Addition theorems for Ck real functions and applications in ordinary differential equations

TL;DR

This paper develops addition theorems and double-angle formulas for Ck real functions, enabling a duplication algorithm for solving ODEs that converges uniformly and performs well compared to standard methods.

Contribution

It introduces new addition theorems and double-angle formulas for Ck functions and proposes a duplication algorithm as an alternative numerical method for ODEs.

Findings

The duplication algorithm converges uniformly on compact domains.

Numerical simulations show the algorithm performs well compared to standard methods.

Necessary and sufficient conditions for addition formulas are established.

Abstract

This work establishes the existence of addition theorems and double-angle formulas for Ck real scalar functions. Moreover, we determine necessary and sufficient conditions for a bivariate function to be an addition formula for a Ck real function. The double-angle formulas allow us to generate a duplication algorithm, which can be used as an alternative to the classical numerical methods to obtain an approximation for the solution of an ordinary differential equation. We demonstrate that this algorithm converges uniformly in any compact domain contained in the maximal domain of that solution. Finally, we carry out some numerical simulations showing a good performance of the duplication algorithm when compared with standard numerical methods