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 EFup n (x, ω). 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 Fup n (x), contain the tension parameter ω, which gives them additional approximation properties. Exponential monomials up to the nth degree can be described exactly by the linear combination of the functions EFup n (x, ω). The function EFup n for n = 0 is called the "mother" ABF of the exponential type, i.e., EFup 0 (x, ω) ≡ Eup(x, ω). In other words, the functions EFup n (x, ω) are elements of the linear vector space EUP n and retain all the properties of their "mother" function Eup(x, ω). 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 Eup (x, ω). This paper presents, in an analogous way, the development and application of the exponential basis functions EFup n (x, ω). Here, for the first time, expressions for calculating the values of the functions EFup n (x, ω) 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.