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Moghadam Dizaj Herik, Leila, Javidi, Mohammad, Shafiee, Mahmoud. (1400). A new numerical fractional differentiation formula to approximate the Caputo-Fabrizio fractional derivative: error analysis and stability. سامانه مدیریت نشریات علمی, 10(1), 12-27. doi: 10.22034/cmde.2020.37595.1664
Leila Moghadam Dizaj Herik; Mohammad Javidi; Mahmoud Shafiee. "A new numerical fractional differentiation formula to approximate the Caputo-Fabrizio fractional derivative: error analysis and stability". سامانه مدیریت نشریات علمی, 10, 1, 1400, 12-27. doi: 10.22034/cmde.2020.37595.1664
Moghadam Dizaj Herik, Leila, Javidi, Mohammad, Shafiee, Mahmoud. (1400). 'A new numerical fractional differentiation formula to approximate the Caputo-Fabrizio fractional derivative: error analysis and stability', سامانه مدیریت نشریات علمی, 10(1), pp. 12-27. doi: 10.22034/cmde.2020.37595.1664
Moghadam Dizaj Herik, Leila, Javidi, Mohammad, Shafiee, Mahmoud. A new numerical fractional differentiation formula to approximate the Caputo-Fabrizio fractional derivative: error analysis and stability. سامانه مدیریت نشریات علمی, 1400; 10(1): 12-27. doi: 10.22034/cmde.2020.37595.1664

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

Computational Methods for Differential Equations
مقاله 2، دوره 10، شماره 1، فروردین 2022، صفحه 12-27    اصل مقاله (234.98 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2020.37595.1664
نویسندگان
Leila Moghadam Dizaj Herik1؛ Mohammad Javidi* 1، 2؛ Mahmoud Shafiee1
1Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran.
2Factulty of Mathematical Sciences, University of Tabriz, Tabriz, Iran.
چکیده
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 α, (0 < α < 1) is developed. It is established by means of the quadratic interpolation approximation using three points (tj−2,y(tj−2)),(tj−1,y(tj−1)), and (tj, y(tj)) on each interval [tj−1,tj] for (j ≥ 2), while the linear interpolation approximation are applied on the first interval [t0,t1]. 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 examples 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.
کلیدواژه‌ها
Fractional differential equation؛ Stability؛ Caputo-Fabrizio fractional derivative؛ Numerical methods؛ Error analysis
مراجع
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