Second Order Non Linear Differential Equation

Question

I'm trying to solve the following differential equation numerically:

s = NDSolve[{
      (z + 2 r[z] r'[z]) (1 + r'[z]^2) -z r[z] r''[z] == 0, 
      r[0.01] == 0.0001, 
      r'[0.01] == 10
     }, 
     r, {z, 0.01, 10}
    ]

But, when I set the initial condition to $z=0$, there are singularities, so I tried to set it as 0.01. Also, for $r(z=0.01)=0$ there is a singularity, so I set $r(z=0.01)=0.0001$.

However, when I try to solve the equation for any value that I put in the boundary conditions, I'm getting "step size is effectively zero; singularity or stiff system suspected." Is there any method to sort this out?

Answer 1

This problem seems to be really stiff as whuber has pointed out. Using the BDF method and allowing NDSolve to ruminate about the order, didn't quite change the solution (z value where stiffness was encountered).

NDSolve[{(z + 2 r[z] r'[z]) (1 + r'[z]^2) - z r[z] r''[z] == 0, 
  r[0.01] == 0.0001, r'[0.01] == 10}, r, {z, 0.01, 10}, 
 MaxStepFraction -> 1/101, 
 Method -> {"BDF", "MaxDifferenceOrder" -> 5}]

Try these commands out as pointed out by several authors here and here:

These answers helped me out with issues I had with stiff equations.

You can look around and find heaps more information.

?NDSolve`BDF
?NDSolve`LSODA