Second order spline method for fractional Bagley-Torvik equation with variable coefficients and Robin boundary conditions

A fractional Bagley-Torvik equation of variable coefficients with Robin boundary conditions is considered in this  paper. We prove the existence of the solution which is demonstrated by converting the boundary value problem into a Volterra integral equation of the second kind and also  prove the uniqueness of the solution  by using the minimum principle. We propose a numerical method that combines the second order spline approximation for the Caputo derivative and the central difference scheme for the second order derivative term. Meanwhile,   the Robin boundary conditions is approximated by three-point endpoint formula. It is to be proved that this method is of second order convergent. Numerical examples are provided to demonstrate the accuracy and efficiency of the method.