In recent years, many new definitions of fractional derivatives have been proposed and used to develop mathematical models for a wide variety of real-world systems containing memory, history, or nonlocal effects. The main purpose of the present paper is to develop and analyze a Caputo–Fabrizio fractional derivative model for the HIV/AIDS epidemic which includes an antiretroviral treatment compartment. The existence and uniqueness of the system of solutions of the model are established using a fixed-point theorem and an iterative method. The model is shown to have a disease-free and an endemic equilibrium point. Conditions are derived for the existence of the endemic equilibrium point and for the local asymptotic stability of the disease-free equilibrium point. The results confirm that the disease-free equilibrium point becomes increasingly stable as the fractional order is reduced. Numerical simulations are carried out using a three-step Adams–Bashforth predictor method for a range of fractional orders to illustrate the effects of varying the fractional order and to support the theoretical results.

This study focuses on the comprehensive analysis of the thermal characterization of an electrically conducting nanofluid flow containing TiO₂ nanoparticles dissolved in two different common liquids (H₂O and C₂H₆O₂) and contained within a continuously extending disk. The heat and concentration formulation are modified by incorporating the standard interpretations of the Joule heating, energy source, dissipation, and modified activation energy with binary chemical reactions. The Darcy–Forchheimer addition is used to highlight the significance of porous media. The Brownian dispersion’s effects are clearly discernible in the current investigation. Furthermore, as a novelty, special thermophysical models of viscosity and thermal conductivity are included. The obtained differential formulations are tackled through the analytical procedure HAM. The veracity of the finding is verified through a numerical scheme (ND-solve technique). Using several graphs and charts, the perception trends of model factors are evaluated. The influence of $C_{fz},N{u}_z,$ and $S{h}_z$ are estimated based on flow characteristics. The $F^{\prime }(\zeta )$ curve pattern for both nanofluids is declined by the increasing magnitude of $M,$ $\varphi ,$ ${\beta }_p^{*},$ and $F_r^{*},$ while thermal profile. The drag force and heat transmission rate are shown to optimize with $M$ as well as $\varphi$ in the pictorial simulations produced from the existing model. The $\Phi (\zeta )$ can also be strengthened by the presence of a high degree $Nb.$ The outcomes are validated by previous studies and a good degree of consistency is revealed.

We apply the G′/G2-expansion method to construct exact solutions of three interesting problems in physics and nanobiosciences which are modeled by nonlinear partial differential equations (NPDEs). The problems to which we want to obtain exact solutions consist of the Benny-Luke equation, the equation of nanoionic currents along microtubules, and the generalized Hirota-Satsuma coupled KdV system. The obtained exact solutions of the problems via using the method are categorized into three types including trigonometric solutions, exponential solutions, and rational solutions. The applications of the method are simple, efficient, and reliable by means of using a symbolically computational package. Applying the proposed method to the problems, we have some innovative exact solutions which are different from the ones obtained using other methods employed previously.

The major purpose of this article is to seek for exact traveling wave solutions of the nonlinear space-time Sharma–Tasso–Olver equation in the sense of conformable derivatives. The novel $\left(\frac{G^{\prime }}{G}\right)$ -expansion method and the generalized Kudryashov method, which are analytical, powerful, and reliable methods, are used to solve the equation via a fractional complex transformation. The exact solutions of the equation, obtained using the novel $\left(\frac{G^{\prime }}{G}\right)$ -expansion method, can be classified in terms of hyperbolic, trigonometric, and rational function solutions. Applying the generalized Kudryashov method to the equation, we obtain explicit exact solutions expressed as fractional solutions of the exponential functions. The exact solutions obtained using the two methods represent some physical behaviors such as a singularly periodic traveling wave solution and a singular multiple-soliton solution. Some selected solutions of the equation are graphically portrayed including 3-D, 2-D, and contour plots. As a result, some innovative exact solutions of the equation are produced via the methods, and they are not the same as the ones obtained using other techniques utilized previously.

In this paper, the $(G^{\prime }/G,1/G)$ -expansion method is applied to acquire some new, exact solutions of certain interesting, nonlinear, fractional-order partial differential equations arising in mathematical physics. The considered equations comprise the time-fractional, (2+1)-dimensional extended quantum Zakharov-Kuznetsov equation, and the space-time-fractional generalized Hirota-Satsuma coupled Korteweg-de Vries (KdV) system in the sense of the conformable fractional derivative. Applying traveling wave transformations to the equations, we obtain the corresponding ordinary differential equations in which each of them provides a system of nonlinear algebraic equations when the method is used. As a result, many analytical exact solutions obtained of these equations are expressed in terms of hyperbolic function solutions, trigonometric function solutions, and rational function solutions. The graphical representations of some obtained solutions are demonstrated to better understand their physical features, including bell-shaped solitary wave solutions, singular soliton solutions, solitary wave solutions of kink type, and so on. The method is very efficient, powerful, and reliable for solving the proposed equations and other nonlinear fractional partial differential equations with the aid of a symbolic software package.