Properties of sums of some elementary functions and modeling of transitional and other processes

TL;DR

This paper generalizes properties of sums of elementary functions like polynomials, powers, and exponentials, providing new methods to analyze solutions and characteristic points, with applications in modeling transitional processes across sciences.

Contribution

It introduces a novel approach relating equations of different function types and a Descartes-like rule for real solutions, enhancing modeling accuracy.

Findings

Established properties of sums of elementary functions.

Developed a method to determine maximum real solutions.

Linked solutions of different function-type equations.

Abstract

The article presents mathematical generalization of results which originated as solutions of practical problems, in particular, the modeling of transitional processes in electrical circuits and problems of resource allocation. However, the presented findings have broader meaning and can be used for approximation of transitional and other processes in different areas of science and technology. We present discovered properties of sums of polynomial, power, and exponential functions of one variable. It is shown that for an equation composed of one type of function there is a corresponding equation composed of functions of the other type. The number of real solutions of such equations and the number of characteristic points of certain appropriate corresponding functions are closely related. In particular, we introduce a method similar to Descartes Rule of Signs that allows finding the maximum number of real solutions for the power equation and equation composed of sums of exponential functions. The discovered properties of these functions allow us to improve the adequacy of mathematical models of real phenomena.