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

Abstract

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