In this study, we establish a novel version of Hermite-Hadamard inequalities through neoteric generalized Riemann-Liouville fractional integrals (RLFIs). For functions with the convex absolute values of derivatives, we create a variety of midpoint and trapezoid form inequalities, including the generalized RLFIs. Moreover, multiple fractional inequalities can be produced as special cases of the findings of this study.

This paper is devoted to proving some new fractional inequalities via recent generalized fractional operators. These inequalities are in the Hermite–Hadamard and Minkowski settings. Many previously documented inequalities may clearly be deduced as specific examples from our findings. Moreover, we give some comparative remarks to show the advantage and novelty of the obtained results.

In this article, we utilize recent generalized fractional operators to establish some fractional inequalities in Hermite–Hadamard and Minkowski settings. It is obvious that many previously published inequalities can be derived as particular cases from our outcomes. Moreover, we articulate some flaws in the proofs of recently affiliated formulas by revealing the weak points and introducing more rigorous proofs amending and expanding the results.

We look at fractional Langevin equations (FLEs) with generalized proportional Hadamard–Caputo derivative of different orders. Moreover, nonlocal integrals and nonperiodic boundary conditions are considered in this paper. For the proposed equations, the Hyres–Ulam (HU) stability, existence, and uniqueness (EU) of the solution are defined and investigated. In implementing our results, we rely on two important theories that are Krasnoselskii fixed point theorem and Banach contraction principle. Also, an application example is given to bolster the accuracy of the acquired results.

This work is devoted to providing new kinds of deterministic and stochastic solutions of one of the famous nonlinear equations that depends on time, called the Schrödinger–Hirota equation. A new and straightforward methodology is offered to extract exact wave solutions of the stochastic nonlinear evolution equations (NEEs) with generalized differential conformable operators (GDCOs). This methodology combines the features of GDCOs, some instruments of white noise analysis, and the generalized Kudryashov scheme. To demonstrate the usefulness and validity of our methodology, we applied it to extract diversified exact wave solutions of the Schrödinger–Hirota equation, particularly in a Wick-type stochastic space and with GDCOs. These wave solutions can be turned into soliton and periodic wave solutions that play a main role in numerous fields of nonlinear physical sciences. Moreover, three-dimensional, contour, and two-dimensional graphical visualizations of some of the extracted solutions are exhibited with some elected functions and parameters. According to the results, our new approach demonstrates the impact of random and conformable factors on the solutions of the Schrödinger–Hirota equation. These findings can be applied to build new models in plasma physics, condensed matter physics, industrial studies, and optical fibers. Furthermore, to reinforce the importance of the acquired solutions, comparative aspects connected to some former works are presented for these types of solutions.

Modeling human immunodeficiency virus (HIV) via fractional operators has several benefits over the classical integer-order HIV model. The reason is that the fractional HIV model relies not only on the recent status but also on the former conduct of the model. Thus, we are motivated to introduce and analyze a new fractional HIV model. This article focuses on a novel fractional HIV model under the proportional Hadamard-Caputo fractional operators. The study of this model involves the existence and uniqueness (EU) of its solution and the stability examination. We employ Leray–Schauder nonlinear alternative (L-SNLA) and Banach’s fixed point theorems to analyze the EU results. In addition, for this provided model, we develop several forms of Ulam’s stability findings. As a special case of our results, we give and analyze a new fractional HIV model with Hadamard-Caputo operators. Moreover, by appropriate choice of the fractional parameters, the obtained outcomes are valid for analysis of the fractional HIV models formed by several fractional operators defined in the past literature.