A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications

Abstract

In the present work, first, a new fractional numerical differentiation formula (called the L1-2 formula) to approximate the Caputo fractional derivative of order α (0<α<1) is developed. It is established by means of the quadratic interpolation approximation using three points (tj−2,f(tj−2)),(tj−1,f(tj−1)) and (tj,f(tj)) for the integrand f(t) on each small interval [tj−1,tj] (j⩾2), while the linear interpolation approximation is applied on the first small interval [t0,t1]. As a result, the new formula can be formally viewed as a modification of the classical L1 formula, which is obtained by the piecewise linear approximation for f(t). Both the computational efficiency and numerical accuracy of the new formula are superior to that of the L1 formula. The coefficients and truncation errors of this formula are discussed in detail. Two test examples show the numerical accuracy of L1-2 formula. Second, by the new formula, two improved finite difference schemes with high order accuracy in time for solving the time-fractional sub-diffusion equations on a bounded spatial domain and on an unbounded spatial domain are constructed, respectively. In addition, the application of the new formula into solving fractional ordinary differential equations is also presented. Several numerical examples are computed. The comparison with the corresponding results of finite difference methods by the L1 formula demonstrates that the new L1-2 formula is much more effective and more accurate than the L1 formula when solving time-fractional differential equations numerically.