Suggest me How i solve given problem with cubic spline or finete difference method or finite difference method with cubic spline.
f"'+f'f"=0
B.C. f(0)=constant , f'(0)=0 and f'(infinity)=1 ?
First of all, we have to write problem in a bounded domain using say artificial boundary condition. Sometimes, if theoretically one can not prove the error committed by artificial boundary condition is small, one can do so by varying the right hand boundary condition and see its effect numerically. Then the problem is posed on a bounded domain. One can take say $f' =u$ and obtain a nonlinear two point boundary value problem and apply fdm which may give rise to nonlinear system and solve .
But why did you want to apply numerical method. May be try to do it analytically
after obtaining $u$: one finds : $u''+ (u^2/2)'=0$ Then integrate from 0 to \eta. Then
and possibly one can integrate again.
your problem is nonlinear, so we may get a difficult to solve, so Cubic Spline when satisfied with using Taylor series, may be you can take it a solution approximately
Your problem is solved in more general case in the attached file by using "Shooting method". It might have useful to solve your problem by "Difference method".
in cubic B-spline collocation method you can use Crank-nicolson approach to deal with f and its derivatives
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