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I am trying to numerically solve a non-linear ODE with variable initial conditions, specifically it is :

$$ \partial_{t}R (r,t) ^2 + 2 R(r,t) \partial_{t} ^2 R(r,t) +r^2 k(r) = 0 $$ and my initial conditions are $$ R(r,1) = r (1-\frac{r}{10})^{2/3} \ \ \ \partial_{t} R(r,1) = \frac{2}{3} \frac{r}{(1-\frac{r}{10})^{1/3}}.$$

Here $t >1 $ and $0\leq r \leq 1$ and $k(r) = \frac{(1-(r^2-1-10^{-4})^{20}}{r^2}. $

I solve this equation two ways :

1) By treating $R$ as a function of $t$ alone, and defining a solution function using NDSolve:

sol[r_]:= NDSolve[{R'[t]^2+ 2 R[t] R''[t]==-r^2 k[r], R[1]== r(1-r/10)^2/3, 
    R'[1]==(2/3) 4/(1-r/10)^(1/3)},R,{t,1,3}, WorkingPrecision-> 50,
    AccuracyGoal->40, PrecisionGoal->40, InterpolationOrder->All]

Then I can just compute $sol[r][t]$ which is really $R[r,t]$ for any $r$. When I plot the left hand side of the differential equation for any $r$, by setting the WorkingPrecision to around 20 I exactly get zero, as I should. The disadvantage here is that I cannot treat $sol[r][t]$ as a function of $r$. I cannot differentiate it with respect to $r$, or at least I don't know how.

2) I solve it using

sol= NDSolve[{D[R[r,t],t]^2 + 2 R[r,t] D[R[r,t],{t,2}]==-r^2 k[r], R[r,1]== r(1-r/10)^2/3,
    Derivative[0,1][R][r,1]==(2/3) 4/(1-r/10)^(1/3)},R,{t,1,3}, {r,9/10,1}, 
    WorkingPrecision-> 50, AccuracyGoal->40, PrecisionGoal->40, InterpolationOrder->All]

Then I am in trouble because when I evaluate the left hand side using the numerical solution I do not get zero for all values of $r$.

I think something gets screwed up as it multiplies interpolating functions together, but I have no idea how to fix it.

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    $\begingroup$ Try using ParametricNDSolve,with r as the parameter. According to the documentation, it is possible to differentiate with respect to the parameter. $\endgroup$ – bbgodfrey Jul 11 '15 at 0:51
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With

k[r] = (1 - (r^2 - 1 - 10^-4)^20)/r^2

Use

sol = ParametricNDSolveValue[{R'[t]^2 + 2 R[t] R''[t] == -r^2 k[r], 
   R[1] == r (1 - r/10)^2/3, R'[1] == (2/3) 4/(1 - r/10)^(1/3)}, 
   R, {t, 1, 3}, {r}];
dsol = Derivative[1][sol]

and then plot, for example, R[1][t], R'[1][t], and (to verify that this really is the derivative with respect to r), (R[1.1]-R[1])/.1.

Plot[{sol[1][t], dsol[1][t], (sol[1.1][t] - sol[1.0][t])/.1}, {t, 1, 3}]

enter image description here

A 3D plot shows that the function is smooth except as r approaches 0.

Plot3D[sol[r][t], {t, 1, 3}, {r, 0.006, 1}, AxesLabel -> {t, r, R}, 
 LabelStyle -> Directive[Black, Bold, 12]]

enter image description here

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