A non-iterative transformation method for Newton's free boundary problem |
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| Institution: | 1. COMIMSA, Corporación Mexicana de Investigación en Materiales, Ciencia y Tecnología No. 790, Fracc. Saltillo 400, C. P. 25290, Saltillo, Coah., Mexico;2. CIIDIT, Centro de Innovación, Investigación y Desarrollo en Ingeniería y Tecnología, Universidad Autónoma de Nuevo León, Nueva carretera al Aeropuerto Internacional de Monterrey, Km. 10 Apodaca, Nuevo León, Mexico;1. State Key Laboratory for Chemo/Biosensing and Chemometrics, Hunan Key Laboratory of Two-Dimensional Materials, and College of Chemistry and Chemical Engineering, Hunan University, Changsha 410082, China;2. School of Physics and Electronics, Hunan University, Changsha 410082, China;1. Acoustics & Vibration Unit, School of Engineering and Information Technology, The University of New South Wales Canberra, Canberra, ACT 2600, Australia;2. Centre for Audio, Acoustics and Vibration, Faculty of Engineering and Information Technology, University of Technology Sydney, Sydney, NSW 2007, Australia;1. MVZ Labor Dr. Limbach & Kollegen GbR, Heidelberg, Germany;2. MVZLM - Medizinisches Versorgungszentrum für Labormedizin und Mikrobiologie Ruhr GmbH, Essen, Germany;3. Roche Diagnostics GmbH, Mannheim, Germany;4. Roche Diagnostics GmbH, Penzberg, Germany;5. Institute for Pharmacology and Preventive Medicine, Mahlow, Germany;6. Roche Diagnostics International Ltd. Rotkreuz, Switzerland |
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| Abstract: | In book II of Newton's Principia Mathematica of 1687 several applicative problems are introduced and solved. There, we can find the formulation of the first calculus of variations problem that leads to the first free boundary problem of history. The general calculus of variations problem is concerned with the optimal shape design for the motion of projectiles subject to air resistance. Here, for Newton's optimal nose cone free boundary problem, we define a non-iterative initial value method which is referred in the literature as a transformation method. To define this method we apply invariance properties of Newton's free boundary problem under a scaling group of point transformations. Finally, we compare our non-iterative numerical results with those available in the literature and obtained via an iterative shooting method. We emphasize that our non-iterative method is faster than shooting or collocation methods and does not need any preliminary computation to test the target function as the iterative method or even provide any initial iterate. Moreover, applying Buckingham Pi-Theorem we get the functional relation between the unknown free boundary and the nose cone radius and height. |
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| Keywords: | Newton's free boundary problem Scaling invariance Non-iterative numerical method Buckingham Pi-Theorem |
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