Exploring dynamics in RLC circuits: a novel approach utilizing the $(k,\varphi )$-Hilfer proportional fractional operator

In this study, the theory of fractional calculus is applied to the electrical circuits. In this work, we investigated the Langevin-type differential equations under the $(k,\varphi )$-Hilfer proportional fractional derivative. By utilizing the bivariate Mittag-Leffler function and the $\psi$-Laplace transform, we designed a representation of an explicit analytical solution for the linear system corresponding to the considered model. We explored Ulam–Hyers stability results with the Mittag-Leffler function and their generalizations to confirm Ulam stability by applying the extended Gronwall inequality under the context of the $(k,\varphi )$-proportional fractional operators. Finally, the RLC electrical circuit model was chosen as the application's agent to validate the accuracy of our theoretical results. Our results offer additional analytical choices due to the wider range of parameter values compared to earlier studies.