A method of lines for solving the nonlinear time- and space-fractional Schrödinger equation via stable Gaussian radial basis function interpolation

Sepehrian, Behnam; Shamohammadi, Zahra

doi:10.22034/cmde.2024.54090.2265

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	A method of lines for solving the nonlinear time- and space-fractional Schrödinger equation via stable Gaussian radial basis function interpolation
Computational Methods for Differential Equations
مقاله 4، دوره 13، شماره 1، فروردین 2025، صفحه 41-60 اصل مقاله (564.48 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2024.54090.2265
نویسندگان
Behnam Sepehrian^* ؛ Zahra Shamohammadi
Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran.
چکیده
The stable Gaussian radial basis function (RBF) interpolation is applied to solve the time and space-fractional Schrödinger equation (TSFSE) in one and two-dimensional cases. In this regard, the fractional derivatives of stable Gaussian radial basis function interpolants are obtained. By a method of lines, the computations of the TSFSE are converted to a coupled system of Caputo fractional ODEs. To solve the resulting system of ODEs, a high-order finite difference method is proposed, and the computations are reduced to a coupled system of nonlinear algebraic equations, in each time step. Numerical illustrations are performed to certify the ability and accuracy of the new method. Some comparisons are made with the results in other literature.
کلیدواژه‌ها
Caputo derivative؛ Nonlinear؛ Fractional Schrödinger equation؛ Radial basis functions؛ Riesz derivative

مراجع
[1] X. Antonie, Q. Tang, and J. Zhang, On the numerical solution and dynamical laws of nonlinear fractional Shrödinger/Gross-Pitaevskii equations, International journal of computer mathematics, 6-7 (2018), 1423–1443. [2] M. D. Buhman, Spectral convergence of multiquadratic interpolation, Proc. Edinburg Math. Soc., 36 (1993), 319–333. [3] J. X. Cao, C. P. Li, and Y. Q. Chen, High-order approximation to Caputo derivatives and Caputo-type advectiondiffusion equations (II), Fractional Calculus and Applied Analysis, 18(3) (2015), 735-761. [4] R. E. Carlson and T. A. Foley, The parameter r2 in multiquadratic interpolation, Comput. Math. Appl., 21 (1991), 29–42. [5] R. Cavoretto, G. E. Fasshauer, and M. McCourt, An introduction to the Hilbert-Schmidt SVD using iterated Brownian bridge kernels, Numer. Algorithms, 68(2) (2015), 393-422. [6] E. H. Dohaa, A. H. Bhrawy, and S. S. Ezz-Eldien, A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order, Computers and Mathematics with Applications, 62 (2011), 2364–2373. [7] G. E. Fasshauer and M. J. MacCourt, Stable evaluation of Gaussian radial basis function interpolants, SIAM J. Sci. Comput., 34(2) (2012), A737–A762. [8] R. P. Feynman and A. R. Hibbs, Quantum mechanics and path integrals, McGraw-Hill, New York, NY, USA, 1965. [9] B. Guo, Y. Han, and J. Xin, Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation, Applied Mathematics and Computation, 204(1) (2008), 468–477. [10] M. A. E. Herzallah and K. A. Gepreel, Approximate solution to the time-space fractional cubic nonlinear Schr¨odinger equation, Appl. Math. Model., 36 (2012), 5678–5685. [11] J. Hu, J. Xin, and H. Lu, The global solution for a class of systems of fractional nonlinear Schrödinger equations with periodic boundary condition, Computers and Mathematics with Applications, 62(3) (2011), 1510–1521. [12] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006. [13] M. Kirane, Finite element method for time- space- fractional Schödinger equation, Electronic Journal of Differential Equations,166 (2017), 1–18. [14] N. Laskin, Fractional Schr¨odinger equation, Phys. Rev. E, 66 (2002), 056108. [15] N. Laskin, Fractals and quantum mechanics, Chaos, 10 (2000), 780–790. [16] Q. Liu, Y. T. Gu, P. Zhuang, F. Liu, and Y. F. Nie, An implicit RBF meshless approach for time fractional diffusion equations, Comput. Mech., 48, (2011), 1–12. [17] W. K. Liu and W. M. Han, Reproducing kernel element method. Part I: Theoritical information, Comput. Methods Appl. Mech. Engrg., 193 (2004), 933–951. [18] N. Liu and W. Jiang, A numerical method for solving the time fractional Schrödinger equation, Adv Comput Math.,44 (2018), 1235–1248. [19] Q. Liu, F. Zeng, and Ch. Li, Finite difference method for time-space-fractional Schödinger equation, International Journal of Computer Mathematics, (2014), 1439–1451. [20] J. M. Melenk and I. Babuska, The partition of unity method: basic theory and applications, Comput. Methods Appl. Mech. Engrg., 139 (1996), 289–314. [21] M. Naber, Time fractional Schrödinger equation, J. Math. Phys., 45(8) (2004), 3339–3352. [22] M. Pazouki and R. Schaback, Bases for kernel-based spaces, Journal of Computational and Applied Mathematics,236(4) (2011), 575–588. [23] C. Piret and E. Hanert, A radial basis functions method for fractional diffusion equations, J. Comput. Phys., 238 (2013), 71–81. [24] I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999. [25] A. Quarteroni, Riccardo Sacco, and Fausto Saleri, Numerical mathematics, Springer, 2000. [26] J. Rashidinia, G. Fasshauer, and M. Khasi, A stable method for the evaluation of Gaussian radial basis function solutions of interpolation and collocation problems, Comput. Math. Appl., 72(1) (2016), 178–193. [27] J. Rashidinia and M. Khasi, Stable Gaussian radial basis function method for solving Helmholtz equations, Computational Methods for Differential Equations, 7(1) (2019), 138–151. [28] J. Rashidinia, M. Khasi, and G. E. Fasshauer, A stable Gaussian radial basis function method for solving nonlinear unsteady convection-diffusion-reaction equations, Computers and Mathematics with Applications, 75 (2018), 1831–1850. [29] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional integrals and derivatives: Theory and applications, Gordon and Breach, New York, 1993. [30] L. S. Schulman, Techniques and applications of path integration, John Wiley and Sons, New York, NY, USA, 1981. [31] P. Wang and C. Huang, An energy conservative difference scheme for the nonlinear fractional Schrödinger equations, J. Comput. Phys., 293 (2015), 238–251. [32] D. Wang, A. Xiao, and W. Yang, Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative, J. Comput. Phy., 242 (2013), 670–681. [33] D. Wang, A. Xiao, and W. Yang, Maximum-norm error analysis of a difference scheme for the space fractional CNLS, Appl. Math. Comput., 257 (2015), 241–251. [34] S. W. Wang and M. Y. Xu, Generalized fractional Schrödinger equation with space-time fractional derivatives, J. Math. Phys., 48 (2007), 043502. [35] X. Zhao, Z. Z. Sun, and Z. P. Hao, A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrödinger equation, SIAM J. Sci. Comput., 36(6) (2014), A2865–A2886.
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A method of lines for solving the nonlinear time- and space-fractional Schrödinger equation via stable Gaussian radial basis function interpolation