Re: NDSolve, three 2-d order ODE, 6 initial conditions

*To*: mathgroup at smc.vnet.net*Subject*: [mg115197] Re: NDSolve, three 2-d order ODE, 6 initial conditions*From*: DrMajorBob <btreat1 at austin.rr.com>*Date*: Tue, 4 Jan 2011 04:24:14 -0500 (EST)

No error here, in version 8. In any version, there's REALLY no reason to type in d[x[t],t] instead of x'[t], D[x[t], {t, 2}] instead of x''[t], or D[x[t], t] /. {t -> 0} instead of x'[0]. The long version is what the short one means, so type the short one. Bobby On Mon, 03 Jan 2011 14:16:31 -0600, michael partensky <partensky at gmail.com> wrote: > * > > * > *On Mon, Jan 3, 2011 at 2:16 PM, DrMajorBob <btreat1 at austin.rr.com> > wrote: > * >> >> Perhaps a call to ndSol with some parameters (known only to you) will >> cause >> an error message... but the statement you posted, as it is, gives NO >> error >> message. >> >> *Sorry, Bob. The error message came with the evaluation (see below), but > it was not sensitive to the choice of parameters. > > * > >> That said, it seems to me that {t, t1} needs replacement with something >> like {t, t1, t2}, as NDSolve needs lower AND upper limits. >> >> *By default, {t,t1} = {t,0,t1} (I checked it in Help). However, the >> later > version includes* > *tmin, tmax, and the error message was the same. > > * > >> {x[t], y[t], z[t]} should be {x, y, z}, since NDSolve will return >> InterpolatingFunction results, not symbolic ones. >> > *Yes, I fixed it. The outcome did not change. * > >> >> Though it's not necessary, I'd also replace D[x[t], {t, 2}] with x''[t], >> D[x[t], t] with x'[t], and (D[x[t], t] /. {t -> 0}) with x'[0]. >> > *I used both. Just thought (apparently it was wrong) that D[ ] is > bullet-proof.* > *Thanks for the advise.* > >> >> The result is >> >> ndSol[w_,w0_,w1_,x0_,y0_,z0_,v0x_,v0y_,v0z_,t1_,t2_]:=NDSolve[{-w Sin[t >> w] >> (x^\[Prime])[t]+w Cos[t w] (y^\[Prime])[t]+Cos[t w] >> (x^\[Prime]\[Prime])[t]+Sin[t w] (y^\[Prime]\[Prime])[t]==(w-w0) (Sin[t >> w] >> (x^\[Prime])[t]-Cos[t w] (y^\[Prime])[t]),-Sin[t w] >> (x^\[Prime]\[Prime])[t]+Cos[t w] (y^\[Prime]\[Prime])[t]==(w-w0) (Cos[t >> w] >> (x^\[Prime])[t]+Sin[t w] (y^\[Prime])[t])+w1 (z^\[Prime])[t],0==w1 >> (Sin[t w] >> (x^\[Prime])[t]-Cos[t w1] >> (y^\[Prime])[t]),(x^\[Prime])[0]==v0x,(y^\[Prime])[0]==v0y,(z^\[Prime])[0]==v0z,x[0]==x0,y[0]==y0,z[0]==z0},{x, >> y, z},{t,t1,t2}]; >> >> That will fail if w1 == 0, since it causes one of the equations to be >> trivially True, and the equation is unnecessarily complicated when w1 >> is NOT >> zero. >> >> In case w1 == 0, z is also uncoupled from x and y, and it will be simply >> >> z[t_] = z0 + v0z t >> > *Absolutely. This is when the normal (oscillating) magnetic field is > absent, > and the projection on the stationary field (along the z axis) is > conserved. > I can not ignore w1. Effectively, this is the normal component of B1 > (w1=B1/gamma, denominator being giro-magnetic ratio) rotating with the > angular frequency w, and it introduces the resonance (unless I screwed > the > equations up- will check it later) * > >> >> Hence, we could redefine ndSol as: >> >> Clear[ndSol] >> >> ndSol[w_,w0_,0,x0_,y0_,z0_,v0x_,v0y_,v0z_,t1_,t2_]:= >> Append[NDSolve[{-w Sin[t w] x'[t]+w Cos[t w] y'[t]+Cos[t w] >> x''[t]+Sin[t w] >> y''[t]==(w-w0) (Sin[t w] x'[t]-Cos[t w] y'[t]),-Sin[t w] x''[t]+Cos[t w] >> y''[t]==(w-w0) (Cos[t w] x'[t]+Sin[t w] >> y'[t]),x'[0]==v0x,y'[0]==v0y,x[0]==x0,y[0]==y0},{x, >> y},{t,t1,t2}],z->z0+# >> v0z&] >> >> ndSol[w_,w0_,w1_,x0_,y0_,z0_,v0x_,v0y_,v0z_,t1_,t2_]:=NDSolve[{-w Sin[t >> w] >> x'[t]+w Cos[t w] y'[t]+Cos[t w] x''[t]+Sin[t w] y''[t]==(w-w0) (Sin[t w] >> x'[t]-Cos[t w] y'[t]),-Sin[t w] x''[t]+Cos[t w] y''[t]==(w-w0) (Cos[t w] >> x'[t]+Sin[t w] y'[t])+w1 (z')[t],0==Sin[t w] x'[t]-Cos[t w1] >> y'[t],x'[0]==v0x,y'[0]==v0y,(z')[0]==v0z,x[0]==x0,y[0]==y0,z[0]==z0},{x, >> y, >> z},{t,t1,t2}]; >> >> *Thanks. * > *My last version was * > * > * > *ndSol[w_, w0_, w1_, x0_, y0_, z0_, v0x_, v0y_, v0z_, t1_, t2_] :=* > * NDSolve[{Cos[w t ] D[x[t], {t, 2}] + Sin[ w t] D[y[t], {t, 2}] - w > Sin[w > t] D[x[t], t] + w Cos[w t] D[y[t], t] == (w - w0) ( Sin[w t ] D[x[t], t] > - > Cos[w t] D[y[t], t]),* > * -Sin[w t] D[x[t], {t, 2}] + Cos[w t] D[y[t], {t, 2}] == (w - w0) > (Cos[w > t] D[x[t], t] + Sin[w t] D[y[t], t]) + w1 D[z[t], t],* > * D[z[t], {t, 2}] == w1 (Sin[w t] D[x[t], t] - Cos[w1 t] D[y[t], t]), > (D[x[t], t] /. {t -> 0} ) == v0x, (D[y[t], t] /. {t -> 0} ) == v0y, > (D[z[t], > t] /. {t -> 0}) == v0z, x[0] == x0, y[0] == y0, z[0] == z0 }, {x, y, z}, > {t, t1, t2}];* > * > * > * > * > *I evaluated it in Mathematica 7.0 with the following parameters > ndSol[10,10.1,1,0,0,0,1,0,1,0,10]* > * > * > *The message was:* > NDSolve::ndnco: The number of constraints (6) (initial conditions) is not > equal to the total differential order of the system (5). >> > * > * > *I just heard from Daniel Lichtblau that the message from Mathematica > 7.1 > was different, and that Mathematica 8. did solve it without troubles. > Hence, > Daniel suggested that this was a bug.* > *I am surprised by encountering a bug so effortlessly. Previously, the > errors have always been mine. :) Is it possible that I did introduce and > error, but Mathematica 8. is simply more flexible in its defaults > (functions > are more "overloaded" bla-bla ?). * > *What do you think? > > Btw, I have just installed M. 8 but did not give it a try yet.* > * > * > *Thanks* > *Michael* > > Whether that is solvable for the parameters you'd like to pass, I can't >> venture to guess. >> >> Bobby >> >> >> On Mon, 03 Jan 2011 02:56:52 -0600, michael partensky >> <partensky at gmail.com> >> wrote: >> >> Hi, group! >>> >>> An attempt to demonstrate a (restricted) analogy between the Bloch >>> (magnetic resonance) equation and the motion equation for a charged >>> particle >>> in the magnetic field leads to the following equation: >>> >>> ndSol[w_, w0_, w1_, x0_, y0_, z0_, v0x_, v0y_, v0z_, t1_] := >>> NDSolve[{Cos[w t ] D[x[t], {t, 2}] + Sin[ w t] D[y[t], {t, 2}] - w >>> Sin[w >>> t] D[x[t], t] + w Cos[w t] D[y[t], t] == (w - w0) ( Sin[w t ] D[x[t], >>> t] - >>> Cos[w t] D[y[t], t]), >>> -Sin[w t] D[x[t], {t, 2}] + Cos[w t] D[y[t], {t, 2}] == (w - w0) >>> (Cos[w >>> t] D[x[t], t] + Sin[w t] D[y[t], t]) + w1 D[z[t], t], >>> D[z, {t, 2}] == w1 (Sin[w t] D[x[t], t] - Cos[w1 t] D[y[t], t]), >>> (D[x[t], t] /. {t -> 0} ) == v0x, (D[y[t], t] /. {t -> 0} ) == v0y, >>> (D[z[t], >>> t] /. {t -> 0}) == v0z, x[0] == x0, y[0] == y0, z[0] == z0 }, {x[t], >>> y[t], >>> z[t]}, {t, t1}]; >>> >>> Apparently there is an error - u will see the message. Could you please >>> help >>> catching it? >>> Thanks >>> Michael Partenskii >>> >>> >>> >> >> -- >> DrMajorBob at yahoo.com >> -- DrMajorBob at yahoo.com