High-order schemes based on extrapolation for semilinear fractional differential equation

By 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 \(\alpha \in (1,2).\) The error has the asymptotic expansion \( \big ( d_{3} \tau ^{3- \alpha } + d_{4} \tau ^{4-\alpha } + d_{5} \tau ^{5-\alpha } + \cdots \big ) + \big ( d_{2}^{*} \tau ^{4} + d_{3}^{*} \tau ^{6} + d_{4}^{*} \tau ^{8} + \cdots \big ) \) at any fixed time \(t_{N}= T, N \in {\mathbb {Z}}^{+}\), where \(d_{i}, i=3, 4,\ldots \) and \(d_{i}^{*}, i=2, 3,\ldots \) denote some suitable constants and \(\tau = T/N\) denotes the step size. Based on this discretization, a new scheme for approximating the linear fractional differential equation of order \(\alpha \in (1,2)\) 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.