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.

This work comprehensively analyzed the monkeypox virus utilizing a deterministic mathematical model within a constant proportional-Caputo derivative framework. The suggested model considered the interplay of human and rodent populations by incorporating certain realistic vaccination parameters. Our study was a testament to the thoroughness of this work. We explored the uniqueness result using Banach's contraction principle. The solution's positivity and boundedness were studied in detail, as were the basic reproduction number and the stability analysis of the system's equilibrium conditions. We performed a variety of Ulam's stability analyses to guarantee the solution existed. Additionally, we implemented a decomposition formula to obtain the numerical scheme. This numerical approach allowed for numerical simulation as a graphical representation for certain real data sets and different parameter values in order to understand the model's dynamic behavior.

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.