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Chebyshev polynomials for the numerical solution of fractal–fractional model of nonlinear Ginzburg–Landau equation

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Abstract

This paper introduces a new version for the nonlinear Ginzburg–Landau equation derived from fractal–fractional derivatives and proposes a computational scheme for their numerical solutions. The fractal–fractional derivative is defined in the Atangana–Riemann–Liouville sense with Mittage–Leffler kernel. The proposed approach is based on the shifted Chebyshev polynomials (S-CPs) and the collocation scheme. Through the way, a new operational matrix (OM) of fractal–fractional derivative is derived for the S-CPs and used in the presented method. More precisely, the unknown solution is separated into their real and imaginary parts, and then, these parts are expanded in terms of the S-CPs with undetermined coefficients. These expansions are substituted into the main equation and the generated operational matrix is utilized to extract a system of nonlinear algebraic equations. Thereafter, the yielded system is solved to obtain the approximate solution of the problem. The accuracy of the proposed approach is examined through some numerical examples. Numerical results confirm the suggested approach is very accurate to provide satisfactory results.

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References

  1. Atangana A (2016) Derivative with two fractional orders: a new avenue of investigation toward revolution in fractional calculus. Eur Phys J Plus 131:373

    Article  Google Scholar 

  2. Atangana A (2018) Non validity of index law in fractional calculus: a fractional differential operator with Markovian and non-Markovian properties. Phys A 505:688–706

    Article  MathSciNet  Google Scholar 

  3. Hosseininia M, Heydari MH, Roohi R, Avazzadehd 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 

  4. Roohi R, Heydari MH, Sun HG (2019) Numerical study of unsteady natural convection of variable-order fractional Jeffrey nanofluid over an oscillating plate in a porous medium involved with magnetic, chemical and heat absorption effects using Chebyshev cardinal functions. Eur Phys J Plus 134:535

    Article  Google Scholar 

  5. Engheta N (1996) On fractional calculus and fractional multipoles in electromagnetism. Antennas Propag 44:554–566

    Article  MathSciNet  Google Scholar 

  6. Kulish VV, Lage JL (2002) Application of fractional calculus to fluid mechanics. J Fluids Eng 124(3):803–806

    Article  Google Scholar 

  7. Bagley RL, Torvik PJ (1985) Fractional calculus in the transient analysis of viscoelastically damped structures. AIAA J 23:918–925

    Article  Google Scholar 

  8. Lederman C, Roquejoffre JM, Wolanski N (2004) Mathematical justification of a nonlinear integrodifferential equation for the propagation of spherical flames. Annali di Matematica 183:173–239

    Article  MathSciNet  Google Scholar 

  9. Meral FC, Royston TJ, Magin R (2010) Fractional calculus in viscoelasticity: an experimental study. Commun Nonlinear Sci Numer Simul 15:939–945

    Article  MathSciNet  Google Scholar 

  10. Podlubny I (1999) Fractional differential equations. Academic Press, San Diego

    MATH  Google Scholar 

  11. Oldham KB, Spanier J (1974) The fractional calculus. Academic Press, New York

    MATH  Google Scholar 

  12. Baleanu D, Diethelm K, Scalas E, Trujillo JJ (2012) Fractional calculus models and numerical methods, series on complexity, nonlinearity and chaos. World Scientific, Boston

    Book  Google Scholar 

  13. Diethelm K, Ford NJ (2002) Analysis of fractional differential equations. J Math Anal Appl 265:229–248

    Article  MathSciNet  Google Scholar 

  14. Wess W (1996) The fractional diffusion equation. J Math Phys 27:2782–2785

    Article  MathSciNet  Google Scholar 

  15. Langlands TAM, Henry BI (2005) The accuracy and stability of an implicit solution method for the fractional diffusion equation. J Comput Phys 205(2):719–736

    Article  MathSciNet  Google Scholar 

  16. Heydari MH, Hooshmandasl MR, Mohammadi F (2014) Legendre wavelets method for solving fractional partial differential equations with Dirichlet boundary conditions. Appl Math Comput 234:267–276

    MathSciNet  MATH  Google Scholar 

  17. Heydari MH, Hooshmandasl MR, Mohammadi F, Cattani C (2014) Wavelets method for solving systems of nonlinear singular fractional Volterra integro-differential equations. Commun Nonlinear Sci Numer Simul 19:37–48

    Article  MathSciNet  Google Scholar 

  18. Heydari MH, Hooshmandasl MR, Cattani C (2016) Numerical solution of fractional sub-diffusion and time-fractional diffusion-wave equations via fractional-order Legendre functions. Eur Phys J Plus 131:268–290

    Article  Google Scholar 

  19. Heydari MH (2019) Numerical solution of nonlinear 2D optimal control problems generated by Atangana–Riemann–Liouville fractal–fractional derivative. Appl Num Math. https://doi.org/10.1016/j.apnum.2019.10.020

    Article  MATH  Google Scholar 

  20. Aranson IS, Kramer L (2002) The world of the complex Ginzburg–Landau equation. Rev Mod Phys 74:99–143

    Article  MathSciNet  Google Scholar 

  21. Goyal A, Alka, Raju T S, Kumar C N (2012) Lorentzian-type soliton solutions of AC-driven complex Ginzburg–Landau equation. Appl Math Comput 218:11931–11937

    MathSciNet  MATH  Google Scholar 

  22. Li B, Zhang Z (2015) A new approach for numerical simulation of the time-dependent Ginzburg–Landau equations. J Comput Phys

  23. Yana Y, IraMoxley F, Dai W (2015) A new compact finite difference scheme for solving the complex Ginzburg–Landau equation. Appl Math Comput 260:269–287

    MathSciNet  Google Scholar 

  24. Lopez V (2018) Numerical continuation of invariant solutions of the complex Ginzburg–Landau equation. Commun Nonlinear Sci Numer Simulat 61:248–270

    Article  MathSciNet  Google Scholar 

  25. Shokri A, Bahmani E (2018) Direct meshless local Petrov-Galerkin (DMLPG) method for 2D complex Ginzburg–Landau equation. Eng Anal Bound Elements 1–9

  26. Wang P, Huang C (2016) An implicit midpoint difference scheme for the fractional Ginzburg–Landau equation. J Comput Phys 312(1):31–49

    Article  MathSciNet  Google Scholar 

  27. Li M, Huang C, Wang N (2017) Galerkin finite element method for the nonlinear fractional Ginzburg–Landau equation. Appl Num Math 118:131–149

    Article  MathSciNet  Google Scholar 

  28. Wang N, Huang C (2018) An efficient split-step quasi-compact finite difference method for the nonlinear fractional Ginzburg–Landau equations. Comput Math Appl 75(7):2223–2242

    Article  MathSciNet  Google Scholar 

  29. Zeng W, Xiao A, Li X (2019) Error estimate of Fourier pseudo-spectral method for multidimensional nonlinear complex fractional Ginzburg–Landau equations. Appl Math Lett

  30. Heydari M H, Hooshmandasl M R, Maalek Ghaini F M (2014) An efficient computational method for solving fractional biharmonic equation. Comput Math Appl 68(9):269–287

    Article  MathSciNet  Google Scholar 

  31. Heydari MH, Avazzadeh Z, Mahmoudi MR (2019) Chebyshev cardinal wavelets for nonlinear stochastic differential equations driven with variable-order fractional Brownian motion. Chaos Solitons Fractals 124:105–124

    Article  MathSciNet  Google Scholar 

  32. Canuto C, Hussaini M, Quarteroni A, Zang T (1988) Spectral methods in fluid dynamics. Springer, Berlin

    Book  Google Scholar 

  33. Xu X, Zhang CS (2019) A new estimate for a quantity involving the Chebyshev polynomials of the first kind. J Math Anal Appl 476:302–308

    Article  MathSciNet  Google Scholar 

  34. Pereira M, Desassis N (2019) Efficient simulation of Gaussian Markov random fields by Chebyshev polynomial approximation. Spat Stat 31:100359

    Article  MathSciNet  Google Scholar 

  35. Kumar S, Piret C (2019) Numerical solution of space-time fractional PDEs using RBF-QR and chebyshev polynomials. Appl Num Math 143:300–315

    Article  MathSciNet  Google Scholar 

  36. Heydari MH (2019) A direct method based on the Chebyshev polynomials for a new class of nonlinear variable-order fractional 2D optimal control problems. J Franklin Inst (in press)

  37. Roohi R, Heydari MH, Bavi O, Emdad H (2019) Chebyshev polynomials for generalized couette flow of fractional Jeffrey nanofluid subjected to several thermochemical effects. Eng Comput. https://doi.org/10.1007/s00366-019-00843-9

  38. Atangana A (2017) Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system. Chaos, Solitons Fractals 102:396–406

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  40. Khader MM (2011) On the numerical solutions for the fractional diffusion equation. Commun Nonlinear Sci Numer Simulat 16:2535–2542

    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 Natural and Agricultural Sciences, University of the Free State, Bloemfontein, South Africa

    A. Atangana

  3. Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, Suzhou, 215123, Jiangsu, China

    Z. Avazzadeh

  4. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

    A. Atangana

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

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Heydari, M.H., Atangana, A. & Avazzadeh, Z. Chebyshev polynomials for the numerical solution of fractal–fractional model of nonlinear Ginzburg–Landau equation. Engineering with Computers 37, 1377–1388 (2021). https://doi.org/10.1007/s00366-019-00889-9

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