Numerical approaches to Caputo-Hadamard fractional derivatives with applications to long-term integration of fractional differential systems

In this paper, three kinds of numerical formulas are proposed for approximating the Caputo-Hadamard fractional derivatives, which are called L1-2 formula, L2-1(sigma) formula, and H2N2 formula, respectively. Among them, the numerical formulas L1-2 and L2-1(sigma) are for order alpha is an element of(0, 1) with (3-alpha)-th order convergence, and H2N2 formula is for order alpha is an element of(1, 2) with (3 - alpha)-th order convergence too, where the theoretical convergence order has been verified by some illustrative examples. Finally, these three new formulas are applied to long-term integration of fractional differential systems. (C) 2021 Elsevier B.V. All rights reserved.