Sesión Ecuaciones Diferenciales y aplicacionesTwo-Parameter Local Operators: A New Tool for Differential Analysis
Miguel Vivas-Cortez
Pontificia Universidad Católica del Ecuador, Ecuador - Esta dirección de correo electrónico está siendo protegida contra los robots de spam. Necesita tener JavaScript habilitado para poder verlo.
This work introduces a new local differential operator called the biparametric V-derivative, which generalizes the previously proposed deformable derivative. It is defined for parameters $\varphi \geq 0$ and $\psi \gt 0$ by
\[ V^{\varphi, \psi}(g(t)) := \lim_{h \to 0} \dfrac{(\psi + h(\psi - \varphi)) \, g\left(t + h \dfrac{\varphi}{\psi}\right) - \psi g(t)}{\psi \cdot h} \]
whenever the limit exists.
Fundamental properties of this new derivative are established, including generalized versions of Rolle’s Theorem and the Mean Value Theorem. An integral operator associated with the V-derivative is introduced, which enables the formulation of a version of the Fundamental Theorem of Calculus.
A generalized Laplace transform is also defined using this new integral operator. Moreover, biparametric fractional differential equations are solved to illustrate the applicability of the proposed framework.
Unlike the deformable derivative, the biparametric V-derivative does not require the restriction $\varphi + \psi = 1$, offering greater flexibility in applications. It is also shown that classical and deformable derivatives are recovered for specific parameter values.
This framework opens new directions for research, such as defining partial derivatives of order $(\varphi, \psi)$, exploring new integral transforms, and extending the V-derivative to complex domains.
Referencias
[1] M. Vivas-Cortez, J. Velasco, H. Jarrin, ”A New Generalized Local Derivative of Two Parameters,” Applied Mathematics & Information Sciences, vol. 19, no. 3, pp. 713-723, May 2025. doi:10.18576/amis/190319.
[2] Ahuja, Priyanka, Fahed Zulfeqarr, and Amit Ujlayan. ”Deformable fractional derivative and its applications.” AIP Conference Proceedings, vol. 1897, no. 1, 2017. AIP Publishing.
[3] Vivas-Cortez, M. and J. E. N. Valdés. ”On the generalized Laplace transform.” Applied Mathematics and Information Sciences, vol. 15, no. 5, pp. 667–675, 2021
[4] Vivas-Cortez, M., M. P. Árciga, J. C. Najera, and J. E. Hernández. ”On some conformable boundary value problems in the setting of a new generalized conformable fractional derivative.” Demonstration Mathematica, vol. 56, no. 1, Article 0212, 2023. https://doi.org/10.1515/dema-2022-0212.
[5] Vivas-Cortez, M., L. M. Lugo, J. E. N. Valdés, and M. E. Samei. ”A Multi-Index Generalized Derivative; Some Introductory Notes.” Applied Mathematics and Information Sciences, vol. 16, no. 6, pp. 883–890, 2022. doi:10.18576/amis/160604.