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An efficient wavelet-based approximation method for the coupled nonlinear fractal–fractional 2D Schrödinger equations

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Abstract

In this study, a fast semi-discrete approach based on the two-dimensional Legendre wavelets (2D LWs) is developed to provide approximate solutions of a new class of the coupled nonlinear time fractal–fractional Schrödinger equations. Although the proposed method can be applied for any fractional derivative, we focus on the Atangana–Reimann–Liouville derivative with Mittag–Leffler kernel due to its privileges. To carry out the method, first the fractal–fractional derivatives need to be discretized through the finite difference scheme and the \(\theta \)-weighted method to derive a recurrent algorithm. Then, the unknown solution of the intended problem is expanded in terms of the 2D LWs. Finally, by applying the differentiation operational matrices in each time step, the problem becomes reduced to a linear system of algebraic equations. In the proposed method, acceptable approximate solutions are achieved by employing only a few number of the basis functions. To illustrate the validity and accuracy of this practicable wavelet method, some numerical test examples are provided. The achieved numerical results manifest the exponential accuracy of approximate solutions of the new fractal–fractional model.

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References

  1. Hasegawa A (1989) Optical solitons in fibers. Springer, Berlin

    Book  Google Scholar 

  2. Scott A (1999) Nonlinear science: emergence and dynamics of coherent structures. Oxford University Press, Oxford

    MATH  Google Scholar 

  3. Kivshar Y, Agrawal G (2003) Optical solitons: from fibers to photonic crystals. Academic Press, New York

    Google Scholar 

  4. Gibbon J, Dodd R, Eilbeck J, Morris H (1982) Solitons and nonlinear wave equations. Academic Press, New York

    MATH  Google Scholar 

  5. Arnold A (1998) Numerically absorbing boundary conditions for quantum evolution equations. VLSI Design 6:313–319

    Article  Google Scholar 

  6. Lévy M (2000) Parabolic equation methods for electromagnetic wave propagation. IEE

  7. Tappert FD (1977) The parabolic approximation method. In: Keller JB, Papadakis JS (eds) Wave propagation and underwater acoustics. Lecture notes in physics, vol 70. Springer, Berlin, pp 224–287

    Chapter  Google Scholar 

  8. Hamed SHM, Yousif EA, Arbab AI (2014) Analytic and approximate solutions of the space-time fractional Schrödinger equations by homotopy perturbation sumudu transform method. Abstract and Applied Analysis

  9. Abdelkawy MA, Bhrawya AH (2015) A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations. J Comput Phys 294:462–483

    Article  MathSciNet  Google Scholar 

  10. Sahadevan R, Bakkyaraj T (2015) Approximate analytical solution of two coupled time fractional nonlinear Schrödinger equations. Int J Appl Comput Math

  11. Abdel-Salama EAB, Yousif EA, El-Aasser MA (2017) On the solution of the space-time fractional cubic nonlinear Schrödinger equation. Physics

  12. Heydari MH, Atangana A (2019) A cardinal approach for nonlinear variable-order time fractional Schrödinger equation defined by Atangana–Baleanu–Caputo derivative Chaos. Solitons Fractals 128:339–348

    Article  MathSciNet  Google Scholar 

  13. Zhang H, Jiang X (2019) Spectral method and bayesian parameter estimation for the space fractional coupled nonlinear Schrödinger equations. Nonlinear Dyn 95(2):1599–1614

    Article  Google Scholar 

  14. Eslami M (2016) Exact traveling wave solutions to the fractional coupled nonlinear Schröodinger equations. Appl Math Comput 285:141–148

    MathSciNet  MATH  Google Scholar 

  15. Wang DL, Xiao AG, Yang W (2014) A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schrödinger equations. J Comput Phys 272:644–655

    Article  MathSciNet  Google Scholar 

  16. Ran MH, Zhang CJ (2016) A conservative difference scheme for solving the strongly coupled nonlinear fractional Schrödinger equations. Commun Nonlinear Sci Numer Simul 41:64–83

    Article  MathSciNet  Google Scholar 

  17. Ran M, Zhang C (2019) Linearized Crank-Nicolson scheme for the nonlinear time-space fractional Schrödinger equations. J Comput Appl Math 355:218–231

    Article  MathSciNet  Google Scholar 

  18. Lenzi EK, de Castro ASM, Mendes RS (2019) Time dependent solutions for fractional coupled Schrödinger equations. Appl Math Comput 346:6228–632

    Google Scholar 

  19. Wei L, Zhang X, Kumar S, Yildirim A (2012) A numerical study based on an implicit fully discrete local discontinuous Galerkin method for the time-fractional coupled Schrödinger system. Comput Math Appl 64(8)

  20. Li M, Gu X, Huang C, Fei M, Zhang G (2018) A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrödinger equations. J Comput Phys 358:256–282

    Article  MathSciNet  Google Scholar 

  21. Wang D, Xiao A, Yang W (2013) Crank-nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative. J Comput Phys 242:670–681

    Article  MathSciNet  Google Scholar 

  22. Heydari MH, Hooshmandasl MR, Mohammadi F (2014) Two-dimensional Legendre wavelets for solving time-fractional telegraph equation. Adv Appl Math Mech 6(2):247–260

    Article  MathSciNet  Google Scholar 

  23. Heydari MH (2016) Wavelets Galerkin method for the fractional subdiffusion equation. J Comput Nonlinear Dyn 11(6):061014–4

    Article  Google Scholar 

  24. Heydari MH, Avazzadeh Z (2018) A new wavelet method for variable-order fractional optimal control problems. Asian J Control 20(5):1–14

    Article  MathSciNet  Google Scholar 

  25. Hosseininia M, Heydari MH, Maalek Ghaini FM, Avazzadeh Z (2018) Two-dimensional Legendre wavelets for solving variable-order fractional nonlinear advection-diffusion equation with variable coefficients. Int J Nonlinear Sci Numer Simul 19(7–8):793–802

    Article  MathSciNet  Google Scholar 

  26. Hosseininia M, Heydari MH, Roohi R, Avazzadeh Z (2019) A computational wavelet method for variable-order fractional model of dual phase lag bioheat equation. J Comput Phys 395:1–18

    Article  MathSciNet  Google Scholar 

  27. Hosseininia M, Heydari MH, Maalek Ghaini FM, Avazzadeh Z (2019) A wavelet method to solve nonlinear variable-order time fractional 2D Klein–Gordon equation. Comput Math Appl 78(2):3713–3730

    Article  MathSciNet  Google Scholar 

  28. Hosseininia M, heydari MH (2019) Legendre wavelets for the numerical solution of nonlinear variable-order time fractional 2D reaction-diffusion equation involving Mittag–Leffler non-singular kernel. Chaos Solitons Fract 127:400–407

    Article  MathSciNet  Google Scholar 

  29. Cai J (2010) Multisymplectic schemes for strongly coupled Schrödinger system. Appl Math Comput 216:2417–2429

    MathSciNet  MATH  Google Scholar 

  30. Atangana A, Qureshi S (2019) Modeling attractors of chaotic dynamical systems with fractal-fractional operators. Chaos Solitons Fract 123:320–337

    Article  MathSciNet  Google Scholar 

  31. Atangana A (2017) Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system. Chaos Solitons Fract 123:1–1

    MathSciNet  MATH  Google Scholar 

  32. Heydari MH, Atangana A, Avazzadeh Z (2019) Chebyshev polynomials for the numerical solution of fractal-fractional model of nonlinear Ginzburg–Landau equation. Eng Comput. https://doi.org/10.1007/s00366-019-00889-9

  33. Quarteroni A, Canuto C, Hussaini M, Zang T (1998) Spectral methods in fluid dynamics. Springer, Berlin

    MATH  Google Scholar 

  34. Song J, Yin F, Lu F (2014) A coupled method of Laplace transform and Legendre wavelets for nonlinear Klein–Gordon equations. Math Methods Appl Sci 37:781–791

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Shiraz University of Technology, Shiraz, Iran

    M. H. Heydari

  2. Faculty of Mathematics, Yazd University, Yazd, Iran

    M. Hosseininia

  3. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Z. Avazzadeh

Authors
  1. M. H. Heydari
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  2. M. Hosseininia
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  3. Z. Avazzadeh
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Correspondence to Z. Avazzadeh.

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Heydari, M.H., Hosseininia, M. & Avazzadeh, Z. An efficient wavelet-based approximation method for the coupled nonlinear fractal–fractional 2D Schrödinger equations. Engineering with Computers 37, 2129–2144 (2021). https://doi.org/10.1007/s00366-020-00934-y

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