Symplectic and symmetric methods for the numerical solution of some mathematical models of celestial objects
Subject Areas : Statistics
1 - Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
2 - Department of Mathematics, Faculty of Sciences, Golestan University,Gorgan, Iran
Keywords: معادلات دیفرانسیل معمولی, دستگاههای هامیلتونی, روش رانگ-کوتا و رانگ-کوتا افر, همتافتگی و تقارن,
Abstract :
In the last years, the theory of numerical methods for system of non-stiff and stiff ordinary differential equations has reached a certain maturity. So, there are many excellent codes which are based on Runge–Kutta methods, linear multistep methods, Obreshkov methods, hybrid methods or general linear methods. Although these methods have good accuracy and desirable stability properties such as A-stability and L-stability, they are not suitable for the numerical solution of special classes of problems arising from different research areas, for example the mathematical models of celestial objects which are Hamiltonian systems. Since the solution of such problems has special geometric property such as symplecticity and usually reversibility. Therefore, it is natural to search for numerical methods that share this property. It is the purpose of this paper to design high order symplectic and symmetric methods. Efficiency and accuracy of the constructed methods are confirmed by implementing on well-known Hamiltonian problems of the motions of celestial objects.
[1] Andersen, P., Petersen, N. C., (1993), A procedure for ranking efficient units in data envelopment analysis Management Science 39, 1261-1264.
[2] Banker, R. D., Chang, H., (2006), The super-efficiency procedure for outlier identification, not for ranking efficient units, Euopean Journal of Operation Research 175 (2), 1311-1320.
[3] Charnes, A., Cooper, W.W. Rhodes, E., (1987), Measuring the efficiency of decisions making units, European Journal of Operational Research 2, 429-444.
[4] Chen, Y., (2004) Ranking efficient units in DEA, Omega 32 (3), 213-219.
[5] Chen, Y., Cook, W. D., Li, N., Zho, J. (2009), Additive efficiency decomposition in two stage DEA, European Journal of Operational Research 196, 1170-1176.
[6] Chen, Y., Cook, W. D., Zhu, J., (2010), Deriving the DEA frontier for two-stage processes, European Journal of Operational Research 202, 138-142.
[7] Chen, Y., Zhu, J., (2004), Measuring Information Technology's Indirect Impact on Firm Performance, Information Technology and Management 5, 9-22.
[8] Halkos, G. E., Tzermes, N. G., Kourtzidis, S. A., (2014), A unified classification of two-stage DEA models, Surveys in Operations Research and Management Science 19, 1-16.
[9] Li, S., Jahanshahloo, G.R., Khodabakhshi, M., (2007), A super-efficiency model for ranking efficient units in data envelopment analysis, Applied Mathematics and Computation 184 (2), 638-648.
[10] Liu, F.H.F., Peng, H. H., (2006), Ranking of units on the DEA frontier with common weights, Computer & Operation Research 35, 1624- 1637.
[11] Lovell, C. A. K, Rouse, A. P. B., (2003), Equivalent standard DEA models to provide superefficiency scores, Journal of the Operational Reseach Society 54 (1), 101-108.
[12] Obata, T., Ishii, H., (2003), A method of discriminating efficient candidates with ranked voting data, European Jouurnal of Operational Research 151, 233-237.
[13] Seiford, L. M., Zhu, J., (1999), Infeasibility of super-efficiency data envelopment analysis models, INFOR 37 (2), 174-187.
[14] Seiford, L. M., Zhu, J., (1999), Profitability and marketability of the top 55 US commercial banks, Management Science 45 (9), 1270-1288.
[15] Sexton, T. R., Silkman, R. H., Hogan, A. J., (1986), Data envelopment analysis; Critique and extensions, in; R. H. Silkman (Ed.), Measuring Efficienency An Assessment of Data Envelopment Analysis, Jossey-Bass, San Francisco, CA, 73-105.
