Generalized Newtonian fractional model for the vertical motion of a particle |
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| Institution: | 1. Department of Mathematics, Faculty of Sciences and Humanities, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia;2. Department of Basic Engineering Science, Faculty of Engineering, Menofia University, Shebin El-Kom 32511, Egypt;3. Department of Mathematics, College of Science and Humanities at Howtat Sudair, Majmaah University, Majmaah 11952, Saudi Arabia;4. Department of Mathematics, Faculty of Science, University of Tabuk, P.O.Box 741, Tabuk 71491, Saudi Arabia;5. Institute of Engineering, Polytechnic of Porto, Rua Dr. António Bernardino de Almeida, 431, Porto 4249-015, Portugal;1. School of Reliability and Systems Engineering, Beihang University, Beijing 100083, China;2. Science and Technology on Reliability and Environmental Engineering Laboratory, Beijing 100083, China;1. Institute of Petroleum Geology and Geophysics SB RAS, 3 Koptug ave., Novosibirsk 630090, Russia;2. Sobolev Institute of Mathematics SB RAS, 4 Koptug ave., Novosibirsk 630090, Russia;3. Novosibirsk State University, 2 Pirogova st., Novosibirsk 630090, Russia;4. Mathematical Center in Akademgorodok, 2 Pirogova st. & 4 Koptug ave., Novosibirsk 630090, Russia;1. Center of Materials Science and Optoelectronics Engineering, College of Materials Science and Opto-Electronic Technology, University of Chinese Academy of Sciences, Beijing 100049, China;2. Qiushi Honors College, Tianjin University, Tian jin 300350, China;3. McGill Metals Processing Centre, McGill University, Montreal, Quebec H3A 2B2 Canada;1. Department of Mathematics, School of Sciences, Zhejiang Sci-Tech University, Hangzhou, Zhejiang, 310018, China;2. Department of Mathematics and Statistics, University of Turku, Turku FIN-20014, Finland;1. Department of Mathematics, Shaheed Bhagat Singh College, University of Delhi, India;2. Government P.G. College, Mushafirkhana, Uttar Pradesh, India |
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| Abstract: | 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 × 103 seconds. |
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