Skip to main content
SpringerLink
Log in

Numerical solutions of Schrödinger wave equation and Transport equation through Sinc collocation method

  • Original paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This study carries the novel applications of the Sinc collocation method to investigate the numerical computing paradigm of Schrödinger wave equation and Transport equation as a great level of accuracy and precision. A global collocation-based Sinc function is embedded with a cardinal expansion to discretize initially time derivatives by finite difference method and secondly spatial derivatives are approximated with \(\theta \)-weighted scheme. Sinc collocation method (SCM) is found to be a more robust approach in order to avoid singularities in proposed problems and for yielding accurate numerical results. The governing PDEs are transformed with the help of Sinc function into algebraic system of equations, and further, these algebraic equations are solved with the help of computational iteration scheme to obtain the numerical results. The scheme provides a reliable and excellent procedure for adaptation of finding unknown in Sinc function for these problems. An extensive stability analysis is done to validate the convergence, accuracy and exactness of the proposed scheme.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Olver, P.J.: Introduction to Partial Differential Equations, pp. 182–184. Springer, New York (2014)

    Book  Google Scholar 

  2. Saadatmandi, A., Dehghan, M.: The use of Sinc-collocation method for solving multi-point boundary value problems. Commun. Nonlinear Sci. Numer. Simul. 17(2), 593–601 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Stenger, F.: Numerical Methods Based on Sinc and Analytic Functions. Springer, New York (1993)

    Book  MATH  Google Scholar 

  4. Dehghan, Mehdi: Saadatmandi: the numerical solution of a nonlinear system of second-order boundary value problems using the Sinc-collocation method. Math. Comput. Model. 46(11), 1434–1441 (2007)

    Article  MATH  Google Scholar 

  5. Sugihara, M., Matsuo, T.: Recent developments of the Sinc numerical methods. J. Comput. Appl. Math. 164, 673–689 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Alali, A., Schöffel, P.J., Herb, J., Macian, R.: Numerical investigations on the coupling of the one-group interfacial area transport equation and subcooled boiling models for nuclear safety applications. Ann. Nucl. Energy 120, 155–168 (2018)

    Article  Google Scholar 

  7. Mohammadi, V., Dehghan, M., Khodadadian, A., Wick, T.: Numerical investigation on the transport equation in spherical coordinates via generalized moving least squares and moving kriging least squares approximations. Eng. Comput. 37, 1231–1249 (2019)

    Article  Google Scholar 

  8. Ackleh, A.S., Saintier, N., Skrzeczkowski, J.: Sensitivity equations for measure-valued solutions to transport equations. Math. Biosci. Eng 17, 514–537 (2020)

    Article  MathSciNet  Google Scholar 

  9. Sharp, M.K., Carare, R.O., Martin, B.A.: Dispersion in porous media in oscillatory flow between flat plates: applications to intrathecal, periarterial and paraarterial solute transport in the central nervous system. Fluids Barriers CNS 16(1), 13 (2019)

    Article  Google Scholar 

  10. Jiakuan, X.U., Lei, Q.I.A.O., Junqiang, B.A.I.: Improved local amplification factor transport equation for stationary crossflow instability in subsonic and transonic flows. Chin. J. Aeronaut. 33, 3073–3081 (2020)

    Article  Google Scholar 

  11. McGaughey, A.J., Jain, A., Kim, H.Y., Fu, B.: Phonon properties and thermal conductivity from first principles, lattice dynamics, and the Boltzmann transport equation. J. Appl. Phys. 125(1), 011101 (2019)

    Article  Google Scholar 

  12. Moore, W.: Schrodinger. Cambridge University Press, Cambridge (2015)

    Book  MATH  Google Scholar 

  13. Musa, A.E., Elammeen, G.E., Mahmoud, H.M., Mohammed, E.E., Abdallah, M.D., Ahmed, S.A.E.: Energy quantization of electrons for spherically symmetric atoms and nano particles according to Schrödinger equation

  14. Lowney, J.D.: Manipulating and probing angular momentum and quantized circulation in optical fields and matter waves (2016)

  15. Lu, D., Seadawy, A.R., Khater, M.M.: Structures of exact and solitary optical solutions for the higher-order nonlinear Schrödinger equation and its applications in mono-mode optical fibers. Mod. Phys. Lett. B 33(23), 1950279 (2019)

    Article  Google Scholar 

  16. Seadawy, A.R., Cheemaa, N.: Applications of extended modified auxiliary equation mapping method for high-order dispersive extended nonlinear Schrödinger equation in nonlinear optics. Mod. Phys. Lett. B 33(18), 1950203 (2019)

    Article  Google Scholar 

  17. Arshad, M., Seadawy, A.R., Lu, D., Jun, W.: Optical soliton solutions of unstable nonlinear Schröodinger dynamical equation and stability analysis with applications. Optik 157, 597–605 (2018)

    Article  Google Scholar 

  18. Sultan, A.M., Lu, D., Arshad, M., Rehman, H.U., Saleem, M.S.: Soliton solutions of higher order dispersive cubic-quintic nonlinear Schrödinger equation and its applications. Chin. J. Phys. 67, 405–413 (2020)

    Article  Google Scholar 

  19. Seadawy, A.R.: Nonlinear wave solutions of the three-dimensional Zakharov–Kuznetsov–Burgers equation in dusty plasma. Physica A 439, 124–131 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Dehghan, M.: Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math. Comput. Simul. 71(1), 16–30 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Triki, H., Biswas, A.: Dark solitons for a generalized nonlinear Schrödinger equation with parabolic law and dual-power law nonlinearities. Math. Methods Appl. Sci. 34(8), 958–962 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhang, L.H., Si, J.G.: New soliton and periodic solutions of (1+ 2)-dimensional nonlinear Schrödinger equation with dual-power law nonlinearity. Commun. Nonlinear Sci. Numer. Simul. 15(10), 2747–2754 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Dehghan, M., Mohammadi, V.: A numerical scheme based on radial basis function finite difference (RBF-FD) technique for solving the high-dimensional nonlinear Schrödinger equations using an explicit time discretization: Runge–Kutta method. Comput. Phys. Commun. 217, 23–34 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Berezin, F.A., Shubin, M.: The Schrödinger Equation, vol. 66. Springer, Berlin (2012)

    Google Scholar 

  25. Mio, K., Ogino, T., Minami, K., Takeda, S.: Modified nonlinear Schrödinger equation for Alfvén waves propagating along the magnetic field in cold plasmas. J. Phys. Soc. Jpn. 41(1), 265–271 (1976)

    Article  MATH  Google Scholar 

  26. Saadatmandi, A., Razzaghi, M.: The numerical solution of third-order boundary value problems using Sinc-collocation method. Commun. Numer. Methods Eng. 23, 681–689 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sulaiman, T.A., Bulut, H., Atas, S.S.: Optical solitons to the fractional Schrödinger–Hirota equation. Appl. Math. Nonlinear Sci. 4(2), 535–542 (2019)

    Article  MathSciNet  Google Scholar 

  28. Modanli, M., Akgül, A.: On solutions of fractional order telegraph partial differential equation by Crank–Nicholson finite difference method. Appl. Math. Nonlinear Sci. 5(1), 163–170 (2020)

    Article  MathSciNet  Google Scholar 

  29. Aksoy, N.Y.: The solvability of first type boundary value problem for a Schrödinger equation. Appl. Math. Nonlinear Sci. 5(1), 211–220 (2020)

    Article  MathSciNet  Google Scholar 

  30. Wu, X., Kong, W., Li, C.: Sinc collocation method with boundary treatment for two-point boundary value problem. J. Comput. Appl. Math. https://doi.org/10.1016/j.cam.2005.09.003

  31. Khan, R.A., Usman, M.: Eventual periodicity of forced oscillations of the Korteweg–de Vries type equation. Appl. Math. Model. 36(2), 736–742 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Bialecki, B.: Sinc-collocation methods for two-point boundary value problems. IMA J. Numer. Anal. 11, 357–375 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  33. Dehghan, M., Emami-Naeini, F.: The Sinc-collocation and Sinc-Galerkin methods for solving the two-dimensional Schrödinger equation with nonhomogeneous boundary conditions. Appl. Math. Model. 37(22), 9379–9397 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Parand, K., Pirkhedri, A.: Sinc-collocation method for solving astrophysics equations. New Astron. 15(6), 533–537 (2010)

    Article  Google Scholar 

  35. Parand, K., Dehghan, M., Pirkhedri, A.: Sinc-collocation method for solving the Blasius equation. Phys. Lett. A 373(44), 4060–4065 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. Babaei, A., Moghaddam, B.P., Banihashemi, S., Machado, J.A.T.: Numerical solution of variable-order fractional integro-partial differential equations via Sinc collocation method based on single and double exponential transformations. Commun. Nonlinear Sci. Numer. Simul. 82, 104985 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  37. Fariborzi, M.A., Noeiaghdam, S.: Valid implementation of the Sinc-collocation method to solve linear integral equations by the CADNA library. J. Math. Model. 7(1), 63–84 (2019)

    MathSciNet  MATH  Google Scholar 

  38. Qiu, W., Xu, D., Guo, J.: Numerical solution of the fourth-order partial integro-differential equation with multi-term kernels by the Sinc-collocation method based on the double exponential transformation. Appl. Math. Comput. 392, 125693 (2020)

    MathSciNet  MATH  Google Scholar 

  39. Kong, D., Xu, Y., Zheng, Z.: A hybrid numerical method for the KdV equation by finite difference and sinc collocation method. Appl. Math. Comput. 355, 61–72 (2019)

    MathSciNet  MATH  Google Scholar 

  40. Lund, J., Bowers, K.: Sinc Methods for Quadrature and Differential Equations. SIAM, Philadelphia, PA (1992)

    Book  MATH  Google Scholar 

  41. Bialecki, B.: Sinc-collocation methods for two-point boundary value problems. IMA J. Numer. Anal. 11, 357–375 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  42. Mundewadi, R.A., Kumbinarasaiah, S.: Numerical solution of Abel’s integral equations using Hermite Wavelet. Appl. Math. Nonlinear Sci. 4(2), 395–406 (2019)

    Article  MathSciNet  Google Scholar 

  43. Cordero Barbero, A., Jaiswal, J.P., Torregrosa Sánchez, J.R.: Stability analysis of fourth-order iterative method for finding multiple roots of nonlinear equations. Appl. Math. Nonlinear Sci. 4(1), 43–56 (2019)

    Article  MathSciNet  Google Scholar 

  44. Baskonus, H.M., Bulut, H., Sulaiman, T.A.: New complex hyperbolic structures to the Lonngren-wave equation by using sine–Gordon expansion method. Appl. Math. Nonlinear Sci. 4(1), 141–150 (2019)

    MathSciNet  Google Scholar 

  45. Stenger, F.: Numerical Methods Based on Sinc and Analytic Functions, vol. 20. Springer, Berln (2012)

    MATH  Google Scholar 

  46. Stenger, F.: Numerical methods based on Whittaker cardinal, or sinc functions. SIAM Rev. 23(2), 165–224 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  47. Stenger, F.: Bounds on the error of Gauss type quadratures. Numer. Math. 8, 150–160 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  48. Lund, J., Bowers, K.L.: Sinc Methods for Quadrature and Differential Equations. Society for Industrial and Applied Mathematics, Philadelphia (1992)

    Book  MATH  Google Scholar 

  49. Zarebnia, M., Nikpour, Z.: Solution of linear Volterra integro-differential equations via Sinc functions. Int. J. Appl. Math. Comput 2, 1–10 (2010)

    Google Scholar 

  50. Pindza, E., Maré, E.: Sinc collocation method for solving the Benjamin-Ono equation. J. Comput. Methods Phys. 2014, 392962 (2014)

  51. Stenger, F.: Numerical Methods Based on Sinc and Analytic Functions, vol. 20. Springer, Berlin (2012)

    MATH  Google Scholar 

  52. Stenger, F.: Matrices of Sinc method. J. Comput. Appl. Math. 86(1), 297–310 (1997)

  53. Al-Khaled, K.: Sinc numerical solution for solitons and solitary waves. J. Comput. Appl. Math. 130(1–2), 283–292 (2001)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This research was partially funded by Ministerio de Ciencia, Innovación y Universidades, Grant Number PGC2018-097198-B-I00, and by Fundación Séneca of Región de Murcia, Grant Number 20783/PI/18. This research was supported by the National Research Program for Universities (NRPU), Higher Education Commission, Pakistan, No.  8103/Punjab/NRPU/R and D/HEC/2017.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Gujrat, Gujrat, 50700, Pakistan

    Iftikhar Ahmad, Syed Ibrar Hussain, Hira Ilyas & Shabnam Rehmat

  2. Department of Applied Mathematics and Statistics, Technical University of Cartagena, Cartagena, Spain

    Juan Luis García Guirao

  3. Institute for Mathematics, Technical University, Clausthal, Germany

    Adeel Ahmed

  4. Department of Mathematics, Faculty of Science, King Abdulaziz University, 80203, Jeddah, 21589, Saudi Arabia

    Juan Luis García Guirao & Tareq Saeed

Authors
  1. Iftikhar Ahmad
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Syed Ibrar Hussain
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hira Ilyas
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Juan Luis García Guirao
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Adeel Ahmed
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Shabnam Rehmat
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Tareq Saeed
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Juan Luis García Guirao.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ahmad, I., Hussain, S.I., Ilyas, H. et al. Numerical solutions of Schrödinger wave equation and Transport equation through Sinc collocation method. Nonlinear Dyn 105, 691–705 (2021). https://doi.org/10.1007/s11071-021-06596-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-021-06596-9

Keywords

Search

Navigation