@Article{icces.2023.09393,
AUTHOR = {Zhuojia Fu, Qiang Xi, Wenzhi Xu},
TITLE = {Research Advances on the Collocation Methods Based on the PhysicalInformed Kernel Functions},
JOURNAL = {The International Conference on Computational \& Experimental Engineering and Sciences},
VOLUME = {27},
YEAR = {2023},
NUMBER = {1},
PAGES = {1--1},
URL = {http://www.techscience.com/icces/v27n1/54092},
ISSN = {1933-2815},
ABSTRACT = {In the past few decades, although traditional computational methods such as finite element have been 
successfully used in many scientific and engineering fields, they still face several challenging problems such 
as expensive computational cost, low computational efficiency, and difficulty in mesh generation in the 
numerical simulation of wave propagation under infinite domain, large-scale-ratio structures, engineering 
inverse problems and moving boundary problems. This paper introduces a class of collocation discretization 
techniques based on physical-informed kernel function (PIKF) to efficiently solve the above-mentioned 
problems. The key issue in the physical-informed kernel function collocation methods (PIKFCMs) is to 
construct the related basis functions, which includes the physical information of the considered differential
governing equation. Based on these physical-informed kernel functions (PIKFs), these methods do not 
need/only need a few collocation nodes to discretize the considered differential governing equations, which 
may effectively improve the computational efficiency. In this paper, several typical physical-informed kernel 
functions (PIKFs) that satisfy common-used homogeneous differential equations, such as the fundamental 
solutions, the harmonic functions, the radial Trefftz functions and the T-complete functions and so on, are 
firstly introduced. After that, the ways to construct the physical-informed kernel functions (PIKFs) for 
nonhomogeneous differential equations, inhomogeneous differential equations, unsteady-state differential 
equations and implicit differential equations are introduced in turn. Then according to the characteristics of 
the considered problems, the global collocation scheme or the localized collocation scheme is selected to 
establish the corresponding physical-informed kernel function collocation method (PIKFCM). Finally, four 
typical examples are given to verify the effectiveness of the physical-informed kernel function collocation 
methods (PIKFCMs) proposed in this paper.},
DOI = {10.32604/icces.2023.09393}
}