Re: Rule replacement doesn't work after NDSolve?

*To*: mathgroup at smc.vnet.net*Subject*: [mg123976] Re: Rule replacement doesn't work after NDSolve?*From*: A Retey <awnl at gmx-topmail.de>*Date*: Tue, 3 Jan 2012 05:28:44 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <jdrnpb$93i$1@smc.vnet.net>

Hi, > Can anyone explain why the three rule replacement steps after NDSolve don't work, but the three Plot3D commands do? Thanks. > > n = 2; > tf = 20; > varsx = Table[Subscript[x, i][z, t], {i, n}]; > varsy = Table[y[z, t], {i, 1, 1}]; > > eqnsx = Table[{\!\( > \*SubscriptBox[\(\[PartialD]\), \(t\)]\( > \(\*SubscriptBox[\(x\), \(i\)]\)[z, t]\)\) == y[z, t] \!\( > \*UnderoverscriptBox[\(\[Sum]\), \(j = 1\), \(n\)]\(\((i - j)\)\ > \(\*SubscriptBox[\(x\), \(j\)]\)[z, t]\)\), > Subscript[x, i][z, 0] == Sin[2 i Pi z]}, {i, n}]; > > eqnsy = Table[{\!\( > \*SubscriptBox[\(\[PartialD]\), \(t\)]\(y[z, t]\)\) == - y[z, t] + > Exp[-(t - tf/4)^2], y[z, 0] == 0}, {j, 1, 1}]; > > vars = Join[varsx, varsy]; > eqns = Join[eqnsx, eqnsy]; > > sol = NDSolve[eqns, vars, {t, 0, tf}, {z, 0, 10}, DependentVariables -> vars, MaxStepSize -> 0.01, MaxSteps -> 10^5, Method -> "ExplicitRungeKutta"][[1]] > > Subscript[x, 1][0.1, 3] /. sol > Subscript[x, 2][0.1, 1] /. sol > y[0.5, 1] /. sol That's the order of evaluation, the LHS of the rules in sol just don't match the expressions you try to replace. You need e.g.: y[z, t] /. sol /. {z -> 0.5, t -> 1} > Grid[{{Plot3D[y[z, t] /. sol, {t, 0, tf}, {z, 0, 1}, PlotLabel -> "y[z,t]", AxesLabel -> {"t", "z"}, PlotRange -> All, ImageSize -> 400], > Plot3D[Subscript[x, 1][z, t] /. sol, {t, 0, tf}, {z, 0, 1}, PlotLabel -> "\!\(\*SubscriptBox[\(x\), \(1\)]\)[z,t]]", AxesLabel -> {"t", "z"}, PlotRange -> All, ImageSize -> 400]}, > {Plot3D[Subscript[x, 2][z, t] /. sol, {t, 0, tf}, {z, 0, 1}, PlotLabel -> "\!\(\*SubscriptBox[\(x\), \(2\)]\)[z,t]", AxesLabel -> {"t", "z"}, PlotRange -> All, ImageSize -> 400],}}] to avoid such problems, I usually find it much better to define the dependent variables in NDSolve without arguments, like here: sol = NDSolve[eqns, Head /@ vars, {t, 0, tf}, {z, 0, 10}, MaxStepSize -> 0.01, MaxSteps -> 10^5, Method -> "ExplicitRungeKutta"][[1]] then the rest of your original code will work just fine. Is there a special reason why you use DependentVariables here? It seems to make no sense when you do return all of them anyway... hth, albert