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Using NDSolve while solving system of ODE's. Here is my try

Eqn1 = -C  f'''''[x] + f'''[x] + f[x]  f''[x] - f'[x]  f'[x] + r^2 + λ  T[x] == 0   

Eqn2 = 
  T''[x] + p1 f[x] T'[x] + 
    p2/(3 (1 - tw)) (12 (tw + (1 - tw) T[x] ) + tw)^2 (1 - tw) T'[x] T'[x] +  
    4 ((1 - tw) T[x] + tw)^3 (1 - tw) T''[x] == 0

BC1 = f[0] == 0
BC2 = f'[0] == 1
BC3 = f'[10] == r
BC4 = f''[10] == 0
BC5 = f'''[10] == 0
BC6 = T'[0] == -B (1 - T[0])
BC7 = T[10] == 0

param = {λ -> 1, p1 -> 10, B -> 10, tw -> 1.5, p2 -> 0.5, 
  C -> 1, r -> 1}

Sol = 
  NDSolve[{Eqn1, Eqn2, BC1, BC2, BC3, BC4, BC5, BC6, BC7} /. param, 
    {f, T}, {x, 0, 10}, 
    Method -> 
      {"Shooting", 
       "StartingInitialConditions" -> 
        {f[0] == 0, f'[0] == 0, f''[0] == 1, f'''[0] == 0, f''''[0] == 0, T[0] = 0}}]

But I'm getting this error

NDSolve::ndode: Input is not an ordinary differential equation.

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1 Answer 1

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The specific reason for the error message is that "StartingInitialConditions" contains T[0] = 0. It should be T[0] == 0. The fact that f'[0] == 0 is inconsistent with BC2 and that there are too few "StartingInitialConditions" for T also may cause difficulties. Certainly, with f'[0] == 0 and T'[0] == -10 added, the integration proceeds further before encountering

NDSolveValue::ndsz: At x == 9.991371232789929`, step size is effectively zero; singularity or stiff system suspected. >>

This last issue suggests that the NDSolve Method may need to be reconsidered.

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  • $\begingroup$ Thanks dear for pointing out the syntax error. But NDSolve is facing issues to solve the odes. $\endgroup$
    – zhk
    Commented Mar 1, 2015 at 4:26
  • 1
    $\begingroup$ Integration will fail, if 1 + 4*(1 - tw)*(tw + T[x] - tw*T[x])^3 vanishes within the domain of integration. $\endgroup$
    – bbgodfrey
    Commented Mar 1, 2015 at 5:06
  • $\begingroup$ if I take p2 = 0, that should make it solvable but I still get 'singularity or stiff system suspected. $\endgroup$
    – zhk
    Commented Mar 1, 2015 at 6:37
  • $\begingroup$ I see that you made progress in 76135. Congratulations. $\endgroup$
    – bbgodfrey
    Commented Mar 1, 2015 at 12:57
  • $\begingroup$ yes I did but with asymptomatic boundary at 3. I would like to see the results for at least 12. $\endgroup$
    – zhk
    Commented Mar 1, 2015 at 13:13

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