In this paper, we discuss the numerical solution of special class of fractional boundary value
problems of order 2. The method of solution is based on a conjugating collocation and
spline analysis combined with shooting method. A theoretical analysis about the existence
and uniqueness of exact solution for the present class is proven. Two examples involving
Bagley–Torvik equation subject to boundary conditions are also presented; numerical
results illustrate the accuracy of the present scheme.

We present a method for solving the boundary value problems for the generalized Bagley-
Torvik (BGT) equation. The problem involves Caputo fractional derivatives of order β, 0< β<
2. The existence and the uniqueness of the solution are presented. For the numerical
solution, we apply the collocation-shooting method. A numerical investigation is given on an
example with a non-polynomial exact solution. Results are presented via tables and figures
illustrating the efficiency in accuracy of the method.