Generalized forms of fractional Euler and Runge–Kutta methods using non-uniform grid

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2022-07-08 DOI:10.1515/ijnsns-2021-0278

Pushpendra Kumar, V. S. Erturk, M. Murillo‐Arcila, C. Harley

引用次数: 8

Abstract

Abstract In this article, we propose generalized forms of three well-known fractional numerical methods namely Euler, Runge–Kutta 2-step, and Runge–Kutta 4-step, respectively. The new versions we provide of these methods are derived by utilizing a non-uniform grid which is slightly different from previous versions of these algorithms. A new generalized form of the well-known Caputo-type fractional derivative is used to derive the results. All necessary analyses related to the stability, convergence, and error bounds are also provided. The precision of all simulated results is justified by performing multiple numerical experiments, with some meaningful problems solved by implementing the code in Mathematica. Finally, we give a brief discussion on the simulated results which shows that the generalized methods are novel, effective, reliable, and very easy to implement.

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使用非均匀网格的分数Euler和Runge–Kutta方法的广义形式

摘要本文分别给出了欧拉法、龙格-库塔2步法和龙格-库塔4步法这三种著名的分数阶数值方法的推广形式。我们提供的这些方法的新版本是通过使用非均匀网格派生的，这与这些算法的先前版本略有不同。利用著名的caputo型分数阶导数的一种新的推广形式来推导结果。文中还对稳定性、收敛性和误差界进行了必要的分析。通过多次数值实验验证了所有模拟结果的精度，并在Mathematica中实现了一些有意义的问题。最后，对仿真结果进行了简要讨论，结果表明，该方法新颖、有效、可靠，易于实现。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.