Re: better example on method of lines and system of ode's
- To: mathgroup at smc.vnet.net
- Subject: [mg41252] Re: better example on method of lines and system of ode's
- From: Selwyn Hollis <selwynh at earthlink.net>
- Date: Sat, 10 May 2003 04:01:57 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
Sean,
The reason for the error you're getting is that you have a couple of
L[i][t]'s lurking in your equations for LRout and Rout. Here's a neat
way to see it:
eqns /. t -> 0 /. Flatten[ToRules /@ initc]
Jens-Peer's question is a good one. Do you intend for the
"nondiffusing" components to be uniform through out the interval?
That's a diffusion rate of Infinity. Or do you want their diffusion
rate to be 0? If so, those equations must be discretized in space as
well.
---Selwyn
On Friday, May 9, 2003, at 03:28 AM, sean kim wrote:
> Hello group,
>
> I think I have made a big mistake in using someone else's example. I
> figured the group might be more familiar with it. But I don't fully
> understand what the other users were posting about... How could I use
> those as examples( though one seemed like typical coupled diffusion and
> the other belousov-zhbotinsky reaction)
>
> Please disregard the previous examples. Here's a system that I have
> been working on using the information I have learned from the group. As
> I have said in the other post, I have a single second order derivative
> in a system of otherwise ODE's.
>
> once the system is discretized using center difference method, it
> should be solvable as a system of ode's; but for some reason,
> mathematica isn't doing that. And I can't figure out what I'm doing
> wrong here. I know the numerical soution exists for this system( It's
> an old system that has been solved by other people)
>
> Could the group please look over my codes for this system, and make
> some suggestions as to how to solve it using mathematica? I don't have
> any problems with normal initial value problems. It's when I have
> intial boundary value problems and discretizing, I struggle a lot. I
> need to use this for my own simulated annealing algorithm which I have
> implemented, so it's pretty important that I solve this system, unless
> I have a solution to the system, i can't use it in my algorithm.
>
> (*here is the system before discretizing.*)
>
> {D[L[t], t] == koff LRout[t] - kon L[t] Rout[t] + df D[D[L[x,t],x],x],
> D[LRout[t], t] == kon L[t] Rout[t] + kout LRin[t] - koff LRout[t] - kin
> LRout[t],
> D[ LRin[t], t] == kin LRout[t] - kout LRin[t] - kdeg LRin[t],
> D [Rout[t], t] == krout Rin[t] - krin Rout[t] - kon L[t] Rout[t] +
> koff LRout[t],
> D[ Rin[t], t] == wmax + krin Rout[t] - krdeg Rin[t] - krout Rin[t]}
>
> np = {v -> 5.05* 10^-12, Ro -> 3.00* 10^-08, kon -> 1.40* 10^+06,
> kin -> 6.00* 10^-04, koff -> 1.00* 10^-05, kout -> 6.70* 10^-05,
> kdeg -> 3.30* 10^-05, krin -> 1.37* 10^-03, krout -> 5.90* 10^-04,
> krdeg -> 1.00* 10^-04, wmax -> 6.95* 10^-12, df -> 1* 10^-7}
>
>
> (*and here's my discretized system, I think the system should only have
> ode's, which then can be treated as initial value problem, but it
> appears mathematica can't solve it.*)
>
> xmin = 0; xmax = 100;
> bins = 5;
> dx = Abs[(xmax - xmin)/(bins)];
>
> eqns =
> Join[Table[D[L[i][t], t] == koff LRout[t] - kon L[i][t] Rout[t] + df
> ((L[i + 1][t] - 2L[i][t] + L[i - 1][t])/(dx^2)), {i, 1, bins}] /.
> {L[0][t] -> -(krdeg krin (v + (koff (kdeg + kout) v)/kdeg kin)/(kon
> (krdeg + krout) v - kon krout wmax)) /. np, L[bins + 1][t] -> 0},
> {D[LRout[t], t] == kon L[i][t] Rout[t] + kout LRin[t] - koff LRout[t]
> - kin LRout[t],
> D[LRin[t], t] == kin LRout[t] - kout LRin[t] - kdeg LRin[t],
> D[Rout[t], t] == krout Rin[t] - krin Rout[t] - kon L[i][t] Rout[t] +
> koff LRout[t],
> D[Rin[t], t] == wmax + krin Rout[t] - krdeg Rin[t] - krout Rin[t]}] /.
> np
>
> (*here're the initial coniditons*)
>
> initc = Join[ Table[ L[i][0] == 0, {i, 1, bins}],
> {LRout[0] == 0,
> LRin[0] == 0 ,
> Rout[0] == (krout wmax)/(krdeg krin) /. np ,
> Rin[0] == wmax/(krdeg) /. np}]
>
> variables =
> Join[Table[L[i][t], {i, 1, bins}], {LRout[t], LRin[t], Rout[t],
> Rin[t]}]
>
> (*now here's the solution. *)
>
> discretesystem = Join[eqns, initc]
>
> dsoln = NDSolve[discretesystem, variables, {t, 0, 2000}]
>
> (* plotting down here. but the solution didn't work so no need for
> this*)
>
> << Graphics`MultipleListPlot`
> solnlists = Table[Flatten[Table[Evaluate[(L[j][t] /. dsoln)], {t, 0,
> 20, 4}]], {j, 1, 5}]
>
> tsollist = Transpose[solnlists]
>
> MultipleListPlot[Transpose[solnlists], PlotRange -> All,
> PlotJoined -> True]
>
> and of course, by the way I'm doing it... it gives me the following
> error.( which I do not understand... What does this error mean? )
>
> From In[1160]:=
> NDSolve::"ndnum": "Encountered non-numerical value for a derivative at
> \
> \!\(t\) == \!\(1.5631463870599635`*^-294\)."
> Out[1175]=
> {{L[1][t] -> InterpolatingFunction[{{0., 0.}}, "<>"][t],
> L[2][t] -> InterpolatingFunction[{{0., 0.}}, "<>"][t],
> L[3][t] -> InterpolatingFunction[{{0., 0.}}, "<>"][t],
> L[4][t] -> InterpolatingFunction[{{0., 0.}}, "<>"][t],
> L[5][t] -> InterpolatingFunction[{{0., 0.}}, "<>"][t],
> LRout[t] -> InterpolatingFunction[{{0., 0.}}, "<>"][t],
> LRin[t] -> InterpolatingFunction[{{0., 0.}}, "<>"][t],
> Rout[t] -> InterpolatingFunction[{{0., 0.}}, "<>"][t],
> Rin[t] -> InterpolatingFunction[{{0., 0.}}, "<>"][t]}}
>
> I would love to hear from anyone who can get this to work. I mean, any
> help is truly appreciated. Thank you all very much in advance. ( once
> again)
>
> PS. Thank you Dr. Hollis. your package seems like a fantastic tool for
> people who work on coupled diffusion problems. unfortunately, I can't
> use it since I don't have a coupled diffusion problem... i wish there
> was another package like it for systems such as mine. Is there a way to
> modify your package or some ways to apply it to my system as shown
> above?
>
> And for Dr. Malek-Madani. Thank you for the great book by the way. I
> took it out of the my school library--albeit it's a bit over my head.
> I'm not a mathematician or an engineer. I'm a biology grad student
> working on some modeling projects(and it appears that there are a few
> of us biologists posting and askig questions in here...I know from the
> questions they ask, they seem familiar.)
>
> The boundary condition is dirichlet. I'm specifying what the value of
> the solution is. I have obtained this boundary condition after making
> some assumptions such as steady states(setting the time derivatives to
> 0) and solving the system algebraically for L[t] at the boundaries at
> steady state.
>
>
>
>
> =====
> 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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