The Class of Atomic Exponential Basis Functions EFupn(x,ω)-Development and Application

Abstract

The purpose of this paper is to present the class of atomic basis functions (ABFs) which are of exponential type and are denoted by . While ABFs of the algebraic type are already represented in the numerical modeling of various problems in mathematical physics and computational mechanics, ABFs of the exponential type have not yet been sufficiently researched. These functions, unlike the ABFs of the algebraic type , contain the tension parameter , which gives them additional approximation properties. Exponential monomials up to the th degree can be described exactly by the linear combination of the functions . The function for is called the “mother” ABF of the exponential type, i.e., . In other words, the functions are elements of the linear vector space and retain all the properties of their “mother” function . Thus, this paper, in terms of its content and purpose, can be understood as a sequel of the article by Brajčić Kurbaša et al., which shows the basic properties and application of the basis function This paper presents, in an analogous way, the development and application of the exponential basis functions . Here, for the first time, expressions for calculating the values of the functions and their derivatives are given in a form suitable for application in numerical analyses, which is shown in the verification examples of the approximations of known functions.