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Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices

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  • Dehghan, Mehdi

Abstract

Several finite difference schemes are discussed for solving the two-dimensional Schrodinger equation with Dirichlet’s boundary conditions. We use three fully implicit finite difference schemes, two fully explicit finite difference techniques, an alternating direction implicit procedure and the Barakat and Clark type explicit formula. Theoretical and numerical comparisons between four families of methods are described. The main advantage of the alternating direction implicit finite difference technique is that the bandwidth of the sets of equations is a fixed small number that depends only on the nature of the computational molecule. This allows the use of very efficient and very fast techniques for solving the resulting tridiagonal systems of linear algebraic equations. The unique advantage of the Barakat and Clark technique is that it is unconditionally stable and is explicit in nature. Numerical results are presented followed by concluding remarks.

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  • Dehghan, Mehdi, 2006. "Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 71(1), pages 16-30.
    • Handle: RePEc:eee:matcom:v:71:y:2006:i:1:p:16-30
    DOI: 10.1016/j.matcom.2005.10.001
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    1. Dehghan, Mehdi, 1999. "Implicit locally one-dimensional methods for two-dimensional diffusion with a non-local boundary condition," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 49(4), pages 331-349.
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    Cited by:

    1. Pathak, Maheshwar & Joshi, Pratibha & Nisar, Kottakkaran Sooppy, 2022. "Numerical study of generalized 2-D nonlinear Schrödinger equation using Kansa method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 200(C), pages 186-198.
    2. Dehghan, Mehdi & Saadatmandi, Abbas, 2009. "Variational iteration method for solving the wave equation subject to an integral conservation condition," Chaos, Solitons & Fractals, Elsevier, vol. 41(3), pages 1448-1453.
    3. Kaur, Navneet & Joshi, Varun, 2024. "Kuramoto-Sivashinsky equation: Numerical solution using two quintic B-splines and differential quadrature method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 220(C), pages 105-127.
    4. Guo, Geyang & Lü, Shujuan & Liu, Bo, 2015. "Unconditional stability of alternating difference schemes with variable time steplengthes for dispersive equation," Applied Mathematics and Computation, Elsevier, vol. 262(C), pages 249-259.
    5. Nehad Ali Shah & Ioannis Dassios & Essam R. El-Zahar & Jae Dong Chung & Somaye Taherifar, 2021. "The Variational Iteration Transform Method for Solving the Time-Fractional Fornberg–Whitham Equation and Comparison with Decomposition Transform Method," Mathematics, MDPI, vol. 9(2), pages 1-14, January.
    6. Wang, Hanquan & Ma, Xiu & Lu, Junliang & Gao, Wen, 2017. "An efficient time-splitting compact finite difference method for Gross–Pitaevskii equation," Applied Mathematics and Computation, Elsevier, vol. 297(C), pages 131-144.
    7. Kenzu Abdella & Jeet Trivedi, 2020. "Solving Multi-Point Boundary Value Problems Using Sinc-Derivative Interpolation," Mathematics, MDPI, vol. 8(12), pages 1-14, November.
    8. Mohammadi, Reza, 2018. "Smooth Quintic spline approximation for nonlinear Schrödinger equations with variable coefficients in one and two dimensions," Chaos, Solitons & Fractals, Elsevier, vol. 107(C), pages 204-215.
    9. Dehghan, Mehdi & Shokri, Ali, 2008. "A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(3), pages 700-715.
    10. Shi, Dongyang & Liao, Xin & Wang, Lele, 2016. "Superconvergence analysis of conforming finite element method for nonlinear Schrödinger equation," Applied Mathematics and Computation, Elsevier, vol. 289(C), pages 298-310.
    11. Dehghan, Mehdi & Mohebbi, Akbar, 2008. "High-order compact boundary value method for the solution of unsteady convection–diffusion problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(3), pages 683-699.
    12. Gao, Yali & Mei, Liquan & Li, Rui, 2018. "Galerkin methods for the Davey–Stewartson equations," Applied Mathematics and Computation, Elsevier, vol. 328(C), pages 144-161.
    13. Dehghan, Mehdi & Gharibi, Zeinab, 2021. "Numerical analysis of fully discrete energy stable weak Galerkin finite element Scheme for a coupled Cahn-Hilliard-Navier-Stokes phase-field model," Applied Mathematics and Computation, Elsevier, vol. 410(C).
    14. Ghosh, Suchismita & Deb, Anish & Sarkar, Gautam, 2016. "Taylor series approach for function approximation using ‘estimated’ higher derivatives," Applied Mathematics and Computation, Elsevier, vol. 284(C), pages 89-101.
    15. Ballestra, Luca Vincenzo & Pacelli, Graziella & Radi, Davide, 2016. "A very efficient approach for pricing barrier options on an underlying described by the mixed fractional Brownian motion," Chaos, Solitons & Fractals, Elsevier, vol. 87(C), pages 240-248.
    16. Luo, Yidong, 2020. "Galerkin method with trigonometric basis on stable numerical differentiation," Applied Mathematics and Computation, Elsevier, vol. 370(C).

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