Family of the Atomic Radial Basis Functions of Three Independent Variables Generated by Helmholtz-Type Operator

Abstract

The paper presents an algorithm for constructing the family of the atomic radial basis functions of three independent variables generated by Helmholtz-type operator, which may be used as basis functions for the implementation of meshless methods for solving boundary-value problems in anisotropic solids. Helmholtz-type equations play a significant role in mathematical physics because of the applications in which they arise. In particular, the heat equation in anisotropic solids in the process of numerical solution is reduced to the equation that contains the differential operator of the special form (Helmholtz-type operator), which includes components of the tensor of the second rank, which determines the anisotropy of the material. The family of functions is infinitely differentiable and finite (compactly supported) solutions of the functional-differential equation of the special form. The choice of compactly supported functions as basis functions makes it possible to consider boundary-value problems on domains with complex geometric shapes. Functions include the shape parameter , which allows varying the size of the support and may be adjusted in the process of solving the boundary-value problem. Explicit formulas for calculating the considered functions and their Fourier transform are obtained. Visualizations of the atomic functions and their first derivatives with respect to the variables and at the fixed value of the variable for isotropic and anisotropic cases are presented. The efficiency of using atomic functions as basis functions is demonstrated by the solution of the non-stationary heat conduction problem with the moving heat source. This work contains the results of the numerical solution of the considered boundary-value problem, as well as average relative error, average absolute error and maximum error are calculated using atomic radial basis functions and multiquadric radial basis functions.