I'm trying to solve these two coupled 2nd order differential equations, but I'm getting nowhere. Mathematica can't seem to solve these no matter what I do. The equations are:
The $r_{*}$ is the value of the $r$ at midpoint and I have the mentioned boundary value conditions:
and also I have the following symmetry along the $x$ axis where at the midpoint I have $r'(x)=v(x)=0$. I tried to solve these equations with NDsolve
for the following boundary conditions but I had no luck:
m = 1/2 (Tanh[v[x]/(1/3)] + 1);
NDSolve[{r[x]^4/1.1^2 - r[x]^2 -
2 r'[x] v'[x] + (r[x]^2 - m) v'[x]^2 == 0,
r[x]^2 - r[x]^2 v'[x]^2 - r[x] v''[x] + 2 r'[x] v'[x] == 0,
r[1] == 100000, v[1] == 4, v[2] == 4}, {r, v}, {x, 1, 2},
MaxSteps -> Infinity]
I have set $t=0$ and used 10000 instead of $\infty$. but unfortunately, I get the following error mesages:
Power::infy: Infinite expression 1/0. encountered.
Infinity::indet: Indeterminate expression 0. ComplexInfinity encountered.
General::stop: Further output of Power::infy will be suppressed during this calculation.
NDSolve::ndnum: Encountered non-numerical value for a derivative at x == 1.`
I would very much appreciate any help
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