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)*Reply-to*: sean_incali at yahoo.com

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[0]==1,x0'[0]==0},x0[t],t] DSolve[{x1''[t]+x1[t]==Cos[t]^3+2 w1 Cos[t],x1[0]==0,x1'[0]==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[0] == 1, x0'[0] == 0}, > x0[t], t] > > (*The O (epsilon) equation.*) > DSolve[{x1''[t] + x1[t] == Cos[t] Ë?3 + 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 =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. > > Thanks in advance > > Sean > > > >