Second Order L1 Schemes for Fractional Differential Equations

Difference schemes for the numerical solution of fractional differential equations rely on discretizations of the fractional derivative. In earlier work \cite{d24}, we constructed approximations of the first derivative and applied them to fractional derivatives. In this paper, we extend the method from \cite{d24} to develop parameter-dependent approximations of the second derivative and second-order approximations of the fractional derivative based on the weights of the L1 scheme. We derive the second-order expansion formula of the L1 approximation and show that the coefficient of the second derivative is asymptotically equal to a value of the zeta function, as suggested by the generating function. Using this expansion, we construct a second-order approximation of the fractional derivative and the corresponding asymptotic approximation by a suitable choice of parameter. Examples illustrating the application of these approximations to the numerical solution of ordinary differential equations and fractional differential equations are presented. Both approximations of the fractional derivative are shown to yield second-order numerical methods. Numerical experiments are also provided, confirming the theoretical predictions for the accuracy order of the methods.