Yan, YubinGreen, CharlesPani, AmiyaYang, Yuhui2023-12-112023-12-112023-12-11Yang, Y., Green, C. W. H., Pani, A. K., & Yan, Y. (2024). High-order schemes based on extrapolation for semilinear fractional differential equation. Calcolo, 61(2), 1-40. https://doi.org/10.1007/s10092-023-00553-10008-062410.1007/s10092-023-00553-1http://hdl.handle.net/10034/628374The version of record of this article, first published in Calcolo, is available online at Publisher’s website: https://doi.org/10.1007/s10092-023-00553-1By rewriting the Riemann–Liouville fractional derivative as Hadamard finite-part integral and with the help of piecewise quadratic interpolation polynomial approximations, a numerical scheme is developed for approximating the Riemann–Liouville fractional derivative of order α∈(1, 2). The error has the asymptotic expansion (d3τ3-α+d4τ4-α+d5τ5-α+⋯)+(d2∗τ4+d3∗τ6+d4∗τ8+⋯) at any fixed time tN=T, N∈Z+, where di, i=3, 4, … and di∗, i=2, 3, … denote some suitable constants and τ=T/N denotes the step size. Based on this discretization, a new scheme for approximating the linear fractional differential equation of order α∈(1, 2) is derived and its error is shown to have a similar asymptotic expansion. As a consequence, a high-order scheme for approximating the linear fractional differential equation is obtained by extrapolation. Further, a high-order scheme for approximating a semilinear fractional differential equation is introduced and analyzed. Several numerical experiments are conducted to show that the numerical results are consistent with our theoretical findings.Attribution 4.0 Internationalhttps://creativecommons.org/licenses/by/4.0/Caputo fractional derivativeExtrapolationConvergenceArticle1126-5434Calcolo