it's hard to 
tell what the problem is.  One puzzling thing, though, is why you go to 
the trouble to uncouple this system.  You could express it as a single 
vector IVP, of the form dY/dt = F(t,Y(t)), Y(0)=Y0, where, in your 
notation (treating x(0), y_n(0),  and y_0(t) as known),

Y(t) = [x(t), y_1(t), ... , y_N(t)]', (here ' means transpose, not 
derivative)
Y0 = [x(0), y_1(0), ... , y_N(0)]', and
F(t,Y(t)) = [f(t,x(t)), (a(t)+x(t))*y_1(t)+b(t)*y_0(t), ... ,
                               (a(t)+x(t))y_N(t)+b(t)*y_{N-1}(t)]'

Hope this helps,

Thomas Shores