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Get numerical solution of telegraph equation using NDSolve?


I am trying to get a numerical solution of the telegraph equation using NDSolve.

Given a wire with resistance, inductivity and capacity per length of : r,l,c.
We apply a voltage of ArcTan[20 t] 2/Pi to one end.
The other end ist open. I first tried to model this condition by a NeumannValue, but Mathematica complains that the system ist overdetermined, what I do not understand. Amazingly it works without the NeumanValue, sort of.

Here is the system:

r = .1; l = 1; c = 1; tmax = 2.;  
eq = {D[u[x, t], {x, 2}] == r c D[u[x, t], t] + l c D[u[x, t], {t, 2}],u[0, t] == ArcTan[20 t] 2/Pi, u[x, 0] == 0};  
res =u /. NDSolve[eq, {u}, {x, 0, 1}, {t, 0, tmax}][[1]]  
Plot3D[res[x, t], {x, 0, 1}, {t, 0, tmax}, AxesLabel -> Automatic]

Up to tmax=2 I get reasonable results. But if tmax is larger, e.g. tmax=2.1, only garbage is returned. I tried all available options of NDSolve to get a stable integration, but failed.

Can someone give a hint?

Note that tmax=2 is the time when the wave is reflected a second time at x==0. I think the problem has to do with this reflection.

POSTED BY: Daniel Huber
10 months ago

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