一个新的近似Caputo-Fabrizio分数导数的数值分数微分公式：误差分析和稳定性

IF 1.1 Q2 MATHEMATICS, APPLIED

Computational Methods for Differential Equations Pub Date : 2021-01-05 DOI:10.22034/CMDE.2020.37595.1664

Leila Moghadam Dizaj Herik, M. Javidi, M. Shafiee

{"title":"一个新的近似Caputo-Fabrizio分数导数的数值分数微分公式：误差分析和稳定性","authors":"Leila Moghadam Dizaj Herik, M. Javidi, M. Shafiee","doi":"10.22034/CMDE.2020.37595.1664","DOIUrl":null,"url":null,"abstract":"In the present work, first of all, a new numerical fractional differentiation formula (called the CF2 formula) to approximate the Caputo-Fabrizio fractional derivative of order $alpha,$ $(0<alpha<1)$ is developed. It is established by means of the quadratic interpolation approximation using three points $ (t_{j-2},y(t_{j-2}))$, $(t_{j-1},y(t_{j-1})) $ and $ (t_{j},y(t_{j})) $ on each interval $[t_{j-1},t_{j}]$ for $ ( j geq 2 )$, while the linear interpolation approximation is applied on the first interval $[t_{0},t_{1}]$. As a result, the new formula can be formally viewed as a modification of the classical CF1 formula, which is obtained by the piecewise linear approximation for $y(t)$. Both the computational efficiency and numerical accuracy of the new formula is superior to that of the CF1 formula. The coefficients and truncation errors of this formula are discussed in detail. {Two test example show} the numerical accuracy of the CF2 formula. The CF1 formula demonstrates that the new CF2 is much more effective and more accurate than the CF1 when solving fractional differential equations. Detailed stability analysis and region stability of the CF2 are also carefully investigated.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1000,"publicationDate":"2021-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A new numerical fractional differentiation formula to approximate the Caputo-Fabrizio fractional derivative: error analysis and stability\",\"authors\":\"Leila Moghadam Dizaj Herik, M. Javidi, M. Shafiee\",\"doi\":\"10.22034/CMDE.2020.37595.1664\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In the present work, first of all, a new numerical fractional differentiation formula (called the CF2 formula) to approximate the Caputo-Fabrizio fractional derivative of order $alpha,$ $(0<alpha<1)$ is developed. It is established by means of the quadratic interpolation approximation using three points $ (t_{j-2},y(t_{j-2}))$, $(t_{j-1},y(t_{j-1})) $ and $ (t_{j},y(t_{j})) $ on each interval $[t_{j-1},t_{j}]$ for $ ( j geq 2 )$, while the linear interpolation approximation is applied on the first interval $[t_{0},t_{1}]$. As a result, the new formula can be formally viewed as a modification of the classical CF1 formula, which is obtained by the piecewise linear approximation for $y(t)$. Both the computational efficiency and numerical accuracy of the new formula is superior to that of the CF1 formula. The coefficients and truncation errors of this formula are discussed in detail. {Two test example show} the numerical accuracy of the CF2 formula. The CF1 formula demonstrates that the new CF2 is much more effective and more accurate than the CF1 when solving fractional differential equations. Detailed stability analysis and region stability of the CF2 are also carefully investigated.\",\"PeriodicalId\":44352,\"journal\":{\"name\":\"Computational Methods for Differential Equations\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.1000,\"publicationDate\":\"2021-01-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Computational Methods for Differential Equations\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.22034/CMDE.2020.37595.1664\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Methods for Differential Equations","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.22034/CMDE.2020.37595.1664","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

摘要

本文首先提出了一个新的数值分数阶微分公式(称为CF2公式)来近似阶$ α，$ $(0< α <1)$的Caputo-Fabrizio分数阶导数。对于$(j geq 2)$，在每个区间$[t_{j-1}，t_{j}]$上使用$(t_{j}) $， $(t_{j}) $， $(t_{j}) $和$(t_{j}，y(t_{j})) $三个点进行二次插值逼近，而在第一个区间$[t_{0}，t_{1}]$上应用线性插值逼近。因此，新公式可以形式上看作是对经典CF1公式的修正，经典CF1公式是通过y(t)的分段线性近似得到的。新公式的计算效率和数值精度均优于CF1公式。详细讨论了该公式的系数和截断误差。{两个测试实例显示}CF2公式的数值精度。CF1公式表明，在求解分数阶微分方程时，新的CF2比CF1更有效、更精确。对CF2进行了详细的稳定性分析和区域稳定性研究。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

A new numerical fractional differentiation formula to approximate the Caputo-Fabrizio fractional derivative: error analysis and stability

In the present work, first of all, a new numerical fractional differentiation formula (called the CF2 formula) to approximate the Caputo-Fabrizio fractional derivative of order $alpha,$ $(0

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊