Efficient Numerical Methods for Fractional- and Integer-Order Ordinary Differential Equations

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.