       Re: perturbation methods example from stephen lynch's book?

• To: mathgroup at smc.vnet.net
• Subject: [mg100644] Re: perturbation methods example from stephen lynch's book?
• From: sean k <sean_incali at yahoo.com>
• Date: Wed, 10 Jun 2009 05:33:36 -0400 (EDT)

```I just realized that the codes I posted isn't complete.

Not sure what Lynch is doing in his notebook, but his Collect[] line doesn't seem to work. It's probably because he's using Dt[x, {t, 2}]  on just x not x[t]

epsilon="epsilon";
SetAttributes[{w1,epsilon},Constant]

x=x0+epsilon*x1+epsilon^2*x2;

Collect[(1+2 epsilon w1+epsilon^2 (w1^2+2 w2))Dt[x,{t,2}]+x-epsilon x^3,epsilon]

DSolve[{x0''[t]+x0[t]==0,x0==1,x0'==0},x0[t],t]

DSolve[{x1''[t]+x1[t]==Cos[t]^3+2 w1 Cos[t],x1==0,x1'==0},x1[t],t]
Simplify[%]

--- On Tue, 6/9/09, sean_incali at yahoo.com <sean_incali at yahoo.com> wrote:

> From: sean_incali at yahoo.com <sean_incali at yahoo.com>
> Subject: perturbation methods example from stephen lynch's book?
> To: sean_incali at yahoo.com
> Date: Tuesday, June 9, 2009, 4:57 PM
> Hello group,
>
> This message is a it long.
>
> I was reading Stephen Lynch's "dynamical systems with
> applications
> using mathematica" and noticed he's using asymptotics and
> perturbation
> methods in chapter 4, section 4 perturbation methods.
>
> Except he's only showing the code for linstedt-poincare
> methods which
> fails for the example given in the book. (van der pol
> equation)
>
> (*See Example 8:The Lindstedt-Poincare technique.*)
> SetAttributes[{w1, epsilon}, Constant]
>
> x = x0 + epsilon*x1 + epsilonË?2*x2;
>
> Collect[(1 + 2 epsilon w1 + epsilonË?2 (w1Ë?2 + 2 w2))
> Dt[x, {t, 2}] + x
> - epsilon xË?3, epsilon];
>
> (*The O (1) equation.*)
> DSolve[{x0''[t] + x0[t] == 0, x0 == 1, x0' == 0},
> x0[t], t]
>
> (*The O (epsilon) equation.*)
> DSolve[{x1''[t] + x1[t] == Cos[t] Ë?3 + 2 w1 Cos[t], x1
> == 0, x1'
> == 0}, x1[t], t];
> Simplify[%]
>
>
>
> He then discusses method of multiple scales to approximate
> the
> solutions to van der pol equation (x'' + x =Â  epsilon
> (1 - x^2) x',
> given x=a, and x'=0)
>
> I was wondering if anyone worked out that example. (example
> 10 in
> chapter 4). If so, can someone kindly share the codes for
> it?
> ie.
> 1. solution to PDEs, O(1) and O(epsilon), that results from
> changing
> the time scales,
> 2. using TrigReduce to simplify,
> 3. remove the secular terms,
> 4. impose ICs and approximate the one term O(epsilon)
> solution, x_ms
>
> I might be asking a lot, but I was hoping someone has the
> codes for
> it.
>