موجک‌های لژاندر برای حل عددی دستگاهی از معادلات شرودینگر دوبعدی غیرخطی کسری- فراکتالی

Document Type : Research Paper

Authors

1 بخش ریاضی کاربردی، دانشکده علوم ریاضی، دانشگاه یزد، استان یزد، ایران

2 گروه ریاضی کاربردی، دانشکده ریاضی، دانشگاه صنعتی شیراز، استان فارس، ایران

10.22072/wala.2020.119323.1263

Abstract

در این  مقاله، یک روش نیمه‌گسسته بر اساس موجک‌های لژاندر دوبعدی  را برای به‌دست آوردن جواب‌های تقریبی دستگاهی از معادلات شرودینگر غیرخطی کسری فراکتالی ارایه می‌دهیم. با وجود اینکه روش پیشنهادی  را می‌توان برای هر  نوع مشتق کسری (هسته نامنفرد) به‌کار برد،  ولی  در این مقاله روی مشتق ریمان- لیوویل- آتانگانا با هسته نامنفرد میتاگ- لفلر تمرکز می‌کنیم.   در این راستا ابتدا مشتقات کسری- فراکتال زمانی توسط روش تفاضلات متناهی تقریب زده می‌شوند. سپس از روش  تفاضلات متناهی وزن‌دار شده با پارامتر تتا  برای به‌دست آوردن رابطه بازگشتی مساله استفاده می‌شود.

Keywords


[1] A. Hasegawa, Optical Solitons in Fibers, Berlin, Springer-Verlag, 1989.
[2] K. Y and G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, New York, Academic Press, 2003.
[3] J. Gibbon, R. Dodd, J. Eilbeck and H. Morris, Solitons and Nonlinear Wave Equations, New York, Academic Press, 1982.
[4] A. Arnold, Numerically absorbing boundary conditions for quantum evolution equations, VLSI Design, 6 (1998),   
     313-319.
[5] M. L'{e}vy, Parabolic Equation Methods for Electromagnetic Wave Propagation, IEE, 2000.
[6] J. Xin, J. Hu and H. Lu, The global solution for a class of systems of fractional nonlinear Schr"{o}dinger equations with 
      periodic boundary condition, Comput. Math. Appl., 62(3) (2011), 1510-1521.
[7] Khaled A. Gepreel and M. Herzallah, Approximate solution to the time–space fractional cubic nonlinear Schr"{o}dinger 
      equation, Appl. Math. Modelling, 36 (2012), 5678-5685.
[8] S.H.M. Hamed, E.A. Yousif and A.I. Arbab, Analytic and approximate solutions of the space-time fractional
     Schr"{o}dinger equations by homotopy perturbation sumudu transform method, Abstr. Appl. Anal., https://doi.org 
      /10.1155/2014/863015, 2014.
[9] M.A. Abdelkawy and A.H. Bhrawya, A fully spectral collocation approximation for multi-dimensional fractional 
      schr"{o}dinger equations, J. Comput. Phys., 294 (2015), 462-483.
[10] R. Sahadevan and T. Bakkyaraj, Approximate analytical solution of two coupled time fractional nonlinear 
        schr"{o}dinger equations, Int. J. Appl. Comput. Math., 2(1) (2015), 113-135.
[11] E.A-B. Abdel-Salama, E.A. Yousif and M.A. El-Aasser, On the solution of the space-time fractional cubic nonlinear
       schr"{o}dinger equation, Results in Physics, 8 (2018), 702-708.
[12] M. Hosseininia, M.H. Heydari and C. Cattani, A wavelet method for nonlinear variable-order time fractional 2D 
       Schr"{o}dinger equation, Discrete Contin. Dyn. Syst., Ser. S, doi: 10.3934/dcdss.2020295, 2019.
[13] H. Zhang and X. Jiang, Spectral method and bayesian parameter estimation for the space fractional coupled 
       nonlinear schr"{o}dinger equations, Nonlinear Dyn., 95(2) (2019), 1599-1614.   
[14] M.Eslami, Exact traveling wave solutions to the fractional coupled nonlinear schr"{o}odinger equations, Appl. Math. 
       Comput., 285 (2016), 141-148.
[15] D.L. Wang, A.G. Xiao and W. Yang, A linearly implicit conservative difference scheme for the space fractional coupled 
        nonlinear schr"{o}dinger equations, J. Comput. Phys., 272 (2014), 644-655.
[16] M.H. Ran and C.J. Zhang, A conservative difference scheme for solving the strongly coupled nonlinear fractional 
       schr"{o}dinger equations, Commun. Nonlinear Sci. Numer. Simul., 41 (2016), 64-83.
[17] M. Ran and C. Zhang, Linearized crank-nicolson scheme for the nonlinear timespace fractional schr"{o}dinger 
        equations, J. Comput. Appl. Math., 355 (2019), 218-231.
[18] E.K. Lenzi, A.S.M. de Castro and R.S. Mendes, Time dependent solutions for fractional coupled schr"{o}dinger 
       equations, Appl. Math. Comput., 346(C) (2019), 622-632.
[19] L. Wei, X. Zhang, S. Kumar and A. Yildirim, A numerical study based on an implicit fully discrete local discontinuous 
       galerkin method for the time-fractional coupled schr"{o}dinger system, Comput. Math. Appl., 64(8) (2012), 2603-2615.
[20] M. Li, X. Gu, C.~Huang, M. Fei and G. Zhang, A fast linearized conservative finite element method for the strongly 
        coupled nonlinear fractional schr"{o}dinger equations, J. Comput. Phys., 358 (2018), 256-282.
[21] D. Wang, A. Xiao and W. Yang, Crank-nicolson difference scheme for the coupled nonlinear schr"{o}dinger equations 
        with the riesz space fractional derivative, J. Comput. Phys., 242 (2013), 670-681.
[22] M.H. Heydari, M.R. Hooshmandasl and F. Mohammadi, Two-dimensional Legendre wavelets for solving time-
       fractional telegraph equation, Adv. Appl. Math. Mech., 6(2) (2014), 247-260.
[23] M.H. Heydari, M.R. Hooshmandasl, F.M. Maalek Ghaini and C. Cattani, Wavelets method for the time fractional 
       diffusion-wave equation, Phys. Lett., A, 379 (2015), 71-76.
[24] M.H. Heydari, M.R. Hooshmandasl, F.M. Maalek Ghaini and C. Cattani, Wavelets method for solving fractional optimal
       control problems, Appl. Math. Comput., 286 (2016), 139-154.
[25] M.H. Heydari, Wavelets Galerkin method for the fractional subdiffusion equation,J. Comput. Nonlinear Dynam., 11(6) 
        (2016), 1-7.
[26] M.H. Heydari and Z. Avazzadeh, Legendre wavelets optimization method for variable-order fractional Poisson 
        equation, Chaos Solitons Fractals, 112 (2018), 180-190.
[27] M.H. Heydari and Z. Avazzadeh, A new wavelet method for variable-order fractional optimal control problems, Asian 
        J. Control, 20(5) (2018), 1-14.
[28] M. Hosseininia, M.H. Heydari, R. Roohi and Z. Avazzadeh, A computational wavelet method for variable-order 
       fractional model of dual phase lag bioheat equation, J. Comput. Phys., 395 (2019), 1-18.
[29] M. Hosseininia, M.H. Heydari, F.M. Maalek Ghaini and Z. Avazzadeh, A wavelet method to solve nonlinear variable-
       order time fractional 2D Klein–Gordon equation, Comput. Math. Appl., 78 (2019), 3719-3730.
[30] M. Hosseininia and M.H. heydari, Legendre wavelets for the numerical solution of nonlinear variable-order time 
       fractional 2D reaction-diffusion equation involving Mittag-Leffler non-singular kernel, Chaos Solitons Fractals, 127 
       (2019), 400-407.
[31] A. Atangana and S. Qureshi, Modeling attractors of chaotic dynamical systems with fractal-fractional operators, 
       Chaos Solitons Fractals, 123 (2019), 320-337.
[32] A. Atangana, Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to 
       predict complex system, Chaos Solitons Fractals, 102 (2017), 396-406.
[33] A. Quarteroni, C. Canuto, M. Hussaini and T. Zang, Spectral Methods in Fluid Dynamics, Springer, Berlin, 1998.
[34] J.Song, F.Yin and F.Lu, A coupled method of laplace transform and Legendre wavelets for nonlinear Klein-Gordon 
       equations, Math. Methods Appl. Sci., 37 (2014), 781-791.