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Kapoor, Mamta, Arora, Geeta, Joshi, Varun. (1405). Numerical approximation of coupled Schrödinger equations via NUAH B-spline DQM. سامانه مدیریت نشریات علمی, 14(2), 519-548. doi: 10.22034/cmde.2025.64656.2935
Mamta Kapoor; Geeta Arora; Varun Joshi. "Numerical approximation of coupled Schrödinger equations via NUAH B-spline DQM". سامانه مدیریت نشریات علمی, 14, 2, 1405, 519-548. doi: 10.22034/cmde.2025.64656.2935
Kapoor, Mamta, Arora, Geeta, Joshi, Varun. (1405). 'Numerical approximation of coupled Schrödinger equations via NUAH B-spline DQM', سامانه مدیریت نشریات علمی, 14(2), pp. 519-548. doi: 10.22034/cmde.2025.64656.2935
Kapoor, Mamta, Arora, Geeta, Joshi, Varun. Numerical approximation of coupled Schrödinger equations via NUAH B-spline DQM. سامانه مدیریت نشریات علمی, 1405; 14(2): 519-548. doi: 10.22034/cmde.2025.64656.2935

Numerical approximation of coupled Schrödinger equations via NUAH B-spline DQM

Computational Methods for Differential Equations
مقاله 2، دوره 14، شماره 2، تیر 2026، صفحه 519-548    اصل مقاله (1.7 M)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2025.64656.2935
نویسندگان
Mamta Kapoor* 1؛ Geeta Arora2؛ Varun Joshi2
1Marwadi University Research Center, Department of Mathematics, Faculty of Engineering & Technology, Marwadi University, Rajkot, 360003, Gujarat, India.
2Department of Mathematics, Lovely Professional University, Phagwara, Punjab, India-144411.
چکیده
The goal of the current study is to offer a novel method for numerically solving coupled 1D and 2D nonlinear Schrödinger equations. To discretize the spatial partial derivative, we applied the MCNUAH B-spline DQM. The SSP-RK43 technique is used to solve the reduced system of ODEs. Via the matrix method, the stability of the proposed method is investigated, and it is found to be stable. Four experiments are used to confirm the efficiency of the suggested scheme, and data from the literature are compared throughout. It is clear that the obtained results are satisfactory and in strong accord with preceding results. This approach yields superior outcomes and is effective, straightforward, and reasonably simple to use. The graphical abstract is provided as per Figure 1.
کلیدواژه‌ها
Differential quadrature method؛ NUAH B-spline؛ SSP-RK43 scheme؛ Matrix stability method
مراجع
  • [1] R. Abazari and R. Abazari, Numerical study of some coupled PDEs by using differential transformation method, In Proceedings of World Academy of Science, Engineering and Technology, 66 (2010, June), 52-59.
  • [2] M. J. Ablowitz and H. Segur, Solitons and the inverse scattering transform, Siam, 4 (1981).
  • [3] B. Ali, An effective approximation to the dispersive soliton solutions of the coupled KdV equation via combination of two efficient methods, [Journal details not provided].
  • [4] S. Alipour and F. Mirzaee, An iterative algorithm for solving two-dimensional nonlinear stochastic integral equations: A combined successive approximations method with bilinear spline interpolation, Applied Mathematics and Computation, 371 (2020), 124947.
  • [5] G. Arora and B. K. Singh, Numerical solution of Burgers’ equation with modified cubic B-spline differential quadrature method, Applied Mathematics and Computation, 224 (2013), 166-177.
  • [6] G. Arora and V. Joshi, A computational approach using modified trigonometric cubic B-spline for numerical solution of Burgers’ equation in one and two dimensions, Alexandria Engineering Journal, 57(2) (2018), 10871098.
  • [7] G. Arora, V. Joshi, and R. C. Mittal, Numerical Simulation of Nonlinear Schrödinger Equation in One and Two Dimensions, Mathematical Models and Computer Simulations, 11(4) (2019), 634-648.
  • [8] A. Aydin and B. Karasözen, Multi-symplectic integration of coupled non-linear Schro¨dinger system with soliton solutions, International Journal of Computer Mathematics, 86(5) (2009), 864-882.
  • [9] A. Başhan, A mixed methods approach to Schrödinger equation: Finite difference method and quartic B-spline based differential quadrature method, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 9(2) (2019), 223-235.
  • [10] A. Başhan, A novel approach via mixed Crank–Nicolson scheme and differential quadrature method for numerical solutions of solitons of mKdV equation, Pramana, 92(6) (2019), 84.
  • [11] A. Başhan and N. M. Yağmurlu, A mixed method approach to the solitary wave, undular bore and boundary-forced solutions of the Regularized Long Wave equation, Computational and Applied Mathematics, 41(4) (2022), 169.
  • [12] A. Bashan, N. M. Yağmurlu, Y. Uçar, and A. Esen, An effective approach to numerical soliton solutions for the Schrödinger equation via modified cubic B-spline differential quadrature method, Chaos, Solitons & Fractals, 100 (2017), 45-56.
  • [13] A. Başhan, Y. Uçar, N. M. Yağmurlu, and A. Esen, Numerical solution of the complex modified Korteweg-de Vries equation by DQM, In Journal of Physics: Conference Series, 766(1) (2016, October), 012028, IOP Publishing.
  • [14] A. Başhan, Y. Uçar, N. M. Yağmurlu, and A. Esen, A new perspective for quintic B-spline based Crank-Nicolsondifferential quadrature method algorithm for numerical solutions of the nonlinear Schrödinger equation, The European Physical Journal Plus, 133(1) (2018), 12.
  • [15] R. Bellman, B. G. Kashef, and J. Casti, Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations, Journal of Computational Physics, 10(1) (1972), 40-52.
  • [16] J. Cao and G. Z. Wang, Non-uniform B-spline curves with multiple shape parameters, Journal of Zhejiang University Science C, 12(10) (2011), 800.
  • [17] M. Dehghan, M. Abbaszadeh, and A. Mohebbi, Numerical solution of system of N-coupled nonlinear Schrödinger equations via two variants of the meshless local Petrov-Galerkin (MLPG) method, Comput. Model. Eng. Sci, 100(5) (2014), 399-444.
  • [18] M. E. Fang and G. Wang, ωB-splines, Science in China Series F: Information Sciences, 51(8) (2008), 1167-1176.
  • [19] Z. Gao, S. Song, and J. Duan, The application of (2+1)-dimensional coupled nonlinear Schrödinger equations with variable coefficients in optical fibers, Optik, 172 (2018), 953-967.
  • [20] S. Hajji, B. Danouj, and A. Lamnii, UAT B-Splines of Order 4 for Reconstructing the Curves and Surfaces, Applied Mathematical Sciences, 12(21) (2018), 1007-1020.
  • [21] A. Hasegawa, Optical solitons in fibers, In Optical Solitons in Fibers, Springer, Berlin, Heidelberg, 1 (1989), 1-74.
  • [22] M. S. Ismail, A fourth-order explicit scheme for the coupled nonlinear Schrödinger equation, Applied Mathematics and Computation, 196(1) (2008), 273-284.
  • [23] M. S. Ismail, Numerical solution of coupled nonlinear Schrödinger equation by Galerkin method, Mathematics and Computers in Simulation, 78(4) (2008), 532-547.
  • [24] M. S. Ismail, H. A. Ashi, and F. Al-Rakhemy, ADI method for solving the two-dimensional coupled nonlinear Schrödinger equation, In AIP Conference Proceedings, 1648(1) (2015, March), 050008, AIP Publishing LLC.
  • [25] M. S. Ismail and T. R. Taha, A linearly implicit conservative scheme for the coupled nonlinear Schrödinger equation, Mathematics and Computers in Simulation, 74(4-5) (2007), 302-311.
  • [26] M. S. Ismail and T. R. Taha, Numerical simulation of coupled nonlinear Schrödinger equation, Mathematics and Computers in Simulation, 56(6) (2001), 547-562.
  • [27] M. K. Jain, Numerical solution of differential equations, 2nd ed., Wiley, New York, 1 (1983).
  • [28] R. Jiwari, R. C. Mittal, and K. K. Sharma, A numerical scheme based on weighted average differential quadrature method for the numerical solution of Burgers’ equation, Applied Mathematics and Computation, 219(12) (2013), 6680-6691.
  • [29] R. Jiwari, S. Pandit, and R. C. Mittal, A differential quadrature algorithm for the numerical solution of the second-order one dimensional hyperbolic telegraph equation, International Journal of Nonlinear Science, 13(3) (2012), 259-266.
  • [30] R. Jiwari, S. Pandit, and R. C. Mittal, A differential quadrature algorithm to solve the two dimensional linear hyperbolic telegraph equation with Dirichlet and Neumann boundary conditions, Applied Mathematics and Computation, 218(13) (2012), 7279-7294.
  • [31] R. Jiwari, S. Pandit, and R. C. Mittal, Numerical simulation of two-dimensional sine-Gordon solitons by differential quadrature method, Computer Physics Communications, 183(3) (2012), 600-616.
  • [32] A. Korkmaz and I. Dağ, A differential quadrature algorithm for simulations of nonlinear Schrödinger equation, Computers & Mathematics with Applications, 56(9) (2008), 2222-2234.
  • [33] A. Korkmaz and I. Dağ, Cubic B-spline differential quadrature methods for the advection-diffusion equation, International Journal of Numerical Methods for Heat & Fluid Flow, 22(8) (2012), 1021-1036.
  • [34] A. Korkmaz and I. Dağ, Shock wave simulations using sinc differential quadrature method, Engineering Computations, 28(6) (2011), 654-674.
  • [35] A. Korkmaz and I. Dağ, Solitary wave simulations of complex modified Korteweg–de Vries equation using differential quadrature method, Computer Physics Communications, 180(9) (2009), 1516-1523.
  • [36] B. I. Kvasov, GB-splines and their properties, International Journal of Annals of Numerical Mathematics, 3 (1996), 139-149.
  • [37] A. Lamnii, F. Oumellal, and J. Dabounou, Tension Quartic Trigonometric Bézier Curves Preserving Interpolation Curves Shape, International Journal of Mathematical Modelling & Computations, 5(2) (2015), 99-109.
  • [38] Y. Lü, G. Wang, and X. Yang, Uniform hyperbolic polynomial B-spline curves, Computer Aided Geometric Design, 19(6) (2002), 379-393.
  • [39] F. Mirzaee and S. Alipour, A hybrid approach of nonlinear partial mixed integro-differential equations of fractional order, Iranian Journal of Science and Technology, Transactions A: Science, 44(3) (2020), 725-737.
  • [40] F. Mirzaee and S. Alipour, An efficient cubic B-spline and bicubic B-spline collocation method for numerical solutions of multidimensional nonlinear stochastic quadratic integral equations, Mathematical Methods in the Applied Sciences, 43(1) (2020), 384-397.
  • [41] F. Mirzaee and S. Alipour, Bicubic B-spline functions to solve linear two-dimensional weakly singular stochastic integral equation, Iranian Journal of Science and Technology, Transactions A: Science, 45(3) (2021), 965-972.
  • [42] F. Mirzaee and S. Alipour, Cubic B-spline approximation for linear stochastic integro-differential equation of fractional order, Journal of Computational and Applied Mathematics, 366 (2020), 112440.
  • [43] F. Mirzaee and S. Alipour, Quintic B-spline collocation method to solve n-dimensional stochastic Itˆo-Volterra integral equations, Journal of Computational and Applied Mathematics, 384 (2021), 113153.
  • [44] F. Mirzaee, S. Alipour, and N. Samadyar, A numerical approach for solving weakly singular partial integrodifferential equations via two-dimensional-orthonormal Bernstein polynomials with the convergence analysis, Numerical Methods for Partial Differential Equations, 35(2) (2019), 615-637.
  • [45] R. C. Mittal and R. Bhatia, A numerical study of two dimensional hyperbolic telegraph equation by modified B-spline differential quadrature method, Applied Mathematics and Computation, 244 (2014), 976-997.
  • [46] R. C. Mittal and R. Rohila, Numerical simulation of reaction-diffusion systems by modified cubic B-spline differential quadrature method, Chaos, Solitons & Fractals, 92 (2016), 9-19.
  • [47] R. C. Mittal and S. Dahiya, A comparative study of modified cubic B-spline differential quadrature methods for a class of nonlinear viscous wave equations, Engineering Computations, 35(1) (2018), 315-333.
  • [48] R. C. Mittal and S. Dahiya, Numerical simulation of three-dimensional telegraphic equation using cubic B-spline differential quadrature method, Applied Mathematics and Computation, 313 (2017), 442-452.
  • [49] R. C. Mittal and S. Dahiya, Numerical simulation on hyperbolic diffusion equations using modified cubic B-spline differential quadrature methods, Computers & Mathematics with Applications, 70(5) (2015), 737-749.
  • [50] H. Pottmann and M. G. Wagner, Helix splines as an example of affine Tchebycheffian splines, Advances in Computational Mathematics, 2(1) (1994), 123-142.
  • [51] J. Qian and Y. Tang, On non-uniform Algebraic-Hyperbolic (NUAH) B-splines, Numerical Mathematics, 15(4) (2006), 320.
  • [52] J. R. Quan and C. T. Chang, New insights in solving distributed system equations by the quadrature method-I. Analysis, Computers & Chemical Engineering, 13(7) (1989), 779-788.
  • [53] J. R. Quan and C. T. Chang, New insights in solving distributed system equations by the quadrature method-II. Numerical experiments, Computers & Chemical Engineering, 13(9) (1989), 1017-1024.
  • [54] W. Shen and G. Wang, Changeable degree spline basis functions, Journal of Computational and Applied Mathematics, 234(8) (2010), 2516-2529.
  • [55] C. Shu, Differential quadrature and its application in engineering, Springer Science & Business Media, 1 (2012).
  • [56] C. Shu, Generalized differential-integral quadrature and application to the simulation of incompressible viscous flows including parallel computation, (Doctoral dissertation, ProQuest Dissertations & Theses) (1991).
  • [57] C. Shu and B. E. Richards, High resolution of natural convection in a square cavity by generalized differential quadrature, In Proceedings of the 3rd international conference on advances in numeric methods in engineering: theory and application, Swansea, UK, 1 (1990, January), 978-985.
  • [58] C. Shu and H. Xue, Explicit computation of weighting coefficients in the harmonic differential quadrature, Journal of Sound and Vibration, 204(3) (1997), 549-555.
  • [59] H. S. Shukla, M. Tamsir, R. Jiwari, and V. K. Srivastava, A numerical algorithm for computation modelling of 3D nonlinear wave equations based on exponential modified cubic B-spline differential quadrature method, International Journal of Computer Mathematics, 95(4) (2018), 752-766.
  • [60] W. J. Sonnier and C. I. Christov, Strong coupling of Schrödinger equations: Conservative scheme approach, Mathematics and Computers in Simulation, 69(5-6) (2005), 514-525.
  • [61] A. G. Striz, X. Wang, and C. W. Bert, Harmonic differential quadrature method and applications to analysis of structural components, Acta Mechanica, 111(1-2) (1995), 85-94.
  • [62] J. Q. Sun and M. Z. Qin, Multi-symplectic methods for the coupled 1D nonlinear Schrödinger system, Computer Physics Communications, 155(3) (2003), 221-235.
  • [63] J. Q. Sun, X. Y. Gu, and Z. Q. Ma, Numerical study of the soliton waves of the coupled nonlinear Schrödinger system, Physica D: Nonlinear Phenomena, 196(3-4) (2004), 311-328.
  • [64] N. H. Sweilam and R. F. Al-Bar, Variational iteration method for coupled nonlinear Schrödinger equations, Computers & Mathematics with Applications, 54(7-8) (2007), 993-999.
  • [65] S. Tomasiello, Numerical solutions of the Burgers–Huxley equation by the IDQ method, International Journal of Computer Mathematics, 87(1) (2010), 129-140.
  • [66] M. Wadati, T. Iizuka, and M. Hisakado, A coupled nonlinear Schrödinger equation and optical solitons, Journal of the Physical Society of Japan, 61(7) (1992), 2241-2245.
  • [67] H. Wang, Numerical studies on the split-step finite difference method for nonlinear Schrödinger equations, Applied Mathematics and Computation, 170(1) (2005), 17-35.
  • [68] G. Wang, Q. Chen, and M. Zhou, NUAT B-spline curves, Computer Aided Geometric Design, 21(2) (2004), 193-205.
  • [69] Y. W. Wei and G. Z. Wang, An orthogonal basis for non-uniform algebraic-trigonometric spline space, Applied Mathematics-A Journal of Chinese Universities, 29(3) (2014), 273-282.
  • [70] W. X. Wu, C. Shu, and C. M. Wang, Vibration analysis of arbitrarily shaped membranes using local radial basis function-based differential quadrature method, Journal of Sound and Vibration, 306(1-2) (2007), 252-270.
  • [71] J. Xie and J. Tan, Quasi-Cubic B-Spline Curves by Trigonometric Polynomials, Journal of Information & Computational Science, 7(6) (2010), 1215-1221.
  • [72] Y. Xu and C. W. Shu, Local discontinuous Galerkin methods for nonlinear Schrödinger equations, Journal of Computational Physics, 205(1) (2005), 72-97.
  • [73] G. Xu and G. Z. Wang, AHT Bézier curves and NUAHT B-spline curves, Journal of Computer Science and Technology, 22(4) (2007), 597-607.
  • [74] G. Xu and G. Z. Wang, Extended cubic uniform B-spline and α-B-spline, Acta Automatica Sinica, 34(8) (2008), 980-984.
  • [75] J. Yang, Multisoliton perturbation theory for the Manakov equations and its applications to nonlinear optics, Physical Review E, 59(2) (1999), 2393.
  • [76] J. Zhang, C-curves: an extension of cubic curves, Computer Aided Geometric Design, 13(3) (1996), 199-217.
  • [77] J. Zhang, Two different forms of CB-splines, Computer Aided Geometric Design, 14(1) (1997), 31-41.
  • [78] H. Q. Zhang, X. H. Meng, T. Xu, L. L. Li, and B. Tian, Interactions of bright solitons for the (2+1)-dimensional coupled nonlinear Schrödinger equations from optical fibres with symbolic computation, Physica Scripta, 75(4) (2007), 537.
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