Re: Nonautonomous ODEs

*To*: mathgroup at smc.vnet.net*Subject*: [mg72741] Re: [mg72701] Nonautonomous ODEs*From*: "Josef Otta" <josef.otta at gmail.com>*Date*: Wed, 17 Jan 2007 06:52:46 -0500 (EST)

Hi Virgil, You can use this approach for solving your problem: a = 36; b = 12; c = 145; t0 = 0; tmax = 20; jumps = Flatten[ Table[{{1, (i < x < i + 0.3)}, {0, (i + 0.3 <= x < i + 1)}}, {i, 0, tmax - 1}], 1]; F[x_] := Piecewise[jumps] eqn = a x''[t] + b x'[t] + c x[t] == F[t]; Plot[f[t], {t, t0, tmax}] soln = NDSolve[{eqn, x[0] == 0, x'[0] == 0}, x[t], {t, t0, tmax}]; Plot[Evaluate[x[t] /. soln], {t, t0, tmax}]; I hope, that it will solve your problem. Best regards, Josef Otta http://home.zcu.cz/~jotta 2007/1/16, Virgil Stokes <vs at it.uu.se>: > > I am having a problem with the numerical solution of non-autonomous > ODEs. For example, I have the following code for a simple linear 2nd > order ODE with a forcing function, F[t_] > > Clear[x,t] > a = 36; > b = 12; > c = 145; > t0=0; tmax = 20; > eqn := a x''[t] + b x'[t] + c x[t] == F[t] > F[t_] := 100 Exp[-t/6]Cos[2 t]; > > The following code gives its analytical solution and a plot of the > solution for zero initial conditions, > > soln = DSolve[{eqn,x[0]==0,x'[0]==0},x[t],t]//Simplify > truesoln = x[t]/.soln > Plot[truesoln,{t,t0,tmax}]; > > The numerical solution and its plot can be obtained with, > > soln = NDSolve[{eqn,x[0]==0,x'[0]==0},x[t],{t,t0,tmax}]; > Plot[Evaluate[x[t]/.soln],{t,t0,tmax}]; > > My problem, is that suppose the driving function is a sequence of unit > amplitude rectangular pulses, each with width of 0.3 and having a period > 1.0, then how can F[t_] be defined so that NDSolve can be used to obtain > a numerical solution? > > --V. Stokes > >