Purpose

The purpose of this paper is to develop a mesh-free algorithm based on the least square support vector machines method for numerical simulation of the modified Helmholtz equations.

Design/methodology/approach

The proposed method deals with a Cauchy problem for the modified Helmholtz equations. The algorithm converts the problem into a quadratic programming. It can be divided into three steps. First, some training points are allocated. Then, an approximate function is constructed. Finally, the shape parameters are estimated.

Findings

The proposed method's stability is discussed. Numerical experiments are conducted to check the efficiency of the algorithm. The proposed method is found to feasible for the ill-posed problems of the modified Helmholtz equations.

Originality/value

The originality lies in that the proposed method is applied to solve the modified Helmholtz equations for the first time, and the expected results are obtained.

中文翻译：

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使用LS-SVM学习修改的Helmholtz方程的Cauchy问题解

目的

本文的目的是开发一种基于最小二乘支持向量机方法的无网格算法，用于改进的亥姆霍兹方程的数值模拟。

设计/方法/方法

所提出的方法处理了修正的亥姆霍兹方程的柯西问题。该算法将问题转换为二次规划。它可以分为三个步骤。首先，分配一些训练点。然后，构造一个近似函数。最后，估计形状参数。

发现

讨论了该方法的稳定性。进行了数值实验，以检验算法的效率。发现该方法对于修正的亥姆霍兹方程的不适定问题是可行的。

创意/价值

独创性在于，该方法首次应用于修正的Helmholtz方程组的求解，并获得了预期的结果。