Coupled non-linear boundary value ODEs

Questions
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1

How can you solve these 4 coupled non-linear boundary value ODEs (“Von Kármán Swirling flow” problem) in MATHEMATICA?
my equations are:

F" = F^2 + HF' - G^2
G" = 2FG + HG'
H' = -2F
P' = 2FH - 2F'

Boundary Condition: F(0)=0 F(inf)=0 G(0)=1 G(inf)=0 H(0)=0 P(0)=0

Thanks

2

Could you write this in proper Mathematica syntax? A function, say F, should be written with its arguments as F. Also, by HF, od you mean H times F or do you mean the single functin called HF? And os on. By tdoing that you will bothe learn Mathematica syntax so you can follow the discussion and you will let us know what the actual formulation of your problem is.

3

my Functions is F(t), G(t), H(t) and P(t)
and equations are :

F"[t] = (F[t])^2 + H[t]F'[t] - (G[t])^2
G"[t] = 2F[t]G[t] + H[t]G'[t]
H'[t] = -2F[t]
P'[t] = 2F[t]H[t] - 2F'[t]

Boundary Condition: F(0)=0 F(inf)=0 G(0)=1 G(inf)=0 H(0)=0 P(0)=0
4

Note that you need to use == for equality and that it’s usually a good idea to use lower case variables as Mathematica variables are upper case. However, it appears that your problem cannot be solved symbolically.

In[2]:= DSolve[{f''[t] ==  (f[t])^2 + h[t] f'[t] - (g[t])^2,
  g''[t] == 2 f[t] g[t] + h[t] g'[t],
  h'[t] == -2 f[t],
  p'[t] == 2 f[t] h[t] - 2 f'[t]}, {f[t], h[t], g[t], p[t]}, t]

Out[2]= DSolve[{(f^\[Prime]\[Prime])[t] == 
   f[t]^2 - g[t]^2 + h[t] Derivative[1][f][t], (g^\[Prime]\[Prime])[
    t] == 2 f[t] g[t] + h[t] Derivative[1][g][t], 
  Derivative[1][h][t] == -2 f[t], 
  Derivative[1][p][t] == 2 f[t] h[t] - 2 Derivative[1][f][t]}, {f[t], 
  h[t], g[t], p[t]}, t]