Chebyshev polynomials for the numerical solution of fractal–fractional model of nonlinear Ginzburg–Landau equation
Tóm tắt
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.
Tài liệu tham khảo
Atangana A (2016) Derivative with two fractional orders: a new avenue of investigation toward revolution in fractional calculus. Eur Phys J Plus 131:373
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
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
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
Engheta N (1996) On fractional calculus and fractional multipoles in electromagnetism. Antennas Propag 44:554–566
Kulish VV, Lage JL (2002) Application of fractional calculus to fluid mechanics. J Fluids Eng 124(3):803–806
Bagley RL, Torvik PJ (1985) Fractional calculus in the transient analysis of viscoelastically damped structures. AIAA J 23:918–925
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
Meral FC, Royston TJ, Magin R (2010) Fractional calculus in viscoelasticity: an experimental study. Commun Nonlinear Sci Numer Simul 15:939–945
Podlubny I (1999) Fractional differential equations. Academic Press, San Diego
Oldham KB, Spanier J (1974) The fractional calculus. Academic Press, New York
Baleanu D, Diethelm K, Scalas E, Trujillo JJ (2012) Fractional calculus models and numerical methods, series on complexity, nonlinearity and chaos. World Scientific, Boston
Diethelm K, Ford NJ (2002) Analysis of fractional differential equations. J Math Anal Appl 265:229–248
Wess W (1996) The fractional diffusion equation. J Math Phys 27:2782–2785
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
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
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
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
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
Aranson IS, Kramer L (2002) The world of the complex Ginzburg–Landau equation. Rev Mod Phys 74:99–143
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
Li B, Zhang Z (2015) A new approach for numerical simulation of the time-dependent Ginzburg–Landau equations. J Comput Phys
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
Lopez V (2018) Numerical continuation of invariant solutions of the complex Ginzburg–Landau equation. Commun Nonlinear Sci Numer Simulat 61:248–270
Shokri A, Bahmani E (2018) Direct meshless local Petrov-Galerkin (DMLPG) method for 2D complex Ginzburg–Landau equation. Eng Anal Bound Elements 1–9
Wang P, Huang C (2016) An implicit midpoint difference scheme for the fractional Ginzburg–Landau equation. J Comput Phys 312(1):31–49
Li M, Huang C, Wang N (2017) Galerkin finite element method for the nonlinear fractional Ginzburg–Landau equation. Appl Num Math 118:131–149
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
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
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
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
Canuto C, Hussaini M, Quarteroni A, Zang T (1988) Spectral methods in fluid dynamics. Springer, Berlin
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
Pereira M, Desassis N (2019) Efficient simulation of Gaussian Markov random fields by Chebyshev polynomial approximation. Spat Stat 31:100359
Kumar S, Piret C (2019) Numerical solution of space-time fractional PDEs using RBF-QR and chebyshev polynomials. Appl Num Math 143:300–315
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)
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
Atangana A (2017) Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system. Chaos, Solitons Fractals 102:396–406
Atangana A, Qureshi S (2019) Modeling attractors of chaotic dynamical systems with fractal–fractional operators. Chaos Solitons Fractals 123:320–337
Khader MM (2011) On the numerical solutions for the fractional diffusion equation. Commun Nonlinear Sci Numer Simulat 16:2535–2542