A High Order Formula to Approximate the Caputo Fractional Derivative

doi:10.1007/s42967-019-00023-y

Communications on Applied Mathematics and Computation ›› 2020, Vol. 2 ›› Issue (1): 1-29.doi: 10.1007/s42967-019-00023-y

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A High Order Formula to Approximate the Caputo Fractional Derivative

R. Mokhtari, F. Mostajeran

Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran

收稿日期:2018-08-13 修回日期:2019-02-23 出版日期:2020-03-20 发布日期:2020-02-19
通讯作者: R. Mokhtari, F. Mostajeran E-mail:mokhtari@cc.iut.ac.ir;f_mostajeran@modares.ac.ir

A High Order Formula to Approximate the Caputo Fractional Derivative

R. Mokhtari, F. Mostajeran

Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran

Received:2018-08-13 Revised:2019-02-23 Online:2020-03-20 Published:2020-02-19
Contact: R. Mokhtari, F. Mostajeran E-mail:mokhtari@cc.iut.ac.ir;f_mostajeran@modares.ac.ir

摘要/Abstract

摘要： We present here a high-order numerical formula for approximating the Caputo fractional derivative of order α for 0 < α < 1. This new formula is on the basis of the third degree Lagrange interpolating polynomial and may be used as a powerful tool in solving some kinds of fractional ordinary/partial diferential equations. In comparison with the previous formulae, the main superiority of the new formula is its order of accuracy which is 4-α, while the order of accuracy of the previous ones is less than 3. It must be pointed out that the proposed formula and other existing formulae have almost the same computational cost. The efectiveness and the applicability of the proposed formula are investigated by testing three distinct numerical examples. Moreover, an application of the new formula in solving some fractional partial diferential equations is presented by constructing a fnite diference scheme. A PDE-based image denoising approach is proposed to demonstrate the performance of the proposed scheme.

关键词: Caputo fractional derivative, Fractional partial diferential equation, Finite diference scheme

Abstract: We present here a high-order numerical formula for approximating the Caputo fractional derivative of order α for 0 < α < 1. This new formula is on the basis of the third degree Lagrange interpolating polynomial and may be used as a powerful tool in solving some kinds of fractional ordinary/partial diferential equations. In comparison with the previous formulae, the main superiority of the new formula is its order of accuracy which is 4-α, while the order of accuracy of the previous ones is less than 3. It must be pointed out that the proposed formula and other existing formulae have almost the same computational cost. The efectiveness and the applicability of the proposed formula are investigated by testing three distinct numerical examples. Moreover, an application of the new formula in solving some fractional partial diferential equations is presented by constructing a fnite diference scheme. A PDE-based image denoising approach is proposed to demonstrate the performance of the proposed scheme.

Key words: Caputo fractional derivative, Fractional partial diferential equation, Finite diference scheme

中图分类号:

65M15
65M06

R. Mokhtari, F. Mostajeran. A High Order Formula to Approximate the Caputo Fractional Derivative[J]. Communications on Applied Mathematics and Computation, 2020, 2(1): 1-29.

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A High Order Formula to Approximate the Caputo Fractional Derivative

A High Order Formula to Approximate the Caputo Fractional Derivative

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