Re: pde's and method of lines
- To: mathgroup at smc.vnet.net
- Subject: [mg41243] Re: pde's and method of lines
- From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
- Date: Fri, 9 May 2003 03:22:35 -0400 (EDT)
- Organization: Universitaet Leipzig
- References: <b9dnhc$1fr$1@smc.vnet.net>
- Reply-to: kuska at informatik.uni-leipzig.de
- Sender: owner-wri-mathgroup at wolfram.com
Hi,
it depend on your system in your example below
> du/dt = 1-4 u_i + .02 d^2u/dx^2+ (u)^3 v
> dv/dt = 3 u_i - (u)^3 v
it is not clear what you mean, do you mean
D[u[x,t],t]== 1- 4 u[x,t]+G*D[u[x,t],{x,2}]+u[x,t]*v[x,t]
D[v[x,t],t]== 3 u[x,t] -u[x,t]^2 *v[x,t]
(than you need a dicrete v[x,t] even when v[x,t] can not diffuse
it is still local)
or do you mean
D[u[x,t],t]== 1- 4 u[x,t]+G*D[u[x,t],{x,2}]+u[x,t]*v[t]
D[v[t],t]== 3 u[x,t] -U[x,t]^2 *v[t]
and v[t] is infinite fast.
Regards
Jens
sean kim wrote:
>
> hello group,
>
> once again, I catch myself, resorting to asking question to the group.
>
> I have a question regarding pde's and method of lines. ( it appears
> there are a couple of posts regarding this, but not the question I'm
> going to ask)
>
> please consider the following which was p[osted by one of the wolfram
> researcher as an answer to a post regarding a couple diffusion
> problem.(http://forums.wolfram.com/mathgroup/archive/2002/Aug/msg00437.html)
>
> the systems of,
> du/dt = 1-4 u_i + .02 d^2u/dx^2+ (u)^3 v
> dv/dt = 3 u_i + .02 d^2v/dx^2 - (u)^3 v
>
> will be discretized as
>
> n = 10;
> X = N[Range[1, n]/(n + 1)];
> U[t_] = Map[u[#][t] &, Range[1, n]];
> V[t_] = Map[v[#][t] &, Range[1, n]];
>
> eqns = Join[
> Thread[D[U[t], t] ==
> 1 - 4 U[t] +
> 0.02 ListCorrelate[N[{1, -2, 1} n^2], U[t], {2, 2}, 1] +
> U[t]^2 V[t]],
> Thread[ D[V[t], t] ==
> 3 U[t] + 0.02 ListCorrelate[N[{1, -2, 1} n^2], V[t], {2, 2},
> 3] +
> U[t]^2 V[t]], Thread[U[0] == 1 + Sin[2 Pi X]],
> Thread[V[0] == 3 + 0. X]]; NDSolve[eqns, Join[U[t], V[t]], {t, 0,
> 10}]
>
> above is more or less what I have been trying to recreate with my own
> diffusion system except I don't have a couple diffusion as above. Since
> there are two term,s diffusing. both equations need to be discretized.
>
> but let's say we have single diffusible term in a system of ode's. Do I
> still have to discretize all the other equations?
>
> for instance, let's say we have a single diffusible term, as (instead
> of above) the following. this is an hypothetical situation modified
> from above equations.
>
> new system is
>
> du/dt = 1-4 u_i + .02 d^2u/dx^2+ (u)^3 v
> dv/dt = 3 u_i - (u)^3 v
>
> which means that with u[t][x] discretized, this is an system of ode's,
> so if you try to solve them as such doesn't seem to work too well.
>
> In[20]:=
> n = 10;
> X = N[Range[1, n]/(n + 1)];
> U[t_] = Map[u[#][t] &, Range[1, n]];
>
> eqns = Join[
> Thread[D[U[t], t] ==
> 1 - 4 U[t] + 0.02 ListCorrelate[N[{1, -2, 1} n^2], U[t], {2,
> 2}, 1] +
> U[t]^2 v[t]], D[v[t], t] == 3 u[t] + u[t]^2 v[t],
> Thread[U[0] == 1 + Sin[2 Pi X]], v[0] == 3 ]
> ;
> NDSolve[eqns,
> Join[U[t], v[t]], {t, 0, 10}]
>
> above code fails with following error messages.
>
> positions \!\(1\) and \\!\(2\) are expected to be the same."
>
> \!\(1\) and \!\(2\) \ are expected to be the same."
>
> equation and an \ initial condition."
>
> What am I doing wrong here? My first guess is that even when you only
> have a single diffusible term, all the equations needs to be
> discretized as the initial example shows.
>
> This is a bit of problem for me since my actual system have 9
> equations. Even when I use 10 bin's, that's 90 differential equations
> that needs to be solved.
>
> with additional coefficient choices, you can see the problems becomes
> intractable rapidly.( well, at least computationally intensive)
>
> I have used the above as an example. But the idea is the discretize
> second order spatial derivative using method of lines in a system of
> ode's, then solve that system as a initial value problem. I can give
> more examples if the group would like to see.
>
> Could someone please suggest some ways out of this?
>
> It appears lotta biologists(as according to previous posts regarding
> coupled diffusions and catalysis kinetic rates) are now using
> mathematica to do their modeling, maybe this will be a nice exapmple
> for future users also.
>
> as always, all helpful comments are most sincerely welcome.
>
> sean from UCIrvine
>
> =====
> when riding a dead horse, some dismount.
>
> while others...
>
> form a committee to examine the deadness of the horse, then form an oversight committee to examine the validity of the finding of the previous committee.
>
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