Re: NDSolve and Piecewise

*To*: mathgroup at smc.vnet.net*Subject*: [mg92106] Re: NDSolve and Piecewise*From*: "Kevin J. McCann" <Kevin.McCann at umbc.edu>*Date*: Sat, 20 Sep 2008 04:58:49 -0400 (EDT)*Organization*: University System of Maryland*References*: <gavt1g$g61$1@smc.vnet.net>

Marshall, I am sure others will have a better way, but I use (1+Tanh[alpha(t-t0)])/2 to generate a "step" in which the sharpness of the step is determined by alpha. The advantage is that it is differentiable. Here is some code to redefine you u[t,0] condition: Clear[g, step] step[\[Alpha]_, t_,t0_] := (1 + Tanh[\[Alpha] (t - t0)])/2 g[t_] := t/10 (1 - step[7, t,5]) + (10 - t)/10 step[7, t,5] When I used u[t,0]==g[t] in your code, it worked. (I plotted u and it looks "reasonable".) I set alpha =7 for no very good reason. You could do otherwise. Kevin M.G. Bartlett wrote: > Folks, > > I am having some trouble with getting Piecewise and NDSolve to play > nicely together. My problem is to find a solution to the heat flow > equation with an arbitrary time-varying upper boundary condition and a > Neumann-type lower boundary (steady head flow condition). My code > looks like this: > > NDSolve[{D[u[t, x], t] == D[u[t, x], x, x], > u[t, 0] == Piecewise[{{t/10, 0 <= t < 5}, {(10 - t)/10, 5 <= t < > 10}}], > u[0, x] == x/5, (D[u[t, x], x] /. x -> 5) == 1/5}, u, {t, 0, 10}, > {x, 0, 5}] > > This returns and NDSolve::ndum error on my system, which past > experience tells me is usually me leaving some symbolic value hanging > around somewhere it ought not to be. I can't see any such problem > this time. I'm pretty sure that I have written similar code in the > past (using Piecewise and NDSolve, though it was some time ago and I > can't locate the file now) and had it work, and it works if you > replace Piecewise with another functional form (like Sin[t]). Am I > missing something? > > Thanks, > > Marshall >