How to solve this nonlinear differential equation numerically?

[wdlang]

-f''-(2/x)f'+(2/x^2)f+f^3-f=0

the boundary condition is f(x=0)=0 and f(x=\infty)=1

how to solve f numerically?

 

[Eynstone]

This is similar to the Riccati equation. I hope one of the methods for tackling the Riccati equation is of some use.

 

[phyzguy]

Since you have two boundary conditions at two different points, this is usually solved by the shooting method, where you assume a value of f'(0), and then integrate out to see if it satisfies the boundary condition at infinity, then adjust your assumption to match the desired boundary condition. However, I tried this briefly using Mathematica's NDSolve function:

NDSolve[{-D[D[f[x], x], x] - 2/x D[f[x], x] + 2/x^2 f[x] + f[x]^3 -
f[x] == 0, f[.00001] == 0,
f'[.00001] == 1.0}, f[x], {x, .00001, 20}]

and I got oscillatory solutions that tended to zero as x->inf (see attachment). Perhaps there is a particluar value of f' at zero that will give the solution you want?

 

[wdlang]

Since you have two boundary conditions at two different points, this is usually solved by the shooting method, where you assume a value of f'(0), and then integrate out to see if it satisfies the boundary condition at infinity, then adjust your assumption to match the desired boundary condition. However, I tried this briefly using Mathematica's NDSolve function:

NDSolve[{-D[D[f[x], x], x] - 2/x D[f[x], x] + 2/x^2 f[x] + f[x]^3 -
f[x] == 0, f[.00001] == 0,
f'[.00001] == 1.0}, f[x], {x, .00001, 20}]

and I got oscillatory solutions that tended to zero as x->inf (see attachment). Perhaps there is a particluar value of f' at zero that will give the solution you want?

thanks a lot

but i am always concerned with the singularity at the origin

the coefficients diverge there

so i do not know how to deal with it

 

[wdlang]

the derivative f' at the origin has to be scanned

but i am concerned with how to integrate the eq accurately for a given f'

 

[phyzguy]

thanks a lot

but i am always concerned with the singularity at the origin

the coefficients diverge there

so i do not know how to deal with it

There is no singularity at the origin. Assume as x->0 that f looks like f = K x. Then f->0, f^3 ->0, f''->0, and f'->K, and the equation reduces to: -2K/x+2K/x=0. So everything is well behaved. In essence, the two "singular" terms cancel. You can then start your integration at some small distance away from the origin (I chose x=1E-5, but it won't really matter as long as you start where the other terms are negligible).

 

[gato_]

You can simplify things making $$y(x)=u(x)/x$$. Then you have:
$$u''+(1-2/x^{2})u-u^{3}/x^{2}$$
this equation has two independent solutions as $$x\rightarrow \infty$$, namely:
$$u\rightarrow C_{1}Sin(x)+C_{2}Cos(x)$$
You can integrate the equation backwards. I am not very sure about there being no singularity. If you keep all the terms, except for $$u^{3}/x^{2}$$, you still obtain a solution:
$$u(x)\equiv C_{1}(Sin(x)-Cos(x)/x)+C_{2}(Cos(x)-Sin(x)/x)$$
which should be valid before the neglected term is noticeable. I see no way for any solution being bounded at x=0, unless the term with $$u^3$$ is dominant somehow

 

[wdlang]

You can simplify things making $$y(x)=u(x)/x$$. Then you have:
$$u''+(1-2/x^{2})u-u^{3}/x^{2}$$
this equation has two independent solutions as $$x\rightarrow \infty$$, namely:
$$u\rightarrow C_{1}Sin(x)+C_{2}Cos(x)$$
You can integrate the equation backwards. I am not very sure about there being no singularity. If you keep all the terms, except for $$u^{3}/x^{2}$$, you still obtain a solution:
$$u(x)\equiv C_{1}(Sin(x)-Cos(x)/x)+C_{2}(Cos(x)-Sin(x)/x)$$
which should be valid before the neglected term is noticeable. I see no way for any solution being bounded at x=0, unless the term with $$u^3$$ is dominant somehow

i guess the substitution y=xu may be better?

 

[gato_]

The change I suggest is standard. It removes the first order differentiation, allowing to study the decay and the oscillation separately

 

[wdlang]

The change I suggest is standard. It removes the first order differentiation, allowing to study the decay and the oscillation separately

thanks a lot.

any reference？

 

[wdlang]

The change I suggest is standard. It removes the first order differentiation, allowing to study the decay and the oscillation separately

how about guess a taylor expansion of f around x=0

and integrate outside from x_0<<1?

the boundary condition at x=0 indicates that

f(x)=ax + b x^2 + c x^3+ d x^4+...

we can determine the relations between a, b, c, d by the differential equation.