# Dealing with stiff differential equations in NDSolve

I have a problem with solving the following system of differential equations :

$$\frac {d^2 f (x)} {dx^2} + \frac {1} {x}\frac {df (x)} {dx} = \frac {f (x)} {x^2} + f^3 (x) + f (x) g^2 (x) - f (x)$$ and $$\frac {d^2 g (x)} {dx^2} + \frac {1} {x}\frac {dg (x)} {dx} = g^3 (x) + g (x) f^2 (x) - g (x)$$ where the boundary conditions/initial conditions are $$f (0) = 0,f (\infty ) = 1,g^{\prime} (0) = 0,g (\infty ) = 1$$

Since the initial conditions are difficult to implement, I replaced the initial conditions at 0 by a small number to avoid to divide by zero. For the boundaries at infinity, I would like to take a large number instead.

x2 = 3;
NDSolve[{f'[x]/x + f''[x] == f[x]/(x*x) - f[x] + f[x]*f[x]*f[x] + f[x]*g[x]*g[x],
g'[x]/x + g''[x] == -g[x] + g[x]*g[x]*g[x] + f[x]*g[x]*f[x],
f[0.001] == 0, f[x2] == 0.99, g'[0.001] == 0, g[x2] == 0.99},
{f, g}, {x, 0.001, x2}]


If I take x2 = 2, the code works. But if I take x2 = 3, I get an error message (I would like to go up to 10 at least):

NDSolve::ndsz At x == 0.6048954636068331, step size is effectively zero; singularity or stiff system suspected.

This problem can be solved by mapping interval (0,L) to unit interval (0,1), as a result we have system of equations

x1 = 10^-4; x2 = 1;
eqs = {(f'[x]/x + f''[x] - f[x]/(x*x))/L^2 == -f[x] + f[x]^3 +
f[x]*g[x]^2, (g'[x]/x + g''[x])/L^2 == -g[x] + g[x]^3 +
g[x]*f[x]^2}; bc = {f[x1] == 0, f[x2] == 0.99, g'[x1] == 0,
g[x2] == 0.99};


First we solve this system at L=5 and define initial conditions as follows

sol5 = NDSolve[{eqs /. L -> 5, bc}, {f, g}, {x, x1, x2}][[1]];

ini = {f'[x1], g[x1]} /. sol5


Using ini we can solve the system at L=10 as well

sol10 = NDSolve[{eqs /. L -> 10, bc}, {f, g}, {x, x1, x2},
Method -> {"Shooting",
"StartingInitialConditions" -> {f'[x1] == ini[[1]],
g[x1] == ini[[2]]}}][[1]]


Visualization

L0 = 10; Plot[Evaluate[{f[x/L0], g[x/L0]} /. sol10], {x, x1  L0, L0},
PlotLegends -> {"f", "g"}]
`

• Thanks a lot, I didnt expect such a detailled answer. Thats awesome :) Apr 11 at 19:33
• @ZaraReinm. Does this model have anything to do with Young Mills theory? Apr 11 at 20:40
• Actually not, it has something to do with Superfluids and Vortices. Apr 15 at 7:59
• @ZaraReinm. Ah, yes, you are right, this is superfluid model. :) Apr 15 at 13:12