Difference schemes for the numerical solution of fractional differential equations rely on discretizations of the fractional derivative. In this paper, we obtain the second-order expansion formula for the L1 approximation of the Caputo fractional derivative. Second-order approximations of the fractional derivative are constructed based on the expansion formula and parameter-dependent discretizations of the second derivative. Examples illustrating the application of these approximations to the numerical solution of ordinary and partial fractional differential equations are presented, and the convergence and order of the difference schemes are proved. Numerical experiments are also provided, confirming the theoretical predictions for the accuracy of the numerical methods.

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.

This paper proposes numerical methods for solving ordinary differential equations and fractional ordinary differential equations. The proposed methods are based on discretizations of first- and second-order derivatives, the L1 approximation of the Caputo fractional derivative, and a shifted L1-based approximation on a uniform mesh. The discretizations of the integer-order derivatives depend on a free parameter, which enables the construction of numerical schemes with any prescribed order of accuracy in the interval $(0,2]$ and supports the development of efficient, fast algorithms for computation of the solution. The discretizations of fractional derivatives employ the weights of the L1 approximation together with values of the Riemann zeta function. The convergence and accuracy of the numerical methods are analyzed theoretically. Numerical experiments confirm the theoretical results and demonstrate the improvement of the proposed methods over L1 schemes for the numerical solution of ordinary fractional differential equations.

In this paper we use the asymptotic expansions of the binomial coefficients and the weights of the L1 approximation to obtain approximations of order $2-\alpha$ and second-order approximations of the Caputo derivative by modifying the weights of the shifted Gr\"unwald-Letnikov difference approximation and the L1 approximation of the Caputo derivative. A modification of the shifted Gr\"unwald-Letnikov approximation is obtained which allows second-order numerical solutions of fractional differential equations with arbitrary values of the solutions and their first derivatives at the initial point.