New spectral collocation algorithms for one- and two-dimensional Schrödinger equations with a Kerr law nonlinearity

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• 作者：Ali H Bhrawy ; Fouad Mallawi ; Mohamed A Abdelkawy
• 关键词：one ; dimensional Schrödinger equations ; Kerr law nonlinearity ; two ; dimensional space Schrödinger equations ; collocation method ; Gauss ; type quadratures
• 刊名：Advances in Difference Equations
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：2016
• 期：1
• 全文大小：1,899 KB
• 参考文献：1. Canuto, C, Hussaini, MY, Quarteroni, A, Zang, TA: Spectral Methods: Fundamentals in Single Domains. Springer, New York (2006)
  2. Bhrawy, AH: A Jacobi spectral collocation method for solving multi-dimensional nonlinear fractional sub-diffusion equations. Numer. Algorithms (2015). doi:10.​1007/​s11075-015-0087-2
  3. Heinrichs, W: Spectral methods with sparse matrices. Numer. Math. 56, 25-41 (1989) CrossRef MathSciNet MATH
  4. Heinrichs, W: Algebraic spectral multigrid methods. Comput. Methods Appl. Mech. Eng. 80, 281-286 (1990) CrossRef MathSciNet MATH
  5. Bhrawy, AH: An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system. Appl. Math. Comput. 247, 30-46 (2014) CrossRef MathSciNet
  6. Nemati, S: Numerical solution of Volterra-Fredholm integral equations using Legendre collocation method. J. Comput. Appl. Math. 278, 29-36 (2015) CrossRef MathSciNet MATH
  7. Doha, EH, Bhrawy, AH: A Jacobi spectral Galerkin method for the integrated forms of fourth-order elliptic differential equations. Numer. Methods Partial Differ. Equ. 25, 712-739 (2009) CrossRef MathSciNet MATH
  8. Tatari, M, Haghighi, M: A generalized Laguerre-Legendre spectral collocation method for solving initial-boundary value problems. Appl. Math. Model. 38, 1351-1364 (2014) CrossRef MathSciNet
  9. Bhrawy, AH, Zaky, MA: Shifted fractional-order Jacobi orthogonal functions: application to a system of fractional differential equations. Appl. Math. Model. 40, 832-845 (2016) CrossRef MathSciNet
  10. Bhrawy, AH, Zaky, MA: A fractional-order Jacobi tau method for a class of time-fractional PDEs with variable coefficients. Math. Methods Appl. Sci. (2015). doi:10.​1002/​mma.​3600
  11. Bhrawy, AH, Abdelkawy, MA, Mallawi, F: An accurate Chebyshev pseudospectral scheme for multi-dimensional parabolic problems with time delays. Bound. Value Probl. 2015(1), 103 (2015) CrossRef MathSciNet
  12. Bhrawy, AH: A highly accurate collocation algorithm for \(1+1\) and \(2+1\) fractional percolation equations. J. Vib. Control (2015). doi:10.​1177/​1077546315597815​
  13. Nemati, S: Numerical solution of Volterra-Fredholm integral equations using Legendre collocation method. J. Comput. Appl. Math. 278, 29-36 (2015) CrossRef MathSciNet MATH
  14. Bhrawy, AH, Ezz-Eldien, SS: A new Legendre operational technique for delay fractional optimal control problems. Calcolo (2015). doi:10.​1007/​s10092-015-0160-1
  15. Bhrawy, AH, Zaky, MA: A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations. J. Comput. Phys. 281(15), 876-895 (2015) CrossRef MathSciNet
  16. Bhrawy, AH, Doha, EH, Ezz-Eldien, SS, Abdelkawy, MA: A numerical technique based on the shifted Legendre polynomials for solving the time-fractional coupled KdV equation. Calcolo (2015). doi:10.​1007/​s10092-014-0132-x
  17. Bhrawy, AH, Zaky, MA: Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation. Nonlinear Dyn. 80, 101-116 (2015) CrossRef MathSciNet
  18. D’Avenia, P, Montefusco, E, Squassina, M: On the logarithmic Schrödinger equation. Commun. Contemp. Math. 16, 1350032 (2014) CrossRef MathSciNet
  19. Zhang, Z-Y, Li, Y-X, Liu, Z-H, Mia, X-J: New exact solutions to the perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity via modified trigonometric function series method. Commun. Nonlinear Sci. Numer. Simul. 16, 3097-3106 (2011) CrossRef MathSciNet MATH
  20. Abdel Latif, MS: Bright and dark soliton solutions for the perturbed nonlinear Schrödinger’s equation with Kerr law and non-Kerr law nonlinearity. Appl. Math. Comput. 247, 501-510 (2014) CrossRef MathSciNet
  21. Eslami, M: Solitary wave solutions for perturbed nonlinear Schrodinger’s equation with Kerr law nonlinearity under the DAM. Optik 126, 1312-1317 (2015) CrossRef
  22. Biswas, A: Perturbation of solitons with non-Kerr law nonlinearity. Chaos Solitons Fractals 13, 815-823 (2002) CrossRef MathSciNet MATH
  23. Biswas, A: Quasi-stationary optical solitons with dual-power law nonlinearity. Opt. Commun. 235, 183-194 (2004) CrossRef
  24. Akhmediev, NN, Afanasjev, VV, Soto-Crespo, JM: Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation. Phys. Rev. E 53, 1190-1201 (1996) CrossRef
  25. Hefter, EF: Application of the nonlinear Schrödinger equation with a logarithmic inhomogeneous term to nuclear physics. Phys. Rev. A 32, 1201 (1985) CrossRef
  26. Hernandez, ES, Remaud, B: General properties of gausson-conserving descriptions of quantal damped motion. Physica A 105, 130-146 (1980) CrossRef MathSciNet
  27. De Martino, S, Falanga, M, Godano, C, Lauro, G: Logarithmic Schrodinger-like equation as a model for magma transport. Europhys. Lett. 63, 472-475 (2003) CrossRef
  28. Krolikowski, W, Edmundson, D, Bang, O: Unified model for partially coherent solitons in logarithmically nonlinear media. Phys. Rev. E 61, 3122 (2000) CrossRef
  29. Buljan, H, Siber, A, Soljacic, M, Schwartz, T, Segev, M, Christodoulides, DN: Incoherent white light solitons in logarithmically saturable noninstantaneous nonlinear media. Phys. Rev. E 68, 036607 (2003) CrossRef MathSciNet
  30. Doha, EH, Bhrawy, AH, Abdelkawy, MA, Van Gorder, RA: Jacobi-Gauss-Lobatto collocation method for the numerical solution of \(1+1\) nonlinear Schrödinger equations. J. Comput. Phys. 261, 244-255 (2014) CrossRef MathSciNet
  31. Zhang, LW, Liew, KM: An element-free based solution for nonlinear Schrödinger equations using the ICVMLS-Ritz method. Appl. Math. Comput. 249, 333-345 (2014) CrossRef MathSciNet
  32. Atangana, A: On the stability and convergence of the time-fractional variable order telegraph equation. J. Comput. Phys. 293, 104-114 (2015) CrossRef MathSciNet
  33. Atangana, A: Extension of the Sumudu homotopy perturbation method to an attractor for one-dimensional Keller-Segel equations. Appl. Math. Model. 39(10), 2909-2916 (2015) CrossRef MathSciNet
  34. Atangana, A: On the new fractional derivative and application to nonlinear Fisher’s reaction-diffusion equation. Appl. Math. Comput. 273, 948-956 (2016) CrossRef MathSciNet
  35. Tariboon, J, Ntouyas, SK, Agarwal, P: New concepts of fractional quantum calculus and applications to impulsive fractional q-difference equations. Adv. Differ. Equ. 2015, 18 (2015) CrossRef MathSciNet
  36. Zhou, H, Yang, L, Agarwal, P: Solvability for fractional p-Laplacian differential equations with multipoint boundary conditions at resonance on infinite interval. J. Appl. Math. Comput. (2015). doi:10.​1007/​s12190-015-0957-8
  37. Agarwal, P, Jain, S, Chand, M: Finite integrals involving Jacobi polynomials and I-function. In: Theoretical Mathematics & Applications, vol. 1, pp. 115-123 (2011)
  38. Bhrawy, AH, Abdelkawy, MA: A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations. J. Comput. Phys. 294, 462-483 (2015) CrossRef MathSciNet
  39. 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, 9379-9397 (2013) CrossRef MathSciNet
  
• 作者单位：Ali H Bhrawy (1)
  Fouad Mallawi (2)
  Mohamed A Abdelkawy (1) (3)
  
  1. Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt
  2. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia
  3. Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
  
• 刊物主题：Difference and Functional Equations; Mathematics, general; Analysis; Functional Analysis; Ordinary Differential Equations; Partial Differential Equations;
• 出版者：Springer International Publishing
• ISSN：1687-1847

文摘

A shifted Jacobi collocation method in two stages is constructed and used to numerically solve nonlinear Schrödinger equations (NLSEs) with a Kerr law nonlinearity, subject to initial-boundary conditions. An expansion in a series of spatial shifted Jacobi polynomials with temporal coefficients for the approximate solution is considered. The first stage, collocation at the shifted Jacobi Gauss-Lobatto (SJ-GL) nodes, is applied for a spatial discretization; its spatial derivatives occur in the NLSE with a treatment of the boundary conditions. This in all will produce a system of ordinary differential equations (SODEs) for the coefficients. The second stage is to collocate at the shifted Jacobi Gauss-Radau (SJ-GR-C) nodes in the temporal discretization to reduce the SODEs to a system of algebraic equations which is solved by an iterative method. Both stages can be extended to solve the two-dimensional NLSEs. Numerical examples are carried out to confirm the spectral accuracy and the efficiency of the proposed algorithms. Keywords one-dimensional Schrödinger equations Kerr law nonlinearity two-dimensional space Schrödinger equations collocation method Gauss-type quadratures