Research Article
Kuantum mekaniğinde Schrödinger denklemlerini ele almak için varyasyonel iterasyon yönteminin bazı etkili şemaları
Abstract
Atomik düzeyde evreni ve günlük yaşamı tanımlayan temel diferansiyel denklemlerden biri kuantum mekaniğindeki Schrödinger denklemidir. Schrödinger denklemlerinin çözümleri araştırmacıların ilgisini çekmiştir. Bu çalışmada, üç boyutlu lineer ve lineer olmayan zamana bağlı Schrödinger denklemlerini ele almak için varyasyonel iterasyon yöntemi kullanılmıştır. Lagrange çarpanları ile farklı iterasyon formülleri oluşturulmuştur. Varyasyonel iterasyon yönteminin farklı iterasyon formülleri kullanılarak elde edilen yaklaşık çözümlerin doğruluğu sayısal örneklerle verilmiştir. Yaklaşık çözümlerin ve tam çözümlerin karşılaştırmaları grafiklerle gösterilmiştir. Ayrıca bu karşılaştırma sonuçları için mutlak hata tablolarına da yer verilmiştir. Sonuç olarak, varyasyonel iterasyon yönteminin farklı iterasyon formülleri sayesinde tam çözüme daha hızlı yakınsayan yaklaşımlar verdiği görülmektedir.
References
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- Carinena, J. F., Ibort, A., Marmo, G. ve Morandi, G., Geometry from Dynamics, Classical and Quantum, Springer, London, (2015).
- He, J. H., A New Approach to Nonlinear Partial Differential Equations, Communications in Nonlinear Science and Numerical Simulation, 2, 4, 230-235, (1997).
- Abdou, M. A. ve Soliman, A. A., Variational iteration method for solving Burger’s and coupled Burger’s Equation, Journal of Computational and Applied Mathematics, 181, 2, 245-251, (2005).
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- Hemeda, A. A., Variational iteration method for solving wave equation, Computers and Mathematics with Applications, 56, 8, 1948-1953, (2008).
- He, J. H. ve Latifizadeh, H., A General Numerical Algorithm for Nonlinear Differantial Equations by the Variational Iteration Method, International Journal of Numerical Methods for Heat & Fluid Flow, 30, 11, 4797-4810, (2020).
- Mohyud-Din, S. T., Noor, M. A. ve Noor, K. I., Variational Iteration Method for Solving Telegraph Equations, Applications and Applied Mathematics: An International Journal, 4, 1, 114-121, (2009).
- Li, H. L., Xiao, A. G. ve Zhao, Y. X., Variational Iteration Method for Delay Differential-Algebraic Equations, Mathematical and Computational Applications, 15, 5, 834-839, (2010).
- Sweilam, N. H., Variational iteration method for solving cubic nonlinear Schrödinger equation, Journal of Computational and Applied Mathematics, 207, 1, 155-163, (2007).
- Wazwaz, A. M., A study on linear and nonlinear Schrodinger equations by the variational iteration method, Chaos Solitons Fractals, 37, 4, 1136-1142, (2008).
- Hosseinzadeh, Kh., An analytic Approximation to the Solution of Schrödinger Equation by VIM, Applied Mathematical Sciences, 11, 16, 813-818, (2017).
- Alipour, M. ve Vali, M. A., Approximate Optimal Control of Volterra-Fredholm Integral Equations Based on Parametrization and Variational Iteration Method, Mathematical Communications, 26, 1, 107-119, (2021).
- Shihab, M. A., Taha, W. M., Hameed, R. A., Jameel, A. ve Sulaiman, I. M., Implementation of variational iteration method for various types of linear and nonlinear partial differential equations, International Journal of Electrical and Computer Engineering, 13, 2, 2131-2141, (2023).
Some efficient schemes of variational iteration method for handling Schrödinger equations in quantum mechanics
Abstract
At the atomic level, one of the fundamental differential equations that describe the universe and everyday life is the Schrödinger equation in quantum mechanics. Solutions of Schrödinger equations have attracted the attention of researchers. In this study, the variational iteration method is used to handle three dimensional linear and nonlinear time dependent Schrödinger equations. Different iteration formulas have been constructed with the Lagrange multipliers. The accuracy of the approximate solutions obtained by using different iteration formulas of the variational iteration method is given with numerical examples. Comparisons of approximate solutions and exact solutions are shown with graphs. In addition, absolute error tables are included for these comparison results. As a result, it is seen that the variational iteration method gives approximations that converge to the exact solution more rapidly thanks to different iteration formulas.
References
- Yokus, A., Construction of different types of traveling wave solutions of the relativistic wave equation associated with the Schrödinger equation, Mathematical Modelling and Numerical Simulation with Applications, 1, 1, 24-31, (2021).
- Yokus, A., Tuz, M. ve Güngöz, U., On the exact and numerical complex travelling wave solution to the nonlinear Schrödinger equation, Journal of Difference Equations and Applications, 27, 2, 195-206, (2021).
- Yokus, A. ve Yavuz, M., Solutions of Modified Schrödinger Equation by Using Analytical and Numerical Methods, Book of Abstracts, Proceedings, International Conference on Applied Mathematics in Engineering (ICAME), 151, Balikesir, (2021).
- Duran, S., Durur, H., Yavuz, M. ve Yokus, A., Discussion of numerical and analytical techniques for the emerging fractional order murnaghan model in materials science, Optical and Quantum Electronics, 55, 571, (2023).
- Yavuz, M. ve Yokus, A., Analytical and numerical approaches to nerve impulse model of fractional-order, Numerical Methods for Partial Differential Equations, 36, 6, 1348-1368, (2020).
- Pitaevskii, L. ve Stringari, S., Bose-Einstein Condensation, Clarendon Press, Oxford, (2003).
- Carinena, J. F., Ibort, A., Marmo, G. ve Morandi, G., Geometry from Dynamics, Classical and Quantum, Springer, London, (2015).
- He, J. H., A New Approach to Nonlinear Partial Differential Equations, Communications in Nonlinear Science and Numerical Simulation, 2, 4, 230-235, (1997).
- Abdou, M. A. ve Soliman, A. A., Variational iteration method for solving Burger’s and coupled Burger’s Equation, Journal of Computational and Applied Mathematics, 181, 2, 245-251, (2005).
- Abassy, T. A., El-Tawil, M. A. ve El-Zoheiry, H., Modified variational iteration method for Boussinesq equation, Computers and Mathematics with Applications, 54, 7-8, 955-965, (2007).
- He, J. H. ve Wu, X. H., Variational Iteration method: New development and applications, Computers and Mathematics with Applications, 54, 7-8, 881-894, (2007).
- Yang, Y. J. ve Hua, L. Q., Variational Iteration Transform Method for Fractional Differential Equations with Local Fractional Derivative, Abstract and Applied Analysis, 2014, Article ID 760957, (2014).
- Hemeda, A. A., Variational iteration method for solving wave equation, Computers and Mathematics with Applications, 56, 8, 1948-1953, (2008).
- He, J. H. ve Latifizadeh, H., A General Numerical Algorithm for Nonlinear Differantial Equations by the Variational Iteration Method, International Journal of Numerical Methods for Heat & Fluid Flow, 30, 11, 4797-4810, (2020).
- Mohyud-Din, S. T., Noor, M. A. ve Noor, K. I., Variational Iteration Method for Solving Telegraph Equations, Applications and Applied Mathematics: An International Journal, 4, 1, 114-121, (2009).
- Li, H. L., Xiao, A. G. ve Zhao, Y. X., Variational Iteration Method for Delay Differential-Algebraic Equations, Mathematical and Computational Applications, 15, 5, 834-839, (2010).
- Sweilam, N. H., Variational iteration method for solving cubic nonlinear Schrödinger equation, Journal of Computational and Applied Mathematics, 207, 1, 155-163, (2007).
- Wazwaz, A. M., A study on linear and nonlinear Schrodinger equations by the variational iteration method, Chaos Solitons Fractals, 37, 4, 1136-1142, (2008).
- Hosseinzadeh, Kh., An analytic Approximation to the Solution of Schrödinger Equation by VIM, Applied Mathematical Sciences, 11, 16, 813-818, (2017).
- Alipour, M. ve Vali, M. A., Approximate Optimal Control of Volterra-Fredholm Integral Equations Based on Parametrization and Variational Iteration Method, Mathematical Communications, 26, 1, 107-119, (2021).
- Shihab, M. A., Taha, W. M., Hameed, R. A., Jameel, A. ve Sulaiman, I. M., Implementation of variational iteration method for various types of linear and nonlinear partial differential equations, International Journal of Electrical and Computer Engineering, 13, 2, 2131-2141, (2023).
There are 21 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Numerical Solution of Differential and Integral Equations |
| Journal Section | Research Articles |
| Authors | |
| Early Pub Date | January 16, 2025 |
| Publication Date | January 20, 2025 |
| Submission Date | April 19, 2024 |
| Acceptance Date | October 3, 2024 |
| Published in Issue | Year 2025 Volume: 27 Issue: 1 |