During a shooting method you guess initial values from where you start solving the boundary value problem (BVP) as an initial value problem (IVP). You expect the result to be accurate when the right boundary condition (BC at infinity) is fulfilled. Since you solve an IVP, you can set the interval of integration. Choose it sufficiently large. If you use L1<L2 and there is little difference between them, then the truncated interval describes the problem of infinity well.
I solved similar boundary-layer-type equations several times. So I suggest you to try one or more of the followings:
- Solve the BVP with bvp4c or bvp5c
- Use Chebfun
- Perform group analysis on them so that you can use a transformation method that can transform the BVP to an IVP (see e.g. Töpfer-transformation)
- Use a series of transformations (see e.g. Iterative Transformation Method)
- Code the shooting method (but be aware, this can be quite susceptible to the initial conditions)
