Re: NDSolve problem with switching equations

*To*: mathgroup at smc.vnet.net*Subject*: [mg106183] Re: [mg106163] NDSolve problem with switching equations*From*: Haibo Min <haibo.min at gmail.com>*Date*: Mon, 4 Jan 2010 06:00:04 -0500 (EST)*References*: <201001030842.DAA10093@smc.vnet.net>

Thank you very much, Bobby. I think it is really a brilliant solution. As for the slow plotting in the second case, I think it is due to your second explanation (Plot is working hard to identify features). Best regards, Haibo On Mon, Jan 4, 2010 at 9:08 AM, DrMajorBob <btreat1 at austin.rr.com> wrote: > First, we need a square wave of amplitude one, zero up to 0.5, 0 from .05 > to 1.0, with period 1. > > The following will do well enough: > > Plot[UnitStep@Sin[2 Pi t], {t, 0, 2}] > > So here's a solution with NDSolve: > > Clear[y1, y2] > eqns = {y1'[t] == y2[t], > y2'[t] == 1000 (1 - y1[t]^2) (y2[t] - y1[t] UnitStep@Sin[2 Pi t]), > > y1[0] == 0, y2[0] == 1}; > errors = Take[eqns, 2] /. Equal -> Subtract; > {y1, y2} = {y1, y2} /. First@ > NDSolve[eqns, {y1, y2}, {t, 0, 2}]; > Plot[{y1@t, y2@t}, {t, 0, 1}, AxesOrigin -> {0, 2}, > PlotStyle -> {Red, Blue}] > > Accuracy is good for most of the graph: > > Plot[Norm@errors, {t, .015, 2}, PlotRange -> All] > > But not all: > > Plot[Norm@errors, {t, 0, .015}, PlotRange -> All] > > (Very interesting plots!) > > Increasing WorkingPrecision works well: > > Clear[y1, y2] > eqns = {y1'[t] == y2[t], > y2'[t] == 1000 (1 - y1[t]^2) (y2[t] - y1[t] UnitStep@Sin[2 Pi t]), > > y1[0] == 0, y2[0] == 1}; > errors = Take[eqns, 2] /. Equal -> Subtract; > {y1, y2} = {y1, y2} /. First@ > NDSolve[eqns, {y1, y2}, {t, 0, 1}, WorkingPrecision -> 40]; > Plot[{y1@t, y2@t}, {t, 0, 1}, AxesOrigin -> {0, 1}, > PlotStyle -> {Red, Blue}] > > Do the error plots again, and you'll find they're very slow. > > ecause the interpolations involve a lot of points? Because Plot is working > hard to identify features? > > I'm not sure. > > Bobby > > > On Sun, 03 Jan 2010 02:42:43 -0600, Haibo Min <haibo.min at gmail.com> wrote: > > Hello, everyone. >> >> I am trying to solve a switching ND problem using NDSolve. Specifically, >> suppose the system equation is >> >> y1'[t]==y2; >> y2'[t]==1000(1-y1^2)y2-y1; >> >> y1(0)==0;y2(0)==1; >> >> Then every half a second (0.5s), the system equation transforms to >> >> y1'[t]==y2; >> y2'[t]==1000(1+y1^2)y2; >> >> and then, after 0.5s, it transforms back again. >> >> How to address this problem? >> >> Thank you! >> >> haibo >> >> > > -- > DrMajorBob at yahoo.com >

**References**:**NDSolve problem with switching equations***From:*Haibo Min <haibo.min@gmail.com>