[1] Aranson, I.S. and Kramer, L. The world of the complex Ginzburg-Landau equation, Reviews of Modern Physics 74(1) (2002), 99.
[2] Atangana, A. Fractal-fractional differentiation and integration: connect-ing fractal calculus and fractional calculus to predict complex system, Chaos, Solitons and Fractals, 102 (2017), 396–406.
[3] Atangana, A. and Qureshi, S. Modeling attractors of chaotic dynamical systems with fractal–fractional operators, Chaos, Solitons and Fractals 123 (2019), 320–337.
[4] Bhrawy, A.H., Doha, E.H., Baleanu, D. and Ezz-Eldien, S.S. A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations, Journal of Computational Physics 293 (2015), 142–156.
[5] Darehmiraki, M., Farahi, M.H. and Effati, S. A novel method to solve a class of distributed optimal control problems using Bezier curves, Journal of Computational and Nonlinear Dynamics 11(6) (2016).
[6] Ding, H. and Li, C. High-order numerical algorithm and error analysis for the two dimensional nonlinear spatial fractional complex Ginzburg-Landau equation, Communications in Nonlinear Science and Numerical Simulation 120 (2023) , 107160.
[7] Doha, E.H. On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials, Journal of Physics A: Mathematical and General 37(3) (2004), 657.
[8] Doha, E.H., Bhrawy, A.H. and Ezz-Eldien, S.S. A new Jacobi opera-tional matrix: an application for solving fractional differential equations, Applied Mathematical Modelling 36(10) (2012), 4931–4943.
[9] Du, N., Wang, H. and Liu, W. A fast gradient projection method for a constrained fractional optimal control, Journal of Scientific Computing 68(1) (2016), 1–20.
[10] Goyal, A., Raju, T.S. and Kumar, C.N. Lorentzian-type soliton solutions of ac-driven complex Ginzburg –- Landau equation, Applied Mathematics and Computation 218(24) (2012), 11931–11937.
[11] Gu, X.M., Shi, L. and Liu, T. Well-posedness of the fractional Ginzburg–Landau equation, Applicable Analysis 98(14) (2019), 2545–2558.
[12] Gunzburger, M.D., Hou, L.S. and Ravindran, S.S. Analysis and approx-imation of optimal control problems for a simplified Ginzburg-Landau model of superconductivity, Numerische Mathematik 77 (1997), 243–268.
[13] Hasegawa, A. Optical Solitons in Fibers, In Optical solitons in fibers, Springer, Berlin, Heidelberg, 1989, 1–74
[14] Heydari, M.H., Atangana, A. and Avazzadeh, Z. Chebyshev polynomi- als for the numerical solution of fractal-fractional model of nonlinear Ginzburg –- Landau equation, Eng. Comput. (2019), 1–12.
[15] Heydari, M.H., Atangana, A. and Avazzadeh, Z. Chebyshev polynomi-als for the numerical solution of fractal–fractional model of nonlinear Ginzburg –- Landau equation, Engineering with Computers 2021(37) (2021), 1377–1388.
[16] Heydari, M.H., Avazzadeh, Z. and Atangana, A. Shifted Jacobi polynomi-als for nonlinear singular variable-order time fractional Emden–Fowler equation generated by derivative with non-singular kernel, Advances in Difference Equations 2021(1) (2021), 1–15.
[17] Kilicman, A. and Al Zhour, Z.A.A. Kronecker operational matrices for fractional calculus and some applications, Applied Mathematics and Computation 187(1) (2007), 250–265.
[18] Li, B. and Zhang, Z. A new approach for numerical simulation of the time-dependent Ginzburg -– Landau equations, Journal of Computational Physics 303 (2015), 238–250.
[19] Li, M., Huang, C. and Wang, N. Galerkin finite element method for the nonlinear fractional Ginzburg -– Landau equation, Applied Numerical Mathematics 118 (2017), 131–149.
[20] Lopez, V. Numerical continuation of invariant solutions of the complex Ginzburg –- Landau equation, Communications in Nonlinear Science and Numerical Simulation 61 (2018), 248–270.
[21] Luke, Y.L. The special functions and their approximations, Academic press, 53.
[22] Mainardi, F. and Gorenflo, R. On Mittag-Leffler-type functions in frac-tional evolution processes, J. Comput. Appl. Math. 118 (2000), 283–299.
[23] Mophou, G.M. Optimal control of fractional diffusion equation, Com-puters and Mathematics with Applications 61(1) (2011), 68–78.
[24] Shojaeizadeh, T., Mahmoudi, M. and Darehmiraki, M. Optimal control problem of advection-diffusion-reaction equation of kind fractal-fractional applying shifted Jacobi polynomials, Chaos, Solitons and Fractals 143 (2021), 110568.
[25] Shokri, A. and Bahmani, E. Direct meshless local Petrov-–Galerkin (DMLPG) method for 2D complex Ginzburg–Landau equation, Engineer-ing Analysis with Boundary Elements 100 (2019), 195–203.
[26] Tarasov, V. Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media, Berlin: Springer-Verlag, jointly with Higher Education Press, Beijing, 2011.
[27] Toledo-Hernandez, R., Rico-Ramirez, V., Rico-Martinez, R., Hernandez-Castro, S. and Diwekar, U.M. A fractional calculus approach to the dy-namic optimization of biological reactive systems. Part II: Numerical solution of fractional optimal control problems, Chemical Engineering Science 117 (2014), 239–247.
[28] Wang, N. and Huang, C. An efficient split-step quasi-compact finite dif-ference method for the nonlinear fractional Ginzburg–Landau equations, Computers and Mathematics with Applications 75(7) (2018), 2223–2242.
[29] Wang, P. and Huang, C. An implicit midpoint difference scheme for the fractional Ginzburg - Landau equation, Journal of Computational Physics 312 (2016), 31–49.
[30] Weitzner, H. and Zaslavsky, G.M. Commun Nonlinear, Sci. Numer. Sim-ulation 8 (2003), 273–281.
[31] Yan, Y., Moxley III, F.I. and Dai, W. A new compact finite differ-ence scheme for solving the complex Ginzburg–-Landau equation, Applied Mathematics and Computation 260 (2015), 269–287.
[32] Zaslavsky, G. Hamiltonian chaos and fractional dynamics, Oxford: Ox-ford University Press, 2005.
[33] Zeng, W., Xiao, A. and Li, X. Error estimate of Fourier pseudo–spectral method for multidimensional nonlinear complex fractional Ginzburg–-Landau equations, Applied Mathematics Letters 93 (2019), 40–45.
Send comment about this article