Version 1
: Received: 6 September 2024 / Approved: 9 September 2024 / Online: 9 September 2024 (23:40:09 CEST)
How to cite:
Dimitrov, Y.; Georgiev, S.; Todorov, V. First Derivative Approximations and Applications. Preprints2024, 2024090713. https://doi.org/10.20944/preprints202409.0713.v1
Dimitrov, Y.; Georgiev, S.; Todorov, V. First Derivative Approximations and Applications. Preprints 2024, 2024090713. https://doi.org/10.20944/preprints202409.0713.v1
Dimitrov, Y.; Georgiev, S.; Todorov, V. First Derivative Approximations and Applications. Preprints2024, 2024090713. https://doi.org/10.20944/preprints202409.0713.v1
APA Style
Dimitrov, Y., Georgiev, S., & Todorov, V. (2024). First Derivative Approximations and Applications. Preprints. https://doi.org/10.20944/preprints202409.0713.v1
Chicago/Turabian Style
Dimitrov, Y., Slavi Georgiev and Venelin Todorov. 2024 "First Derivative Approximations and Applications" Preprints. https://doi.org/10.20944/preprints202409.0713.v1
Abstract
In this paper we consider constructions of first derivative approximations using the generating function. The weights of the approximations contain the powers of a parameter whose modulus is less than one. The values of the initial weights are determined, and the convergence and order of the approximations are proved. The paper discusses applications of approximations of first derivative for numerical solution of ordinary and partial differential equations and proposes an algorithm for fast computation of the numerical solution. Proofs of the convergence and accuracy of the numerical solutions are presented and the performance of the numerical methods considered is compared with the Euler method. The main goal of constructing approximations for integer-order derivatives of this type is their application in deriving high-order approximations for fractional derivatives, whose weights have specific properties. The paper proposes the construction of an approximation for the fractional derivative and its application for numerically solving fractional differential equations. The theoretical results for the accuracy and order of the numerical methods are confirmed by the experimental results presented in the paper.
Computer Science and Mathematics, Computational Mathematics
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
