Infinite expression encountered in NDSolve

Question

Trying to solve the following ODE:

solw = NDSolve[{0.75 (w[t] w''[t] - w'[t]^2) + w[t]^3 == 1/2 (1 + Tanh[100 t]), 
    w[0] == 0, w'[0] == 0}, w, {t, 0, 2}, Method -> "MethodOfLines"];
wsol[t_] := Evaluate[w[t] /. solw]

However, NDSolve gives an error:

Power::infy: Infinite expression 1/0. encountered.

I can't figure out why this happens and how to overcome this. Any idea will be appreciated.

Reviewed some related questions like this question or this question, but this does not help.

It's due to your initial conditions. Can you start at a different point? — J. M.'s missing motivation, Apr 02 '18 at 08:07
I start from t=0.0001, it gies the same error. I also try to alter w[0] and try with w[0]=0.0001, but the solution is highly oscillating. This should not be the case. — Asatur Khurshudyan, Apr 02 '18 at 08:12
You need to change the left endpoint in {t, 0, 2} as well if you're doing the perturbation. This is a known limitation of NDSolve[]. May I suggest using $MachineEpsilon instead of 0? — J. M.'s missing motivation, Apr 02 '18 at 08:16
Tried with $MachineEpsilon, but the solution amplitude becomes 10^106. — Asatur Khurshudyan, Apr 02 '18 at 08:21
Speaking of... your code did not provide the definition for f[t]; please edit your question to include it. — J. M.'s missing motivation, Apr 02 '18 at 08:24
Thanks! Done. I suspect that there is no any singularity. Mathematica just devides both sides to w[t] to get a Rayleigh equation. Not sure. — Asatur Khurshudyan, Apr 02 '18 at 08:27
"I suspect that there is no any singularity." - it's certainly possible, it's just that NDSolve[] on its own can't cope, as I said earlier. It has similar trouble with e.g. the Bessel equation. — J. M.'s missing motivation, Apr 02 '18 at 08:29
Look at your ode at t = 0. With your ic's you get 0 == 2/3. Even with a small initial t > 0, w''[t] will have to be huge to match the ode initially, which probably is the cause of high oscillatory behavior. For this equation, NDSolve does ok with less extreme ic's. — Bill Watts, Apr 03 '18 at 00:46
@BillWatts well noted. However, when I impose w[0]=0 and w'[0]>0, I will get the same error. Anyway, thanks for your help. — Asatur Khurshudyan, Apr 03 '18 at 01:43

score 5 · Accepted Answer · edited Apr 08 '18 at 03:52

Your initial conditions do not satisfy the ODE:

eqn = 0.75 (w[t] w''[t]-w'[t]^2)+w[t]^3==1/2 (1+Tanh[100 t]);
eqn /. t->0 /. {w[0]->0, w'[0]->0}

False

The only way your initial conditions can satisfy the ODE is if $$\lim_{t\to 0} w(t) w''(t) = \frac{2}{3}$$ which means that $$\lim_{t\to 0} w''(t) \to \infty$$

This is why NDSolve is unable to compute a solution, and why you get Power::infy messages. If you choose initial conditions that can satisfy the ODE, then NDSolve has no problems:

solw = NDSolveValue[{eqn, w[0] == (1/2)^(1/3), w'[0] == 0}, w, {t, 0, 2}];

Plot[solw[t], {t, 0, 2}]

Infinite expression encountered in NDSolve

1 Answers1