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High-order schemes based on extrapolation for semilinear fractional differential equation
Yan, Yubin Green, Charles Pani, Amiya Yang, Yuhui
Yan, Yubin
Green, Charles
Pani, Amiya
Yang, Yuhui
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2023-12-11
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Abstract
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 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.
Citation
Yang, 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/s10092-023-00553-1
Publisher
Springer
Journal
Calcolo
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DOI
10.1007/s10092-023-00553-1
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Article
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The version of record of this article, first published in Calcolo, is available online at Publisher’s website: https://doi.org/10.1007/s10092-023-00553-1
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ISSN
0008-0624
EISSN
1126-5434