  This article proposes new strategies for solving two-point Fractional order
Nonlinear Boundary Value Problems(FN-BVPs) with Robin Boundary
Conditions(RBCs). In the new numerical schemes, a two-point FNBVP is
transformed into a system of Fractional order Initial Value Problems(FIVPs)
with unknown Initial Conditions(ICs). To approximate ICs in the system of
FIVPs, we develop nonlinear shooting methods based on Newton's method and
Halley's method using the RBC at the right end point. To deal with FIVPs in a
system, we mainly employ High-order Predictor-Corrector Methods(HPCMs) with
linear interpolation and quadratic interpolation into Volterra integral
equations which are equivalent to FIVPs. The advantage of proposed schemes with
HPCMs is that even though they are designed for solving two-point FNBVPs, they
can handle both linear and nonlinear two-point Fractional order Boundary Value
Problems(FBVPs) with RBCs and have uniform convergence rates of HPCMs, $O(h^2)$
and $O(h^3)$ for shooting techniques with Newton's method and Halley's method,
respectively. A variety of numerical examples are demonstrated to confirm the
effectiveness and performance of the proposed schemes. Also we compare the
accuracy and performance of our schemes with another method.
