A journal of IEEE and CAA , publishes high-quality papers in English on original theoretical/experimental research and development in all areas of automation
Volume 6 Issue 1
Jan.  2019

IEEE/CAA Journal of Automatica Sinica

  • JCR Impact Factor: 6.171, Top 11% (SCI Q1)
    CiteScore: 11.2, Top 5% (Q1)
    Google Scholar h5-index: 51, TOP 8
Turn off MathJax
Article Contents
Najeeb Alam Khan and Tooba Hameed, "An Implementation of Haar Wavelet Based Method for Numerical Treatment of Time-fractional Schrodinger and Coupled Schrödinger Systems," IEEE/CAA J. Autom. Sinica, vol. 6, no. 1, pp. 177-187, Jan. 2019. doi: 10.1109/JAS.2016.7510193
Citation: Najeeb Alam Khan and Tooba Hameed, "An Implementation of Haar Wavelet Based Method for Numerical Treatment of Time-fractional Schrodinger and Coupled Schrödinger Systems," IEEE/CAA J. Autom. Sinica, vol. 6, no. 1, pp. 177-187, Jan. 2019. doi: 10.1109/JAS.2016.7510193

An Implementation of Haar Wavelet Based Method for Numerical Treatment of Time-fractional Schrodinger and Coupled Schrödinger Systems

doi: 10.1109/JAS.2016.7510193
More Information
  • The objective of this paper is to solve the timefractional Schrödinger and coupled Schrödinger differential equations (TFSE) with appropriate initial conditions by using the Haar wavelet approximation. For the most part, this endeavor is made to enlarge the pertinence of the Haar wavelet method to solve a coupled system of time-fractional partial differential equations. As a general rule, piecewise constant approximation of a function at different resolutions is presentational characteristic of Haar wavelet method through which it converts the differential equation into the Sylvester equation that can be further simplified easily. Study of the TFSE is theoretical and experimental research and it also helps in the development of automation science, physics, and engineering as well. Illustratively, several test problems are discussed to draw an effective conclusion, supported by the graphical and tabulated results of included examples, to reveal the proficiency and adaptability of the method.

     

  • loading
  • [1]
    K. Diethelm, The Analysis of Fractional Differential Equations. Berlin, Germany: Springer-Verlag, 2010.
    [2]
    I. Podlubny, Fractional Differential Equations. San Diego, USA: Academic Press, 1999.
    [3]
    M. G. Sakar, F. Erdogan, and A. Yildirim, "Variational iteration method for the time-fractional Fornberg-Whitham equation, " Computers and Mathematics with Applications, vol. 63, no. 9, pp. 1382-1388, 2012. doi: 10.1016/j.camwa.2012.01.031
    [4]
    J. C. Liu, and G. L. Hou, "Numerical solutions of the space- and time-fractional coupled Burgers equations by generalized differential transform method, " Applied Mathematics and Computation, vol. 217, no. 16, pp. 7001-7008, 2011. doi: 10.1016/j.amc.2011.01.111
    [5]
    N. A. Khan, M. Jamil, and Ara A, "Approximate solutions to time-fractional Schrödinger equation via Homotopy analysis method, " ISRN Mathematical Physics, vol. 2012: Article ID 197068, 2012. https://www.researchgate.net/publication/258403640_Approximate_Solutions_to_Time-Fractional_Schrodinger_Equation_via_Homotopy_Analysis_Method
    [6]
    A. H. Bhrawy, E. H. Doha, S. S. Ezz-Eldien, and R. A. Van Gorder, "A new Jacobi spectral collocation method for solving $</element-citation>+1$ fractional Schrödinger equations and fractional coupled Schrödinger systems, " The European Physical J. Plus, vol. 129, no. 260, pp. 2014.
    [7]
    A. H. Bhrawy and M. A. Zaky, "A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations, " J. Computational Physics, vol. 281, pp. 876-895, 2015. doi: 10.1016/j.jcp.2014.10.060
    [8]
    A. Bhrawy and M. Zaky, "A fractional-order Jacobi Tau method for a class of time-fractional PDEs with variable coefficients, " Mathematical Methods in the Applied Sciences, vol. 39, no. 7, pp. 1765-1779, 2016. doi: 10.1002/mma.v39.7
    [9]
    Z. J. Fu, W. Chen, and H. T. Yang, "Boundary particle method for Laplace transformed time fractional diffusion equations, " J. Computational Physics, vol. 235, pp. 52-66. 2013. doi: 10.1016/j.jcp.2012.10.018
    [10]
    H. Aminikhah, A. R. Sheikhani, and H. Rezazadeh, "An efficient method for time-fractional coupled Schrödinger system, " International J. Partial Differential Equations, vol. 2014: Article ID 137470. 2014. https://www.researchgate.net/publication/267142224_An_Efficient_Method_for_Time-Fractional_Coupled_Schrodinger_System
    [11]
    V. Daftardar-Gejji and A. Babakhani, "Analysis of a system of fractional differential equations, " J. Mathematical Analysis and Applications, vol. 293, no.2, pp. 511-522, 2004. doi: 10.1016/j.jmaa.2004.01.013
    [12]
    V. Daftardar-Gejji and H. Jafari, "Adomian decomposition: a tool for solving a system of fractional differential equations, " J. Mathematical Analysis and Applications, vol. 301, no. 2, pp. 508-518, 2005. doi: 10.1016/j.jmaa.2004.07.039
    [13]
    Y. J. Jiang and J. T. Ma, "High-order finite element methods for time-fractional partial differential equations, " J. Computational and Applied Mathematics, vol. 235, no. 11, pp. 3285-3290, 2011. doi: 10.1016/j.cam.2011.01.011
    [14]
    Y. Z. Hu, Y. Luo, and Z. Y. Lu, "Analytical solution of the linear fractional differential equation by Adomian decomposition method, " J. Computational and Applied Mathematics, vol. 215, no.1, pp. 220-229, 2008. doi: 10.1016/j.cam.2007.04.005
    [15]
    Z. Odibat and S. Momani, "Numerical methods for nonlinear partial differential equations of fractional order, " Applied Mathematical Modelling, vol. 32, no. 1, pp. 28-39, 2008. http://www.sciencedirect.com/science/article/pii/S0307904X06002800
    [16]
    A. K. Gupta and S. S. Ray, "Wavelet methods for solving fractional order differential equations, " Mathematical Problems in Engineering, vol. 2014: Article ID 140453, 2014. https://www.researchgate.net/publication/270674415_Wavelet_Methods_for_Solving_Fractional_Order_Differential_Equations
    [17]
    A. Setia, Y. C. Liu, and A. S. Vatsala, "Solution of linear fractional Fredholm Integro-differential equation by using second kind Chebyshev wavelet, " in Proc. of the 11th International Conference on Information Technology: New Generations (ITNG), Las Vegas: IEEE, 2014, 465-469. https://www.researchgate.net/publication/262525724_Solution_of_linear_fractional_Fredholm_integro-differential_equation_by_using_second_kind_Chebyshev_wavelet
    [18]
    A. H. Bhrawy, M. M. Tharwat, and A. Yildirim, "A new formula for fractional integrals of Chebyshev polynomials: application for solving multi-term fractional differential equations, " Applied Mathematical Modelling, vol. 37, no. 6, pp. 4245-4252, 2013. doi: 10.1016/j.apm.2012.08.022
    [19]
    A. Setia, Y. C. Liu, and A. S. Vatsala, "Numerical solution of Fredholm-volterra fractional Integro-differential equations with nonlocal boundary conditions, " J. Fractional Calculus and Applications, vol. 5, no. 2, pp. 155-165, 2014. http://adsabs.harvard.edu/abs/2017AIPC.1798b0140S
    [20]
    Laskin N, "Fractional quantum mechanics, " Physical Review E, vol. 62, no. 3, Pt A, pp. 3135-3145, 2000. http://d.old.wanfangdata.com.cn/NSTLQK/NSTL_QKJJ0232673214/
    [21]
    M. ur Rehman and R. A. Khan, "Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations, " Applied Mathematics Letters, vol. 23, no. 9, pp. 1038-1044, 2010. doi: 10.1016/j.aml.2010.04.033
    [22]
    M. ur Rehman and R. A. Khan, "Numerical solutions to initial and boundary value problems for linear fractional partial differential equations, " Applied Mathematical Modelling, vol. 37, no. 7, pp. 5233-5244, 2013. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=ffd872eff60990201b68258bea831aad
    [23]
    J. Shen and T. Tang, Mathematics Monograph Series, Vol. 3: Spectral and High-Order Methods with Applications. Beijing, China: Science Press, 2006.
    [24]
    A. H. Bhrawy, T. M. Taha, and J. A. T. Machado, "A review of operational matrices and spectral techniques for fractional calculus, " Nonlinear Dynamics, vol. 81, no. 3, pp. 1023-1052, 2015. doi: 10.1007/s11071-015-2087-0
    [25]
    E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldien, "A new Jacobi operational matrix: an application for solving fractional differential equations, " Applied Mathematical Modelling, vol. 36, no. 10, pp. 4931-4943, 2012. doi: 10.1016/j.apm.2011.12.031
    [26]
    A. H. Bhrawy and M. A. Zaky, "Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation, " Nonlinear Dynamics, vol. 80, no. 1-2, pp. 101-116, 2015. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=f7b300f3927102b867dabb9c54420213
    [27]
    S. G. Mallat, "A theory for multiresolution signal decomposition: the wavelet representation, " IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 11, no. 7, pp. 674-693, 1989. doi: 10.1109/34.192463
    [28]
    S. Mallat, A Wavelet Tour of Signal Processing, USA: Academic Press, 2009.
    [29]
    Ü. Lepik, "Solving PDEs with the aid of two-dimensional Haar wavelets, " Computers and Mathematics with Applications, vol. 61, no. 7, pp. 1873-1879, 2011. doi: 10.1016/j.camwa.2011.02.016
    [30]
    S. Islam, I. Aziz, and Šarler B, "The numerical solution of second-order boundary-value problems by collocation method with the Haar wavelets, " Mathematical and Computer Modelling, vol. 52, no. 9-10, pp. 1577-1590, 2010. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=7b9c64c06d4b3f25a71f76fc1c7ec799
    [31]
    Y. L. Li and W. W. Zhao, "Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, " Applied Mathematics and Computation, vol. 216, no.8, pp. 2276-2285, 2010. doi: 10.1016/j.amc.2010.03.063
    [32]
    L. F. Wang, Y. P. Ma, and Z. J. Meng, "Haar wavelet method for solving fractional partial differential equations numerically, " Applied Mathematics and Computation, vol. 227, pp. 66-76, 2014. doi: 10.1016/j.amc.2013.11.004
    [33]
    A. Mohebbi, M. Abbaszadeh, and M. Dehghan, "The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrödinger equation arising in quantum mechanics, " Engineering Analysis with Boundary Elements, vol. 37, no. 2, pp. 475-485, 2013. doi: 10.1016/j.enganabound.2012.12.002
    [34]
    D. L. Wang, A. G. Xiao, and W. Yang, "Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative, " J. Computational Physics, vol. 242, pp. 670 -681, 2013. doi: 10.1016/j.jcp.2013.02.037
    [35]
    L. L. Wei, X. D. Zhang, S. Kumar, and A. Yildirim, "A numerical study based on an implicit fully discrete local discontinuous Galerkin method for the time-fractional coupled Schrödinger system, " Computers and Mathematics with Applications, vol. 64, no. 8, pp. 2603-2615, 2012. doi: 10.1016/j.camwa.2012.07.004
    [36]
    A. H. Bhrawy and M. A. Abdelkawy, "A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations, " J. Computational Physics, vol. 294, pp. 462-483, 2015. doi: 10.1016/j.jcp.2015.03.063
    [37]
    X. Y. Guo and M. Y. Xu, "Some physical applications of fractional Schrödinger equation, " J. Mathematical Physics, vol. 47, pp. 082104, 2006. doi: 10.1063/1.2235026
    [38]
    Y. C. Wei, "Some solutions to the fractional and relativistic Schrödinger equations, " International J. Theoretical and Mathematical Physics, vol. 5 no. 5, 87-111, 2015. doi: 10.5923.j.ijtmp.20150505.03.html
    [39]
    C. H. Wang, "On the generalization of block pulse operational matrices for fractional and operational calculus, " J. Franklin Institute, vol. 315, no. 2, 91-102, 1983. doi: 10.1016/0016-0032(83)90069-8

Catalog

    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索

    Figures(2)  / Tables(6)

    Article Metrics

    Article views (1483) PDF downloads(26) Cited by()

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return