This paper examines a class of impulsive boundary value problems related to fractional pantograph differential equations governed by the $(k,\psi )$-Hilfer proportional fractional derivative. The problem is reformulated into an equivalent integral equation, which provides a convenient framework for further analysis. The existence and uniqueness of solutions are derived using Banach's fixed-point theorem. Moreover, several forms of Ulam stability are examined. Illustrative examples and graphical computations are included to support the applicability of the theoretical results.

The initial value problem in Cauchy-type under the $(k,\psi )$-Caputo proportional fractional operators was our focus in this paper. An extended Gronwall inequality and its properties were analyzed. The existence and uniqueness results were proven utilizing the fixed point theory of Banach's and Leray-Schauder's types. The qualitative analysis included results for Ulam-Mittag-Leffler stability, which was also investigated. Using a decomposition principle, a novel numerical technique was presented for the $(k,\psi )$-Caputo proportional fractional derivative operator. Finally, theoretical results were supported with numerical examples to demonstrate their practical application, especially to blood alcohol level problems.

In this paper, we developed a nonlinear mathematical model for the transmission of the monkeypox virus among populations of humans and rodents under the fractal-fractional operators in the context of Atangana-Baleanu. For the theoretical analysis, the renowned theorems of fixed points, like Banach's and Krasnoselskii's types, were used to prove the existence and uniqueness of the solutions. Additionally, some results regarding the stability of the equilibrium points and the basic reproduction number were provided. In addition, the numerical schemes of the considered model were established using the Adams-Bashforth method. Our analytical findings were supported by the numerical simulations to explain the effects of changing a few sets of fractional orders and fractal dimensions. Some graphic simulations were displayed with some parameters calculated from real data to understand the behavior of the model.

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.