Some types of rational fractions are considered. We have proposed new formulas for the decomposition of these rational fractions without having to deal with the powers of x as is the case for the Euclidean division theorem, nor as is the case for the method of indeterminate coefficients, nor even by the use a table of values as is the case with the Horner method. Some of these formulas globalize the law of remainder. We have also proposed a formula for the calculations of the successive derivatives of these types of rational fractions previously setting into the decomposed form. Links established between the terms of the decomposed form and certain hyperbolic functions made it possible to read and propose new functional notations. From these functional notations, we were able to develop properties giving the formulas for calculating the derivatives of these functions. Some of these functional notations turned out to be ansatz solutions of certain nonlinear evolution equations like the KdV equation. The graphical representations of these new functions have allowed to identify a modulational parameter following the shape of the signal produced by each of them. On an analytical level and tie-in to the KdV equation, this parameter also appears as a correction parameter. The existence of this twofold-action parameter allowed to observe the hybrid nature of some of these functions which are in fact new prototypes of signals, thus corroborating with our theoretical predictions. To reassure ourselves that the obtained signals must be observable and applicable experimentally, we carried out intense numerical simulations which permitted to note the stable nature of the new exact solutions obtained for the KdV equation in a relatively long time.

explicit formulas for the derivatives and decompositions, polynomial coefficients, Euclidean division theorem, Horner’s method, KdV equation

Received: November 21, 2023; Revised: December 5, 2023; Accepted: December 28, 2023; Published: March 20, 2024

How to cite this article: Clovis Taki Djeumen Tchaho, Explicit formulas for the derivatives and decompositions by the polynomial coefficients and extension to hyperbolic functions as new solution prototypes for KdV equation, Far East Journal of Dynamical Systems 37(1) (2024), 13-49. http://dx.doi.org/10.17654/0972111824002

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

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