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Re: Periodic Rebirth of Hyperbolic Functions by ODE in Mathematica
*To*: mathgroup at smc.vnet.net
*Subject*: [mg66003] Re: Periodic Rebirth of Hyperbolic Functions by ODE in Mathematica
*From*: Maxim <m.r at inbox.ru>
*Date*: Wed, 26 Apr 2006 04:38:17 -0400 (EDT)
*References*: <e2i7qg$8tg$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
On Mon, 24 Apr 2006 10:04:00 +0000 (UTC), Narasimham
<mathma18 at hotmail.com> wrote:
> Solutions of non-linear second order differential equations formed for
> r = Sech[th],Tanh[th] after two differentiations are repeated at
> about th=8.0 which value should actually be Infinity. After this value
> they do not tend to the expected asymptotic value but become periodic
> functions !.
>
> The NDSolve Runge-Kutta numerical algorithm appears to involve a small
> positive error at large argument values that allows build-up of
> function back to its full value as a reflected periodic function.This
> is seen more clearly in a polar plot of r = Sech[th] and Tanh[th] where
> periodic lobes reappear, simulating periodic functions ( 0.02 added in
> plots just to distinguish the curves).
>
> BTW, I would be much enthused if one can identify the ( Jacobi Elliptic
> like ??) functions labeled as Reborn, that result by this Mathematica
> command/program. Or please indicate whether there is a command to
> convert asymptotic functions with introduced second order errors to
> their periodic counterparts where the time period can be numerically
> defined /specified. It appears as an unacceptable error for Sech or
> Tanh because the hyperbolic monotonic nature of the non-linear second
> order differential equation gets numerically depicted or compromised
> as an elliptic periodic one.
>
> Regards, Narasimham
>
> "-- Sech --"
> Clear[r,th,R];
> eq = { r''[th] == r[th]*(1 - 2 r[th]^2), r[0]==1, r'[0]==0};
> NDSolve[ eq, r,{th,0,15}] ;
> R[th_]=r[th]/.First[%];
> Reborn=Plot[%,{th,0,15}];
> soln=Plot[Sech[th]+.02,{th,0,15}];
> Show[Reborn,soln];
> {x,y}={R[th]Cos[th],R[th]Sin[th]};
> ParametricPlot[{x,y},{th,0,15},AspectRatio->Automatic];
> "-- Tanh --"
> Clear[r,th,R];
> eq = { r''[th] == 2*(r[th]^3 - r[th]), r[0]==0, r'[0]==1};
> NDSolve[ eq, r,{th,0,38}] ;
> R[th_]=r[th]/.First[%];
> Reborn=Plot[%,{th,0,38}];
> soln=Plot[Tanh[th]+.02,{th,0,38}];
> Show[Reborn,soln];
> {x,y}={R[th]Cos[th],R[th]Sin[th]};
> ParametricPlot[{x,y},{th,0,15},AspectRatio->Automatic];
>
One way to obtain a more accurate solution is to figure out an invariant
of the system:
In[1]:= inv = Integrate[(r''[th] - r[th]*(1 - 2*r[th]^2))*r'[th], th]
Out[1]= (-(1/2))*r[th]^2 + r[th]^4/2 + (1/2)*Derivative[1][r][th]^2
Hence this quantity should remain constant. Then
sol = First@ NDSolve[
{r''[th] - r[th]*(1 - 2*r[th]^2) == 0, r[0] == 1, r'[0] == 0},
r, {th, 0, 100},
Method -> {Projection, Invariants -> inv},
MaxStepFraction -> .001]
Plot[r[th] - Sech[th] /. sol, {th, 0, 100}, PlotRange -> All]
shows that the error remains bounded (in fact, the accuracy even increases
linearly). We could also get the exact solution from the system {inv == 0,
r[0] == 1} but in general that may be problematic, as the solution to that
system is not unique, there is another solution, namely (r -> (1&)).
This method doesn't work very well for your second system of ODEs. In that
case we can use (1 - r[th]^2)/r'[th] as an invariant but I don't know how
to obtain this form without guessing.
Maxim Rytin
m.r at inbox.ru
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