On the Fractional Derivatives with an Exponential Kernel

doi:10.1007/s42967-022-00233-x

Communications on Applied Mathematics and Computation ›› 2023, Vol. 5 ›› Issue (4): 1655-1673.doi: 10.1007/s42967-022-00233-x

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On the Fractional Derivatives with an Exponential Kernel

Enyu Fan, Jingshu Wu, Shaoying Zeng

Department of Mathematics, Shanghai University, Shanghai 200444, China

收稿日期:2022-10-17 修回日期:2022-11-12 发布日期:2023-12-16
通讯作者: Enyu Fan,E-mail:fanenyu@shu.edu.cn E-mail:fanenyu@shu.edu.cn

On the Fractional Derivatives with an Exponential Kernel

Enyu Fan, Jingshu Wu, Shaoying Zeng

Department of Mathematics, Shanghai University, Shanghai 200444, China

Received:2022-10-17 Revised:2022-11-12 Published:2023-12-16
Contact: Enyu Fan,E-mail:fanenyu@shu.edu.cn E-mail:fanenyu@shu.edu.cn

摘要/Abstract

摘要： The present article mainly focuses on the fractional derivatives with an exponential kernel (“exponential fractional derivatives” for brevity). First, several extended integral transforms of the exponential fractional derivatives are proposed, including the Fourier transform and the Laplace transform. Then, the L2 discretisation for the exponential Caputo derivative with α ∈ (1, 2) is established. The estimation of the truncation error and the properties of the coefficients are discussed. In addition, a numerical example is given to verify the correctness of the derived L2 discrete formula.

关键词: Exponential fractional derivative, Integral transform, L2 discretisation, Truncation error

Abstract: The present article mainly focuses on the fractional derivatives with an exponential kernel (“exponential fractional derivatives” for brevity). First, several extended integral transforms of the exponential fractional derivatives are proposed, including the Fourier transform and the Laplace transform. Then, the L2 discretisation for the exponential Caputo derivative with α ∈ (1, 2) is established. The estimation of the truncation error and the properties of the coefficients are discussed. In addition, a numerical example is given to verify the correctness of the derived L2 discrete formula.

Key words: Exponential fractional derivative, Integral transform, L2 discretisation, Truncation error

中图分类号:

26A33
44A05

Enyu Fan, Jingshu Wu, Shaoying Zeng. On the Fractional Derivatives with an Exponential Kernel[J]. Communications on Applied Mathematics and Computation, 2023, 5(4): 1655-1673.

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On the Fractional Derivatives with an Exponential Kernel

On the Fractional Derivatives with an Exponential Kernel

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