A meshfree method for the solution of two-dimensional cubic nonlinear Schrödinger equation

S. Abbasbandy, H. Roohani Ghehsareh, Ishak Hashim

Research output: Contribution to journalArticle

52 Citations (Scopus)

Abstract

In this paper, an efficient numerical technique is developed to approximate the solution of two-dimensional cubic nonlinear Schrödinger equations. The method is based on the nonsymmetric radial basis function collocation method (Kansa's method), within an operator Newton algorithm. In the proposed process, three-dimensional radial basis functions (especially, three-dimensional Multiquadrics (MQ) and Inverse multiquadrics (IMQ) functions) are used as the basis functions. For solving the resulting nonlinear system, an algorithm based on the Newton approach is constructed and applied. In the multilevel Newton algorithm, to overcome the instability of the standard methods for solving the resulting ill-conditioned system an interesting and efficient technique based on the Tikhonov regularization technique with GCV function method is used for solving the ill-conditioned system. Finally, the presented method is used for solving some examples of the governing problem. The comparison between the obtained numerical solutions and the exact solutions demonstrates the reliability, accuracy and efficiency of this method.

Original languageEnglish
Pages (from-to)885-898
Number of pages14
JournalEngineering Analysis with Boundary Elements
Volume37
Issue number6
DOIs
Publication statusPublished - 2013

Fingerprint

Meshfree Method
Cubic equation
Nonlinear equations
Nonlinear Equations
Basis Functions
Radial Functions
Three-dimensional
Regularization Technique
Tikhonov Regularization
Mathematical operators
Nonlinear systems
Numerical Techniques
Collocation Method
Nonlinear Systems
Exact Solution
Numerical Solution
Operator
Demonstrate

Keywords

  • Cubic nonlinear
  • Meshfree method
  • Newton algorithm
  • Radial basis functions
  • Schrödinger equation

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Computational Mathematics
  • Engineering(all)

Cite this

A meshfree method for the solution of two-dimensional cubic nonlinear Schrödinger equation. / Abbasbandy, S.; Roohani Ghehsareh, H.; Hashim, Ishak.

In: Engineering Analysis with Boundary Elements, Vol. 37, No. 6, 2013, p. 885-898.

Research output: Contribution to journalArticle

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AB - In this paper, an efficient numerical technique is developed to approximate the solution of two-dimensional cubic nonlinear Schrödinger equations. The method is based on the nonsymmetric radial basis function collocation method (Kansa's method), within an operator Newton algorithm. In the proposed process, three-dimensional radial basis functions (especially, three-dimensional Multiquadrics (MQ) and Inverse multiquadrics (IMQ) functions) are used as the basis functions. For solving the resulting nonlinear system, an algorithm based on the Newton approach is constructed and applied. In the multilevel Newton algorithm, to overcome the instability of the standard methods for solving the resulting ill-conditioned system an interesting and efficient technique based on the Tikhonov regularization technique with GCV function method is used for solving the ill-conditioned system. Finally, the presented method is used for solving some examples of the governing problem. The comparison between the obtained numerical solutions and the exact solutions demonstrates the reliability, accuracy and efficiency of this method.

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