Two discretisations of the Ermakov-Pinney equation |
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| Institution: | 1. Institute of Mathematics and Statistics, Cornwallis Building, University of Kent, Canterbury, Kent CT2 7NF, UK;2. Dienst Theoretische Natuurkunde, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium;1. Locum Consultant, East Surrey Hospital, Redhill RH15RH, UK;2. Dept of Orthopaedics, East Surrey Hospital, Redhill RH15RH, UK;1. School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China;2. Center for Interdisciplinary Studies, Lanzhou University, Lanzhou 730000, China;3. Beijing Computational Science Research Center, Beijing 100084, China;1. Department of Science and Technology, Campus Norrkoping, Linkoping University, SE-60174 Norrkoping, Sweden;2. Department of Physics, Chemistry and Biology (IFM), Linköping University, SE-58183 Linköping, Sweden;1. Department of Ophthalmology, CHR Metz-Thionville, Mercy Hospital, 1, allée du Château, 57085 Metz cedex 03, France;2. Clinical Research Support Unit, Metz-Thionville Regional Hospital Center, Mercy Hospital, Metz, France |
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| Abstract: | We propose two candidates for discrete analogues to the nonlinear Ermakov-Pinney equation. The first one based on an association with a two-dimensional conformal mapping defines a second-degree difference scheme. It possesses the same features as in the continuum: a nonlinear superposition principle relating its general solution to a second-order linear difference equation and by direct linearisation a relationship with a third-order difference equation. The second form, which is new, is obtained from a slight improvement of the superposition principle. It has the advantage of leading to a first degree difference scheme and preserves all the nice properties of its linearisation. |
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