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Re: Re: discretization and plotting pde system
*To*: mathgroup at smc.vnet.net
*Subject*: [mg40062] Re: [mg40042] Re: discretization and plotting pde system
*From*: Selwyn Hollis <selwynh at earthlink.net>
*Date*: Tue, 18 Mar 2003 02:21:34 -0500 (EST)
*Sender*: owner-wri-mathgroup at wolfram.com
This may be somewhat more helpful than my previous suggestions...
I'm not sure I'm using exactly the steady-state system you're trying to
solve, but this should serve as a good example. I'm going to assume the
following boundary values at x=0 and x=1: s[0] = c[0] = sc[0] = 0,
s[1]= c[1] = sc[1] = 1.
Here's a function that solves the initial-value problem, for given
values of s'[0], c'[0], sc'[0]. Note that the system of 3 second-order
ODEs has been converted to a system of 6 first-order ODEs.
solns[u_, v_, w_] := First[ {s[x], c[x], sc[x]} /. NDSolve[{
s'[x] == ds[x],
c'[x] == dc[x],
sc'[x] == dsc[x],
1*ds'[x] - c[x]*s[x] - 10*sc[x] == 0,
0.1*dc'[x] - c[x]*s[x] + 10*sc[x] == 0,
1*dsc'[x] + c[x]*s[x] - 10*sc[x] == 0,
s[0] == 0, c[0] == 0, sc[0] == 0,
ds[0] == u, dc[0] == v, dsc[0] == w},
{s[x], c[x], sc[x], ds[x], dc[x], dsc[x]}, {x, 0, 1} ] ]
Now define
endvals[u_,v_,w_] := solns[u,v,w] /. x->1
and use FindRoot as follows:
slopes= {u, v, w} /. FindRoot[ endvals[u, v, w] - {1, 1, 1}, {u,
{0, 1}}, {v, {0, 1}}, {w, {0, 1}}]
{0.0499949, 7.86619, 0.313381}
Now plot the solution:
Plot[Evaluate[solns@@slopes], {x, 0, 1}, PlotStyle -> {Hue[0],
Hue[.7], {}}]
-----
Selwyn Hollis
On Monday, March 17, 2003, at 03:33 AM, Selwyn Hollis wrote:
> If you're intent on doing this kind of thing from scratch, I'd suggest
> that you begin with a basic numerical analysis book that treats finite
> differences and the Crank-Nicolson method. It would also be a good idea
> to start with a simpler problem, something like
>
> dy/dt == d^2y/dx^2 + y(1-y) , 0 < x < 1,
> y(t,0)= 0, y(t,1) = 1,
> y(0,x) = x^2.
>
> On the other hand, you may want to look at the ReactionDiffusionLab
> package found here:
>
> http://www.math.armstrong.edu/faculty/hollis/mmade/RDL/
>
> With it, you can solve problems such as the one you have, but in two
> space dimensions instead of one.
>
>
> -----
> Selwyn Hollis
> http://www.math.armstrong.edu/faculty/hollis
>
>
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