The essence of the generalized Taylor theorem as the foundation of the homotopy analysis method |
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| Institution: | 1. University “G. d’Annunzio” of Chieti-Pescara, Via dei Vestini 31, 66100 Chieti Scalo (CH), Italy;2. Wrocław University of Science and Technology, Wyb. St. Wyspiańskiego 27, 50-370 Wrocław, Poland;1. Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, Southeast University, Nanjing, Jiangsu 210096, China;2. School of Automation, Southeast University, Nanjing, Jiangsu, 210096, China;1. Institute for Problems in Mechanics, Russian Academy of Sciences, 101 Vernadsky Avenue, bldg 1, 119526 Moscow, Russia;2. Bauman Moscow State Technical University, 5 Second Baumanskaya Street, 105005 Moscow, Russia;3. National Research Nuclear University MEPhI, 31 Kashirskoe Shosse, 115409 Moscow, Russia;4. University of Sonora, Blvd. Luis Encinas y Rosales S/N, Hermosillo C.P. 83000, Sonora, Mexico;1. INSA – Lyon, Laboratoire Vibrations-Acoustique (LVA), 25 bis, av. Jean Capelle, F-69621 Villeurbanne Cedex, France;2. GTM – Grup de recerca en Tecnologies Mèdia, La Salle, Universitat Ramon Llull, C/Quatre Camins 30, 08022 Barcelona, Catalonia, Spain |
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| Abstract: | The generalized Taylor theorem is the foundation of the homotopy analysis method proposed by Liao. This theorem is interesting but hard to understand from the mathematical point of view. Especially, there is a key parameter h whose meaning is still unknown. In the paper, we derive the generalized Taylor theorem from a usual way, that is, we prove that the generalized Taylor expansion is equivalent to a different representation of the usual Taylor expansion at different points. Therefore the meaning of the auxiliary parameter h is clarified. These results give a reasonable explanation of the parameter h and uncover the essence of the generalized Taylor theorem from which we can deeply understand the homotopy analysis method. Through the detailed analysis of some examples, we compare the series solution at the different points with the generalized Taylor series solution obtained by the homotopy analysis method. |
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