A Unified Fast Memory-Saving Time-Stepping Method for Fractional Operators and Its Applications

L1 Schemes for time-fractional differential equations: A brief survey and new development

“…1.13e-6 2.61e-5 5.03e-5 6.41e-5 2 8 2.09e-4 1.01 2.82e-7 2.00 1.03e-5 1.35 2.02e-5 1.32 2.58e-5 1.31 2 9 1.04e-4 1.01 7.03e-8 2.00 4.02e-6 1.35 8.06e-6 1.33 1.04e-5 1.32 2 10 5.19e-5 1.00 1.76e-8 2.00 1.57e-6 1.36 3.20e-6 1.33 4.12e-6 1.33 2 11 2.59e-5 1.00 4.39e-9 2.00 6.09e-7 1.36 1.26e-6 1.34 1.63e-6 1.34…”

“…Due to the nonlocality of the fractional operator, the calculation of the L1 method and its variants is costly compared with the integer-order derivative operators. Generally speaking, we need O(N ) storage and O(N 2 ) arithmetic operations for calculating the L1 method δ α t u n , 1 ≤ n ≤ N (see [11,9,22]), while for the integer-order derivative, i.e., α = 1, the L1 method reduces to the firstorder backward formula, which needs O(1) storage and O(N ) arithmetic operations. The fast L1 method has been extensively studied in the literature (see, e.g., [11]).…”

Core Content Selection and Textbook Design for High School Mathematics

“…Find N 1 and N 2 such that e −te jh +(1−β)jh ≤ εt β−1 for j ≤ −N 1 and j ≥ N 2 . So that we can truncate (37) to derive the following kernel approximation [11] (38)…”

“…It has been shown that the method (38) (or Algorithm 1) works only for β < 1, performs badly when β → 1, and cannot work for β ≥ 1 [11]. This drawback can be tackled by using the following property…”

“…We refer interesting readers to [11] for more kernel approximation as (35), where different kernel approximations are compared, the advantages and pitfalls are displayed, and the corresponding fast algorithms are developed and numerically verified.…”