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Re: Periodic Rebirth of Hyperbolic Functions by ODE in Mathematica
*To*: mathgroup at smc.vnet.net
*Subject*: [mg66183] Re: Periodic Rebirth of Hyperbolic Functions by ODE in Mathematica
*From*: Maxim <m.r at inbox.ru>
*Date*: Wed, 3 May 2006 02:45:07 -0400 (EDT)
*References*: <e2i7qg$8tg$1@smc.vnet.net><e2nds9$3a7$1@smc.vnet.net> <e36v68$n5p$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
On Tue, 2 May 2006 06:45:28 +0000 (UTC), Narasimham <mathma18 at hotmail.com>
wrote:
> Thanks. So in effect there is no way except using first order ODE ? and
> BTW, why does not integ(r' r '' ) = r'^2 /2 work for tanh?
>
Yes, ultimately this method is just a reduction of order, but used in an
indirect way. Note that if we simply tried to feed the invariant to
NDSolve it wouldn't work:
In[1]:= inv = Integrate[(r''[th] - r[th]*(1 - 2*r[th]^2))*r'[th], th];
r[1] /. NDSolve[{inv == 0, r[0] == 1}, r, {th, 0, 100}]
Out[2]= {1., 1.}
because NDSolve would only find the solution to the system {r'[th] == 0,
r[0] == 1}.
To make it work for your second example, we need to increase
WorkingPrecision:
In[3]:= ode = r''[th] - 2*(r[th]^3 - r[th]);
inv = Integrate[ode*r'[th], th];
sol = First@ NDSolve[{ode == 0, r[0] == 0, r'[0] == 1},
r, {th, 0, 100},
Method -> {Projection, Invariants -> inv},
MaxStepFraction -> .001,
WorkingPrecision -> 100, PrecisionGoal -> 10];
Plot[r[th] - Tanh[th] /. sol, {th, 0, 100}, PlotRange -> All]
I don't know how to get an accurate solution from NDSolve without using
this additional information about the invariant.
Here's how to obtain the exact solution:
In[7]:= r[th] /. DSolve[{inv == 1/2, r[0] == 0, r'[0] == 1},
r, th] // FullSimplify
Out[7]= {Tanh[th]}
Another standard method is to introduce the new variable p[r] = r':
In[8]:= ode2 = ode //.
Derivative[k_][r][th] :> D[p[r[th]], {th, k - 1}] /.
r[th] :> r
Out[8]= -2*(-r + r^3) + p[r]*Derivative[1][p][r]
In[9]:= sol = First@ DSolve[{ode2 == 0, p[0] == 1}, p, r]
out[9]= {p -> Function[{r}, Sqrt[1 - 2*r^2 + r^4]]}
In[10]:= r[th] /. DSolve[
{r'[th]^2 == (p[r]^2 /. sol /. r -> r[th]), r[0] == 0, r'[0] == 1},
r, th] // FullSimplify
Out[10]= {Tanh[th]}
Maxim Rytin
m.r at inbox.ru
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