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perturbation methods example from stephen lynch's book?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg100640] perturbation methods example from stephen lynch's book?
  • From: sean_incali at yahoo.com
  • Date: Wed, 10 Jun 2009 05:32:51 -0400 (EDT)

Hello group,

This message is a bit 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 + epsilon=882*x2;

Collect[(1 + 2 epsilon w1 + epsilon=882 (w1=882 + 2 w2)) Dt[x, {t, 2}] + x
- epsilon x=883, epsilon];

(*The O (1) equation.*)
DSolve[{x0''[t] + x0[t] == 0, x0[0] == 1, x0'[0] == 0}, x0[t], t]

(*The O (epsilon) equation.*)
DSolve[{x1''[t] + x1[t] == Cos[t] =883 + 2 w1 Cos[t], x1[0] == 0, x1'[0]
== 0}, x1[t], t];
Simplify[%]



He then discusses method of multiple scales to approximate the
solutions to van der pol equation (x'' + x =  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.

Thanks in advance

Sean





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