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Journal of Applied Mathematics
Volume 2012, Article ID 420387, 17 pages
http://dx.doi.org/10.1155/2012/420387
Research Article

Optimizing a Hybrid Two-Step Method for the Numerical Solution of the Schrödinger Equation and Related Problems with Respect to Phase-Lag

1Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
2Laboratory of Computational Sciences, Department of Computer Science and Technology, Faculty of Sciences and Technology, University of Peloponnese, 221 00 Tripolis, Greece

Received 11 January 2012; Accepted 31 January 2012

Academic Editor: Kuppalapalle Vajravelu

Copyright © 2012 T. E. Simos. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Citations to this Article [88 citations]

The following is the list of published articles that have cited the current article.

  • Ibraheem Alolyan, and T.E. Simos, “A new high order two-step method with vanished phase-lag and its derivatives for the numerical integration of the Schrödinger equation,” Journal of Mathematical Chemistry, vol. 50, no. 9, pp. 2351–2373, 2012. View at Publisher · View at Google Scholar
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  • T. E. Simos, “New high order multiderivative explicit four-step methods with vanished phase-lag and its derivatives for the approximate solution of the Schrödinger equation. Part I: Construction and theoretical analysis,” Journal of Mathematical Chemistry, vol. 51, no. 1, pp. 194–226, 2012. View at Publisher · View at Google Scholar
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  • Ibraheem Alolyan, and T. E. Simos, “High order four-step hybrid method with vanished phase-lag and its derivatives for the approximate solution of the Schrödinger equation,” Journal of Mathematical Chemistry, vol. 51, no. 2, pp. 532–555, 2012. View at Publisher · View at Google Scholar
  • Norazak Senu, Mohamed Suleiman, Fudziah Ismail, and Norihan Md Arifin, “New 4(3) Pairs Diagonally Implicit Runge-Kutta-Nyström Method for Periodic IVPs,” Discrete Dynamics in Nature and Society, vol. 2012, pp. 1–20, 2012. View at Publisher · View at Google Scholar
  • Yonglei Fang, Qinghong Li, Qinghe Ming, and Kaimin Wang, “A New Optimized Runge-Kutta Pair for the Numerical Solution of the Radial Schrödinger Equation,” Abstract and Applied Analysis, vol. 2012, pp. 1–15, 2012. View at Publisher · View at Google Scholar
  • Yonglei Fang, Xiong You, and Zhaoxia Chen, “New Phase Fitted and Amplification Fitted Numerov-Type Methods for Periodic IVPs with Two Frequencies,” Abstract and Applied Analysis, vol. 2012, pp. 1–15, 2012. View at Publisher · View at Google Scholar
  • Ibraheem Alolyan, and T. E. Simos, “A new four-step hybrid type method with vanished phase-lag and its first derivatives for each level for the approximate integration of the Schrödinger equation,” Journal of Mathematical Chemistry, 2013. View at Publisher · View at Google Scholar
  • Ibraheem Alolyan, and T. E. Simos, “A new four-step Runge-Kutta type method with vanished phase-lag and its first, second and third derivatives for the numerical solution of the Schrodinger equation,” Journal Of Mathematical Chemistry, vol. 51, no. 5, pp. 1418–1445, 2013. View at Publisher · View at Google Scholar
  • G. A. Panopoulos, and T. E. Simos, “A new optimized symmetric 8-step semi-embedded predictor–corrector method for the numerical solution of the radial Schrödinger equation and related orbital problems,” Journal of Mathematical Chemistry, 2013. View at Publisher · View at Google Scholar
  • Dimitris F. Papadopoulos, and T. E. Simos, “A Modified Runge-Kutta-Nystrom Method by using Phase Lag Properties for the Numerical Solution of Orbital Problems,” Applied Mathematics & Information Sciences, vol. 7, no. 2, pp. 433–437, 2013. View at Publisher · View at Google Scholar
  • Coşar Gözükırmızı, and Metin Demiralp, “Probabilistic evolution approach for the solution of explicit autonomous ordinary differential equations. Part 1: Arbitrariness and equipartition theorem in Kronecker power series,” Journal of Mathematical Chemistry, vol. 52, no. 3, pp. 866–880, 2013. View at Publisher · View at Google Scholar
  • Ibraheem Alolyan, and T. E. Simos, “A Runge–Kutta type four-step method with vanished phase-lag and its first and second derivatives for each level for the numerical integration of the Schrödinger equation,” Journal of Mathematical Chemistry, vol. 52, no. 3, pp. 917–947, 2013. View at Publisher · View at Google Scholar
  • T. E. Simos, “An explicit four-step method with vanished phase-lag and its first and second derivatives,” Journal of Mathematical Chemistry, 2013. View at Publisher · View at Google Scholar
  • G. A. Panopoulos, and T. E. Simos, “An Optimized Symmetric 8-Step Semi-Embedded Predictor-Corrector Method for IVPs with Oscillating Solutions,” Applied Mathematics & Information Sciences, vol. 7, no. 1, pp. 73–80, 2013. View at Publisher · View at Google Scholar
  • Ghazala Akram, and Hamood Ur Rehman, “Solutions of a Class of Sixth Order Boundary Value Problems Using the Reproducing Kernel Space,” Abstract and Applied Analysis, vol. 2013, pp. 1–8, 2013. View at Publisher · View at Google Scholar
  • Jingjun Zhao, Jingyu Xiao, and Yang Xu, “Stability and Convergence of an Effective Finite Element Method for Multiterm Fractional Partial Differential Equations,” Abstract and Applied Analysis, vol. 2013, pp. 1–10, 2013. View at Publisher · View at Google Scholar
  • Ibraheem Alolyan, and T. E. Simos, “A hybrid type four-step method with vanished phase-lag and its first, second and third derivatives for each level for the numerical integration of the Schrödinger equation,” Journal of Mathematical Chemistry, 2014. View at Publisher · View at Google Scholar
  • T. E. Simos, “An explicit linear six-step method with vanished phase-lag and its first derivative,” Journal of Mathematical Chemistry, 2014. View at Publisher · View at Google Scholar
  • Ibraheem Alolyan, and T. E. Simos, “A family of explicit linear six-step methods with vanished phase-lag and its first derivative,” Journal of Mathematical Chemistry, 2014. View at Publisher · View at Google Scholar
  • Ali Shokri, and Hosein Saadat, “Trigonometrically fitted high-order predictor–corrector method with phase-lag of order infinity for the numerical solution of radial Schrödinger equation,” Journal of Mathematical Chemistry, 2014. View at Publisher · View at Google Scholar
  • T. E. Simos, “A new explicit hybrid four-step method with vanished phase-lag and its derivatives,” Journal of Mathematical Chemistry, 2014. View at Publisher · View at Google Scholar
  • Simos, “On the explicit four-step methods with vanished phase-lag and its first derivative,” Applied Mathematics and Information Sciences, vol. 8, no. 2, pp. 447–458, 2014. View at Publisher · View at Google Scholar
  • Ibraheem Alolyan, and T. E. Simos, “A predictor–corrector explicit four-step method with vanished phase-lag and its first, second and third derivatives for the numerical integration of the Schrödinger equation,” Journal of Mathematical Chemistry, 2014. View at Publisher · View at Google Scholar
  • T. E. Simos, “A new explicit four-step method with vanished phase-lag and its first and second derivatives,” Journal of Mathematical Chemistry, 2014. View at Publisher · View at Google Scholar
  • Fei Hui, and T. E. Simos, “A new family of two stage symmetric two-step methods with vanished phase-lag and its derivatives for the numerical integration of the Schrödinger equation,” Journal of Mathematical Chemistry, 2015. View at Publisher · View at Google Scholar
  • Ibraheem Alolyan, and T. E. Simos, “A high algebraic order multistage explicit four-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives for the numerical solution of the Schrödinger equation,” Journal of Mathematical Chemistry, 2015. View at Publisher · View at Google Scholar
  • Ibraheem Alolyan, and T. E. Simos, “Efficient low computational cost hybrid explicit four-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical integration of the Schrödinger equation,” Journal of Mathematical Chemistry, 2015. View at Publisher · View at Google Scholar
  • Yanping Yang, Ke Wu, and Yonglei Fang, “Exponentially fitted TDRK pairs for the Schrödinger equation,” Journal of Mathematical Chemistry, 2015. View at Publisher · View at Google Scholar
  • Ibraheem Alolyan, and T. E. Simos, “A high algebraic order predictor–corrector explicit method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the Schrödinger equation and related problems,” Journal of Mathematical Chemistry, 2015. View at Publisher · View at Google Scholar
  • Kenan Mu, and T. E. Simos, “A Runge–Kutta type implicit high algebraic order two-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of coupled differential equations arising from the Schrödinger equation,” Journal of Mathematical Chemistry, 2015. View at Publisher · View at Google Scholar
  • Hang Ning, and T. E. Simos, “A low computational cost eight algebraic order hybrid method with vanished phase-lag and its first, second, third and fourth derivatives for the approximate solution of the Schrödinger equation,” Journal of Mathematical Chemistry, 2015. View at Publisher · View at Google Scholar
  • Yusuf Dauda Jikantoro, Norazak Senu, and Fudziah Ismail, “Zero-dissipative trigonometrically fitted hybrid method for numerical solution of oscillatory problems,” Sains Malaysiana, vol. 44, no. 3, pp. 473–482, 2015. View at Publisher · View at Google Scholar
  • Y.D. Jikantoro, F. Ismail, and N. Senu, “Zero-dissipative semi-implicit hybrid method for solving oscillatory or periodic problems,” Applied Mathematics and Computation, vol. 252, pp. 388–396, 2015. View at Publisher · View at Google Scholar
  • Y.D. Jikantoro, F. Ismail, and N. Senu, “Higher order dispersive and dissipative hybrid method for the numerical solution of oscillatory problems,” International Journal of Computer Mathematics, pp. 1–13, 2015. View at Publisher · View at Google Scholar
  • Zhou Zhou, and T. E. Simos, “A new two stage symmetric two-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the radial Schrödinger equation,” Journal of Mathematical Chemistry, 2015. View at Publisher · View at Google Scholar
  • Ibraheem Alolyan, and T. E. Simos, “Family of symmetric linear six-step methods with vanished phase-lag and its derivatives and their application to the radial Schrödinger equation and related problems,” Journal of Mathematical Chemistry, 2015. View at Publisher · View at Google Scholar
  • Zacharoula Kalogiratou, Theodoros Monovasilis, and Theodore E. Simos, “Symplectic Runge-Kutta-Nyström methods with phase-lag oder 8 and infinity,” Applied Mathematics and Information Sciences, vol. 9, no. 3, pp. 1105–1112, 2015. View at Publisher · View at Google Scholar
  • Jing Ma, and Simos, “A special high order runge-kutta type method for the solution of the schrödinger equation,” Applied Mathematics and Information Sciences, vol. 9, no. 5, pp. 2559–2577, 2015. View at Publisher · View at Google Scholar
  • Simos, “On the low algebraic order explicit methods with vanished phase-lag and its first and second derivative,” Applied Mathematics and Information Sciences, vol. 9, no. 6, pp. 2905–2916, 2015. View at Publisher · View at Google Scholar
  • Zacharoula Kalogiratou, and Theodoros Monovasilis, “Diagonally implicit symplectic Runge-Kutta methods with special properties,” Applied Mathematics and Information Sciences, vol. 9, no. 1, pp. 11–17, 2015. View at Publisher · View at Google Scholar
  • Ibraheem Alolyan, and T. E. Simos, “A new two stages tenth algebraic order symmetric six-step method with vanished phase-lag and its first and second derivatives for the solution of the radial Schrödinger equation and related problems,” Journal of Mathematical Chemistry, 2016. View at Publisher · View at Google Scholar
  • Ibraheem Alolyan, and T. E. Simos, “A family of two stages tenth algebraic order symmetric six-step methods with vanished phase-lag and its first derivatives for the numerical solution of the radial Schrödinger equation and related problems,” Journal of Mathematical Chemistry, 2016. View at Publisher · View at Google Scholar
  • Ibraheem Alolyan, and T. E. Simos, “A new eight algebraic order embedded explicit six-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the Schrödinger equation,” Journal of Mathematical Chemistry, vol. 54, no. 8, pp. 1696–1727, 2016. View at Publisher · View at Google Scholar
  • Xiaopeng Xi, and T. E. Simos, “A new high algebraic order four stages symmetric two-step method with vanished phase-lag and its first and second derivatives for the numerical solution of the Schrödinger equation and related problems,” Journal of Mathematical Chemistry, 2016. View at Publisher · View at Google Scholar
  • Ibraheem Alolyan, and T. E. Simos, “A family of embedded explicit six-step methods with vanished phase-lag and its derivatives for the numerical integration of the Schrödinger equation: development and theoretical analysis,” Journal of Mathematical Chemistry, 2016. View at Publisher · View at Google Scholar
  • Minjian Liang, and T. E. Simos, “A new four stages symmetric two-step method with vanished phase-lag and its first derivative for the numerical integration of the Schrödinger equation,” Journal of Mathematical Chemistry, 2016. View at Publisher · View at Google Scholar
  • Dmitriy B. Berg, and T. E. Simos, “High order computationally economical six-step method with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation,” Journal of Mathematical Chemistry, 2016. View at Publisher · View at Google Scholar
  • Jianbin Zhao, and T. E. Simos, “A new fourteenth algebraic order finite difference method for the approximate solution of the Schrödinger equation,” Journal of Mathematical Chemistry, 2016. View at Publisher · View at Google Scholar
  • Maxim A. Medvedev, and T. E. Simos, “Two stages six-step method with eliminated phase-lag and its first, second, third and fourth derivatives for the approximation of the Schrödinger equation,” Journal of Mathematical Chemistry, vol. 55, no. 4, pp. 961–986, 2016. View at Publisher · View at Google Scholar
  • Ibraheem Alolyan, and T. E. Simos, “New two stages high order symmetric six-step method with vanished phase–lag and its first, second and third derivatives for the numerical solution of the Schrödinger equation,” Journal of Mathematical Chemistry, 2016. View at Publisher · View at Google Scholar
  • Xiong You, Yanwei Zhang, and Yonglei Fang, “Exponentially fitted multi-derivative linear methods for the resonant state of the Schrödinger equation,” Journal of Mathematical Chemistry, vol. 55, no. 1, pp. 223–237, 2016. View at Publisher · View at Google Scholar
  • Ibraheem Alolyan, and T. E. Simos, “An implicit symmetric linear six-step methods with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the radial Schrödinger equation and related problems,” Journal of Mathematical Chemistry, 2016. View at Publisher · View at Google Scholar
  • Licheng Zhang, and Theodore E. Simos, “An Efficient Numerical Method for the Solution of the Schrödinger Equation,” Advances in Mathematical Physics, vol. 2016, pp. 1–20, 2016. View at Publisher · View at Google Scholar
  • Fei Hui, and Simos, “Runge-Kutta type tenth algebraic order method with vanished phase-lag and its first, second, third, fourth and fifth derivatives for the numerical solution of the Schrödinger equation,” Applied Mathematics and Information Sciences, vol. 10, no. 1, pp. 143–153, 2016. View at Publisher · View at Google Scholar
  • Jingmei Zhou, and Simos, “Hybrid tenth algebraic order method with vanished phase-lag and its first, second, third and fourth derivatives for the approximate solution of the Schrödinger equation,” Applied Mathematics and Information Sciences, vol. 10, no. 1, pp. 193–202, 2016. View at Publisher · View at Google Scholar
  • T.E. Simos, and Ch. Tsitouras, “A new family of 7 stages, eighth-order explicit Numerov-type methods,” Mathematical Methods in the Applied Sciences, 2017. View at Publisher · View at Google Scholar
  • Jie Fang, Chenglian Liu, and T. E. Simos, “A hybrid finite difference pair with maximum phase and stability properties,” Journal of Mathematical Chemistry, 2017. View at Publisher · View at Google Scholar
  • Junfeng Yao, and T. E. Simos, “New finite difference pair with optimized phase and stability properties,” Journal of Mathematical Chemistry, 2017. View at Publisher · View at Google Scholar
  • Vladislav N. Kovalnogov, Ruslan V. Fedorov, Viktor M. Golovanov, Boris M. Kostishko, and T. E. Simos, “A four stages numerical pair with optimal phase and stability properties,” Journal of Mathematical Chemistry, 2017. View at Publisher · View at Google Scholar
  • Ke Yan, and T. E. Simos, “A finite difference pair with improved phase and stability properties,” Journal of Mathematical Chemistry, 2017. View at Publisher · View at Google Scholar
  • Maxim A. Medvedev, and T. E. Simos, “A multistep method with optimal properties for second order differential equations,” Journal of Mathematical Chemistry, 2017. View at Publisher · View at Google Scholar
  • Lan Yang, and T. E. Simos, “An efficient and economical high order method for the numerical approximation of the Schrödinger equation,” Journal of Mathematical Chemistry, 2017. View at Publisher · View at Google Scholar
  • A. Konguetsof, “A generator of families of two-step numerical methods with free parameters and minimal phase-lag,” Journal of Mathematical Chemistry, 2017. View at Publisher · View at Google Scholar
  • T. E. Simos, and Ch. Tsitouras, “Evolutionary generation of high-order, explicit, two-step methods for second-order linear IVPs,” Mathematical Methods in the Applied Sciences, 2017. View at Publisher · View at Google Scholar
  • Jing Ma, and T. E. Simos, “An efficient and computational effective method for second order problems,” Journal of Mathematical Chemistry, 2017. View at Publisher · View at Google Scholar
  • Şeyhmus Yardimci, Esra Kir Arpat, and Çağla Can, “On the structure of discrete spectrum of a non-selfadjoint system of differential equations with integral boundary condition,” Journal of Mathematical Chemistry, vol. 55, no. 5, pp. 1202–1212, 2017. View at Publisher · View at Google Scholar
  • Dmitriy B. Berg, and T. E. Simos, “Three stages symmetric six-step method with eliminated phase-lag and its derivatives for the solution of the Schrödinger equation,” Journal of Mathematical Chemistry, 2017. View at Publisher · View at Google Scholar
  • Dmitriy B. Berg, and T. E. Simos, “An efficient six-step method for the solution of the Schrödinger equation,” Journal of Mathematical Chemistry, 2017. View at Publisher · View at Google Scholar
  • Maxim A. Medvedev, and T. E. Simos, “A new six-step algorithm with improved properties for the numerical solution of second order initial and/or boundary value problems,” Journal of Mathematical Chemistry, 2017. View at Publisher · View at Google Scholar
  • Dmitry B. Berg, T. E. Simos, and Ch. Tsitouras, “Trigonometric fitted, eighth-order explicit Numerov-type methods,” Mathematical Methods in the Applied Sciences, 2017. View at Publisher · View at Google Scholar
  • Jinbin Zheng, Chenglian Liu, and T. E. Simos, “A new two-step finite difference pair with optimal properties,” Journal of Mathematical Chemistry, 2017. View at Publisher · View at Google Scholar
  • Xin Shi, and T. E. Simos, “New five-stages finite difference pair with optimized phase properties,” Journal of Mathematical Chemistry, 2017. View at Publisher · View at Google Scholar
  • Dmitriy B. Berg, and T. E. Simos, “A new multistep finite difference pair for the Schrödinger equation and related problems,” Journal of Mathematical Chemistry, 2017. View at Publisher · View at Google Scholar
  • Zhiwei Wang, and T. E. Simos, “An economical eighth-order method for the approximation of the solution of the Schrödinger equation,” Journal of Mathematical Chemistry, 2017. View at Publisher · View at Google Scholar
  • Ming Dong, and Theodore E. Simos, “A new high algebraic order efficient finite difference method for the solution of the Schrödinger equation,” Filomat, vol. 31, no. 15, pp. 4999–5012, 2017. View at Publisher · View at Google Scholar
  • Zhong Chen, Chenglian Liu, and T. E. Simos, “New three–stages symmetric two step method with improved properties for second order initial/boundary value problems,” Journal of Mathematical Chemistry, 2018. View at Publisher · View at Google Scholar
  • P. I. Stasinos, and Theodore E. Simos, “New 8-step symmetric embedded predictor–corrector (EPCM) method with vanished phase-lag and its first derivative for the numerical integration of the Schrödinger equation,” Journal of Mathematical Chemistry, 2018. View at Publisher · View at Google Scholar
  • T. E. Simos, and Ch. Tsitouras, “Fitted modifications of classical Runge-Kutta pairs of orders 5(4),” Mathematical Methods in the Applied Sciences, 2018. View at Publisher · View at Google Scholar
  • Ibraheem Alolyan, and T. E. Simos, “New three-stages symmetric six-step finite difference method with vanished phase-lag and its derivatives up to sixth derivative for second order initial and/or boundary value problems with periodical and/or oscillating solutions,” Journal of Mathematical Chemistry, 2018. View at Publisher · View at Google Scholar
  • V. N. Kovalnogov, R. V. Fedorov, A. A. Bondarenko, and T. E. Simos, “New hybrid two-step method with optimized phase and stability characteristics,” Journal of Mathematical Chemistry, 2018. View at Publisher · View at Google Scholar
  • Ke Yan, and T. E. Simos, “New Runge–Kutta type symmetric two-step method with optimized characteristics,” Journal of Mathematical Chemistry, 2018. View at Publisher · View at Google Scholar
  • Ch. Tsitouras, and I. Th. Famelis, “A new eighth order exponentially fitted explicit Numerov-type method for solving oscillatory problems,” Journal of Mathematical Chemistry, 2018. View at Publisher · View at Google Scholar
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