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Generalized Newtonian fractional model for the vertical motion of a particle

机译：颗粒垂直运动的广义牛顿分数模型

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摘要

Based on the Riemann-Liouville (R-L) fractional derivative and the generalized Newtonian law of gravitation, the nonlinear fractional differential equation describing the vertical motion of a particle is solved. Such solution is investigated to obtain the escape velocity (EV) following the fractional Newtonian mechanics. It is well known that the EV from the Earth's gravitational field is about 11.18 km/s within the paradigm of the classical Newtonian mechanics using integer derivatives, but its value has not been yet determined in the scope of fractional calculus. Therefore, we can pose the question: Is the classical value of the EV identical when analyzed under the light of the fractional mechanics? The paper answers this question for the first time. It is found that the fractional escape velocity (FEV) depends on the non-integer order α and a parameter σ with dimension of seconds. The general relation between σ and α is established. The results reveal that the values of the FEV approaches the classical one when α → 1 and σ ≈ 5 × 10~3 seconds.
机译：基于RIEMANN-LIOUVILLE（R-L）分数衍生物和广义牛顿的重力定律，解决了描述颗粒垂直运动的非线性分数微分方程。研究了这种解决方案以在分数牛顿力学之后获得逃逸速度（EV）。众所周知，来自地球的引力场的EV在使用整数衍生物的经典牛顿力学的范式内大约11.18公里/秒，但其值尚未确定在分数微积分的范围内。因此，我们可以提出问题：在分数力学的光线下分析时，EV的经典价值是相同的吗？本文首次回答这个问题。发现分数逃逸速度（FEV）取决于非整数α和具有秒的尺寸的参数σ。 σ和α之间的一般关系建立。结果表明，当α→1和Σ≈5×10〜3秒时，FEV的值接近经典的一个。

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  • 来源
    《Applied Mathematical Modelling》 |2020年第12期|652-660|共9页
  • 作者

    E.R. Elzahar; A.A. Gaber; A.F. Aljohani; J. Tenreiro Machado; A. Ebaid;

  • 作者单位

    Department of Mathematics Faculty of Sciences and Humanities Prince Sattam Bin Abdulaziz University Alkharj 11942 Saudi Arabia Department of Basic Engineering Science Faculty of Engineering Menofia University Shebin El-Kom 32511 Egypt;

    Department of Mathematics College of Science and Humanities at Howtat Sudair Majmaah University Majmaah 11952 Saudi Arabia;

    Department of Mathematics Faculty of Science University of Tabuk P.O.Box 741 Tabuk 71491 Saudi Arabia;

    Institute of Engineering Polytechnic of Porto Rua Dr. Antonio Bernardino de Almeida 431 Porto 4249-015 Portugal;

    Department of Mathematics Faculty of Science University of Tabuk P.O.Box 741 Tabuk 71491 Saudi Arabia;

  • 收录信息 美国《科学引文索引》(SCI);美国《工程索引》(EI);
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

    Newtonian mechanics; Nonlinear fractional differential equations; Escape velocity;

    机译：牛顿力学;非线性分数微分方程;逃逸速度;

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