Solve DE by fourth order Runge Kutta code coupled with the shooting method

Questions
1

how to solve this problem in mathematica, using a forth order Runge Kutta code coupled with the shooting method.
A ordinary differential equation is

f''''[y] - 2*a^2*f''[y] - a^2*(a^2 - Ra)*f[y] == 0,

with boundary condition is,

f[0] == 0,

f'''[0] - a^2*f'[0] == 0, f[1] == 0, f''[1] == 0

?

2

You can solve the boundary problem numerically with NDSolve:

With[{a = 2, R = 1}, 
 sol = NDSolveValue[{f''''[y] - 2 a^2 f''[y] - a^2 (a^2 - R a) f[y] == 0, 
   f[0] == 0, f'''[0] - a^2*f'[0] == 0, f[1] == 0, f''[1] == 0}, f, {y,
    0, 1}, Method -> {"Shooting"}];
 Plot[sol[y], {y, 0, 1}]]