Human immunodeficiency virus (HIV) infection is said to be a dangerous, extremely severe, and potentially fatal disease. After weakening a person's immune system, it impedes the body's ability to fend off future illnesses. In this study, we use the Caputo operator's fractional framework to describe how antiretroviral therapy (ART) affects the HIV/AIDS transmission process. Using fractional calculus, the basic characteristics of the formulated system are found. In addition, we investigated the steady states of the fractional system and determined the reproduction parameter $ \mathcal{R}_0 $. A fractional framework has been used for stability analysis. The optimal value of the Caputo fractional-order $ \alpha $ is around $ 9.437300e-01 $, and the system parameters are determined using the least-squares method, demonstrating that the system matches real HIV/AIDS data the best. Average and maximum absolute relative percentile errors have been used to draw comparisons between the classical and Caputo systems. The case summaries obtained demonstrate the Caputo operator's supremacy. A sensitivity test is run using the partial rank correlation coefficient (PRCC) technique to clearly see which parameters are crucial to the formulation of $ \mathcal{R}_0 $. Lastly, a powerful numerical technique is used to run simulations that show the HIV/AIDS time series in the Caputo fractional framework.

In this paper, at least three weak solutions were obtained for a new class of dual non-linear dual-Laplace systems according to two parameters by using variational methods combined with a critical point theory due to Bonano and Marano. Two examples are given to illustrate our main results applications.

The paper studies the global existence and general decay of solutions using Lyapunov functional for a nonlinear wave equation, taking into account the fractional derivative boundary condition and memory term. In addition, we establish the blow-up of solutions with nonpositive initial energy.

The paper deals with the existence of three different weak solutions of [ ] -Laplacian fractional for an overdetermined nonlinear fractional partial Fredholm–Volterra integro-differential system by using variational methods combined with a critical point theorem due to Bonanno and Marano.