Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-16T04:47:10.432Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical Simulation for the Variable-Order Fractional Schrödinger Equation with the Quantum Riesz-Feller Derivative

Part of: General theory in ordinary differential equations Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  18 January 2017

N. H. Sweilam  and
M. M. Abou Hasan
N. H. Sweilam*
Affiliation:
Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt
M. M. Abou Hasan
Affiliation:
Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt
*
*Corresponding author. Email:nsweilam@sci.cu.edu.eg (N. H. Sweilam)
Get access

Abstract

In this paper the space variable-order fractional Schrödinger equation (VOFSE) is studied numerically, where the variable-order fractional derivative is described here in the sense of the quantum Riesz-Feller definition. The proposed numerical method is the weighted average non-standard finite difference method (WANSFDM). Special attention is given to study the stability analysis and the convergence of the proposed method. Finally, two numerical examples are provided to show that this method is reliable and computationally efficient.

Keywords

Variable-order Schrödinger equationquantum Riesz-Feller variable-order definitionweighted average non-standard finite difference methodJon von Neumann stability analysis

MSC classification

Secondary: 65M06: Finite difference methods 65M12: Stability and convergence of numerical methods 34A08: Fractional differential equations
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 4 , August 2017 , pp. 990 - 1011
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Atangana, A. and Cloot, A. H., Stability and convergence of the space fractional variable-order Schrödinger equation, Adv. Difference Equations, 80 (2013), doi:10.1016/j.jcp.2014.08.015.Google Scholar
[2] Bibi, A., Kamran, A., Hayat, U. and Mohyud-Din, S., New iterative method for time-fractional Schrödinger equations, World J. Model. Simulation, 9(2) (2013), pp. 89–95.Google Scholar
[3] Bhrawy, A. H. and Zaky, M. A., Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation, Nonlinear Dyn., 78(4) (2014), doi:10.1007/s11071-014-1854-7.Google Scholar
[4] Arenas, A. J., González-Parrab, G. and Chen-Charpentier, B. M., Construction of nonstandard finite difference schemes for the SI and SIR epidemic models of fractional order, Math. Comput. Simulation, 121 (2016), pp. 48–63.Google Scholar
[5] Nagy, A. and Sweilam, N., An effcient method for solving fractional Hodgkin-Huxley model, Phys. Lett. A, 378(30) (2014), pp. 1980–1984.Google Scholar
[6] Razminia, A., Dizaji, A. F. and Majd, V. J., Solution existence for non-autonomous variable-order fractional differential equations, Math. Comput. Model., 55 (2011), pp. 1106–1117.Google Scholar
[7] Al-Saqabi, B., Boyadjiev, L. and Luchko, YU., Comments on employing the Riesz-Feller derivative in the Schrödinger equation, The European Physical Journal Special Topics, 222 (2013), pp. 1779–1794.Google Scholar
[8] Çelik, C. and Duman, M., Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative, J. Comput. Phys., 231 (2012), pp. 1743–1750.Google Scholar
[9] Chen, C. M., Liu, F., Anh, V. and Turner, I., Numerical schemes with high spatial accuracy for a variable-order anomalous sub-diffusion equation, SIAM J. Sci. Comput., 32(4) (2010), pp. 1740–1760.Google Scholar
[10] Chen, C. M., Liu, F., Anh, V. and Turner, I., Numerical simulation for the variable-order Galilei invariant advection diffusion equation with a nonlinear source term, Appl. Math. Comput., 217 (2011), pp. 5729–5742.Google Scholar
[11] Valério, D. and Costa, José Sá Da, Variable-order fractional derivatives and their numerical approximations, Signal Processing, 91 (2011), pp. 470–483.Google Scholar
[12] Zhu, D., Kinoshita, S., Cai, D. and Cole, J. B., Investigation of structural colors in Morpho butterflies using the nonstandard-finite-difference time-domain method: Effects of alternately stacked shelves and ridge density, Phys. Rev. E, 80(5) (2009).Google Scholar
[13] Silva, F., Marào, J. A. P. F., Alves Soares, J. C. and Capelas De Oliveira, E., Similarity solution to fractional nonlinear space-time diffusion-wave equation, J. Math. Phys., 56 (2015), pp. 1–16.Google Scholar
[14] Smith, G. D., Numerical Solution of Partial Differential Equations: Finite Difference Methods, Oxford Applied Mathematics and Computing Science Series, (1985).Google Scholar
[15] Sun, H. G., Chen, W., Wei, H. and Chen, Y. Q., A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems, Euro. Phys. J. Spec. Top., 193(1) (2011), pp. 185–192.Google Scholar
[16] Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, (1999).Google Scholar
[17] Dong, J. and Xu, M., Solutions to the space fractional Schrödinger equation using momentum representation method, J. Math. Phys., 48 (2007), 072105.Google Scholar
[18] Dong, J. and Xu, M., Space-time fractional Schrödinger equation with time-independent potentials, J. Math. Anal. Appl., 344 (2008), pp. 1005–1017.Google Scholar
[19] Moaddy, K., Hashim, I. and Momani, S., Non-standard finite difference schemes for solving fractional-order Rössler chaotic and hyperchaotic systems, Comput. Math. Appl., 62(3) (2011), pp. 1068–1074.CrossRefGoogle Scholar
[20] Moaddy, K., Momani, S. and Hashim, I., The non-standard finite difference scheme for linear fractional PDEs in fluid mechanics, Comput. Math. Appl., 61(4) (2011), pp. 1209–1216.CrossRefGoogle Scholar
[21] Moaddy, K., Radwan, A. G., Salama, K. N., Momani, S. and Hashim, I., The fractional-order modeling and synchronization of electrically coupled neuron systems, Comput. Math. Appl., 64(10) (2012), pp. 3329–3339.Google Scholar
[22] Morton, K. W. and Mayers, D. F., Numerical Solution of Partial Differential Equations, Cambridge University Press, Cambridge, (1994).Google Scholar
[23] Ciesielski, M. and Leszczynski, J., Numerical solutions to boundary value problem for anomalous diffusion equation with Riesz-Feller fractional operator, J. Theor. Appl. Mech., 44(2) (2006), pp. 393–403.Google Scholar
[24] Laskin, N., Fractional quantum mechanics, Phys. Rev. E, 62 (2000), pp. 3135–3145.CrossRefGoogle ScholarPubMed
[25] Laskin, N., Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 298 (2000), pp. 298–305.Google Scholar
[26] Laskin, N., Fractals and quantum mechanics, Chaos, 10 (2000), pp. 780–790.Google Scholar
[27] Laskin, N., Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108.Google Scholar
[28] Sweilam, N. H., Khader, M. M. and Adel, M., On the stability analysis of weighted average finite difference methods for fractional wave equation, J. Fractional Differential Calculus, 2 (2012), pp. 17–25.Google Scholar
[29] Sweilam, N. H. and Almrawm, H. M., On the numerical solutions of the variable order fractional heat equation, Studies in Nonlinear Sciences, 2(1) (2011), pp. 31–36.Google Scholar
[30] Sweilam, N. H., Khader, M. M. and Almarwm, H. M., Numerical studies for the variable order nonlinear fractional wave equation, FCAA., 15(4) (2012).CrossRefGoogle Scholar
[31] Sweilam, N. H. and Assiri, T. A., Numerical simulations for the space-time variable order nonlinear fractional wave equation, J. Appl. Math., 2013 (2013), Article ID 586870, 8 pages.Google Scholar
[32] Sweliam, N. H., Nagy, A. M., Assiri, T. A. and Ali, N. Y., Numerical simulations for variable order fractional nonlinear delay differential equations, J. Fractional Calculus Appl., 6(1) (2015), pp. 71–82.Google Scholar
[33] Sweilam, N. H. and Almajbri, T. F., Large stability regions method for the two-dimensional fractional diffusion equation, Progress in Fractional Differentiation and Applications, 1(2) (2015), pp. 123–131.Google Scholar
[34] Lin, R., Liu, F., Anh, V. and Turner, I., Stability and convergence of a new explicit finite difference approximation for the variable-order nonlinear fractional diffusion equation, Appl. Math. Comput., 212 (2009), pp. 435–445.Google Scholar
[35] Mickens, R.E., Nonstandard Finite Difference Model of Differential Equations, World Scientific, Singapore, (1994).Google Scholar
[36] Mickens, R. E., Nonstandard finite difference schemes for reactions-diffusion equations, Numer. Methods Partial Differential Equations Fractals, 15 (1999), pp. 201–214.Google Scholar
[37] Mickens, R. E., Application of Nonstandard Finite Difference Schemes, World Scientific Publishing Co. Pte. Ltd., (2000).Google Scholar
[38] Mickens, R. E., Nonstandard finite difference schemes for differential equations, J. Differential Equations Appl., 8(9) (2002), pp. 823–847.Google Scholar
[39] Mickens, R. E., A nonstandard finite difference scheme for a fisher PDF having nonlinear diffusion, Comput. Math. Appl., 45 (2003), pp. 429–436.Google Scholar
[40] Mickens, R. E., A nonstandard finite-difference scheme for the Lotka-Volterra system, Appl. Numer. Math., 45 (2003), pp. 309–314.Google Scholar
[41] Mickens, R. E. and Washington, T. M., A Note on an NSFD Scheme for a Mathematical Model of Respiratory Virus Transmission, Journal of Difference Equations and Applications, 18(3) (2010).Google Scholar
[42] Herrmann, R., Fractional Calculus, An Introduction For Physicists, World Scientific Publishing Co. Pte. Ltd., (2011).Google Scholar
[43] Yuste, S. B., Weighted average finite difference methods for fractional diffusion equations, J. Comput. Phys., 216 (2006), pp. 264–274.Google Scholar
[44] Yuste, S. B. and Acedo, L., On an explicit finite difference method for fractional diffusion equations, Preprint at http://arxiv.org/abs/cs.NA/0311011, (2003).Google Scholar
[45] Banerjee, S., Cole, J. B. and Yatagai, T., Calculation of diffraction characteristics of sub wave-length conducting gratings using a high accuracy nonstandard finite-difference time-domainmethod, Optical Rev., 12(4) (2005), pp. 274–280.CrossRefGoogle Scholar
[46] Elsheikh, S., Ouifki, R. and Patidar, K. C., A non-standard finite difference method to solve amodel of HIV-Malaria co-infection, J. Difference Equations Appl., 20(3) (2014), pp. 354–378.Google Scholar
[47] Samko, S. G. and Ross, B., Integration and differentiation to a variable fractional order, Integral Transform and Special Functions, 1 (1993), pp. 277–300.Google Scholar
[48] Moghadas, S., Alexander, M. and Corbett, B., A non-standard numerical scheme for ageneralized Gause-type predator-prey model, Physica D: Nonlinear Phenomena, 188(1) (2004), pp. 134–151.Google Scholar
[49] Wang, S. and Xu, M., Generalized fractional Schrödinger equation with space-time fractional derivatives, J. Math. Phys., 48 (2007), 043502.Google Scholar
[50] Zhang, S., Existence and uniqueness result of solutions to initial value problems of fractional differential equations of variable-order, J. Frac. Calc. Anal., 4(1) (2013), pp. 82–98.Google Scholar
[51] Zhang, S., Existence result of solutions to differential equations of variable-order with nonlinear boundary value conditions, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), pp. 3289–3297.Google Scholar
[52] Feller, W., On a generalization of Marcel Riesz’ potentials and the semi-groups generated by them, Meddelanden Lunds Universitets Matematiska Seminarium (Comm. Sém. Mathém. Université de Lund), Tome suppl. dédié à M. Riesz, Lund, 73, (1952).Google Scholar
[53] Guo, X. and Xu, M., Some physical applications of fractional Schrödinger equation, J. Math. Phys., 47 (2006), 82104.Google Scholar
[54] Zhao, X., Sun, Z. and Em, G., Second-order approximations for variable order fractional derivatives: algorithms and applications, J. Comput. Phys., (2014). doi :10.1016/j.jcp.2014.08.015.Google Scholar