Numerical solution of fractional ordinary differential equations by local discontinuous Galerkin method

In this paper, we apply the local discontinuous Galerkin method for solving fractional ordinary differential equations, in general.  In this method, choosing the (natural) numerical upwind flux enables us to solve the initial value problems for ordinary fractional equations interval by interval and forward in time. This means that we require to solve a low-order (k+1) × (k+1) system of equations locally in each subinterval, and there is no need to solve the global system; Here k is the degree of the basis functions in each subinterval. To implement the method, we consider the (local) basis functions as the (shifted) Legendre polynomials. This, in turn, makes some of the coefficient matrices in the system of equations sparse and thus accelerates the computations. Also the stability in the infinity norm and the error estimation of the method are discussed. Finally, with a series of linear and nonlinear examples, we show the efficiency and, in particular, the accuracy of the local discontinuous Galerkin method for fractional differential equations.