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A fast and high-order numerical method for nonlinear fractional-order differential equations with non-singular kernel
Applied Numerical Mathematics ( IF 2.8 ) Pub Date : 2021-01-20 , DOI: 10.1016/j.apnum.2021.01.013
Seyeon Lee , Junseo Lee , Hyunju Kim , Bongsoo Jang

Efficient and fast explicit methods are proposed to solve nonlinear Caputo-Fabrizio fractional differential equations, where Caputo-Fabrizio operator is a new proposed fractional derivative with a smooth kernel. The proposed methods produce the second-order for linear interpolation and the third-order accuracy for quadratic interpolation, respectively. The convergence analysis is proved by using discrete Gronwall's inequality. Furthermore, applying the recurrence relation of the memory term, it reduces CPU time executed the proposed methods. The proposed fast algorithm requires approximately O(N) arithmetic operations while O(N2) is required in case of the regular predictor-corrector schemes, where N is the total number of the time step. The following numerical examples demonstrate the accuracy of the proposed methods as well as the efficiency: nonlinear fractional differential equations, time-fraction sub-diffusion, and time-fractional advection-diffusion equation. Numerical experiments also verify the theoretical convergence rates.



中文翻译:

具有非奇异核的非线性分数阶微分方程的快速高阶数值方法

提出了一种快速有效的显式方法来求解非线性Caputo-Fabrizio分数阶微分方程,其中Caputo-Fabrizio算子是一种新提出的具有光滑核的分数阶导数。所提出的方法分别产生用于线性插值的二阶和用于二次插值的三阶精度。通过使用离散Gronwall不等式证明了收敛性分析。此外,应用存储项的递归关系,可以减少执行建议方法的CPU时间。提出的快速算法大约需要Øñ 算术运算 Øñ2在常规预测器-校正器方案的情况下是必需的,其中N是时间步长的总数。以下数值示例说明了所提出方法的准确性和效率:非线性分数阶微分方程,时间分数次扩散和时间分数维对流扩散方程。数值实验也验证了理论收敛速度。

更新日期:2021-01-22
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