# 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 language English 885-898 14 Engineering Analysis with Boundary Elements 37 6 https://doi.org/10.1016/j.enganabound.2013.03.006 Published - 2013

### Fingerprint

Meshfree Method
Cubic equation
Nonlinear equations
Nonlinear Equations
Basis 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
• 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

title = "A meshfree method for the solution of two-dimensional cubic nonlinear Schr{\"o}dinger equation",
abstract = "In this paper, an efficient numerical technique is developed to approximate the solution of two-dimensional cubic nonlinear Schr{\"o}dinger 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.",
keywords = "Cubic nonlinear, Meshfree method, Newton algorithm, Radial basis functions, Schr{\"o}dinger equation",
author = "S. Abbasbandy and {Roohani Ghehsareh}, H. and Ishak Hashim",
year = "2013",
doi = "10.1016/j.enganabound.2013.03.006",
language = "English",
volume = "37",
pages = "885--898",
journal = "Engineering Analysis with Boundary Elements",
issn = "0955-7997",
publisher = "Elsevier Limited",
number = "6",

}

TY - JOUR

T1 - A meshfree method for the solution of two-dimensional cubic nonlinear Schrödinger equation

AU - Abbasbandy, S.

AU - Roohani Ghehsareh, H.

AU - Hashim, Ishak

PY - 2013

Y1 - 2013

N2 - 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.

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.

KW - Cubic nonlinear

KW - Meshfree method

KW - Newton algorithm

KW - Schrödinger equation

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U2 - 10.1016/j.enganabound.2013.03.006

DO - 10.1016/j.enganabound.2013.03.006

M3 - Article

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SP - 885

EP - 898

JO - Engineering Analysis with Boundary Elements

JF - Engineering Analysis with Boundary Elements

SN - 0955-7997

IS - 6

ER -