Certain Relations Between the Main Matrix Condition Number and Multiquadric Shape Parameter in the Non-Symmetric Kansa Method

Abstract

The Kansa method is one of the most popular meshless methods today. Its ease of implementation, high order of interpolation and ease of application to problems with complex geometry constitute its advantage over many other methods for solving partial differential equation-based problems. However, the Kansa method has a significant disadvantage -the need to find the shape parameter value despite these undeniable advantages. There are dozens of algorithms for finding a good shape parameter value, but none of them is proven to be optimal. Therefore, there is still a great scientific need to research new algorithms and improve those already known. In this work, an algorithm based on the study of the oscillation of certain shape parameter functions concerning the problems of two-dimensional heat flow in a material with spatially variable thermophysical parameters was investigated. It has been shown that algorithms of this type allow this class of problems to achieve solutions with high accuracy. At the same time, it was indicated that this direction of development of algorithms for searching for a good value of the shape parameter is auspicious. It is because this algorithm can be extended to a wide range of functions whose oscillation is studied and, consequently, its application to a broader range of problems.