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

机译：二维三次非线性Schroedinger方程求解的无网格方法

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摘要

In this paper, an efficient numerical technique is developed to approximate the solution of two-dimensional cubic nonlinear Schrodinger 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 regular-ization 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.
机译：在本文中，开发了一种有效的数值技术来近似二维立方非线性薛定inger方程的解。该方法基于运算符牛顿算法中的非对称径向基函数并置方法（Kansa方法）。在所提出的过程中，将三维径向基函数（尤其是三维多二次方（MQ）和逆多二次方（IMQ）函数）用作基础函数。为了求解所得的非线性系统，构造并应用了基于牛顿法的算法。在多级牛顿算法中，为克服标准方法求解结果不佳系统的不稳定性，将基于Tikhonov正则化技术和GCV函数方法的有趣而有效的技术用于解决问题不佳系统。最后，所提出的方法用于解决一些治理问题的例子。所得数值解与精确解之间的比较证明了该方法的可靠性，准确性和有效性。

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