In this study, we introduce a nonlinear leukemia dynamical system for a piecewise modified ABC fractional-order derivative and analyze it for the theoretical as well computational works and examine the crossover effect of the model. For the crossover behavior of the operators, we presume a division of the period of study $[0,t_2]$ in two subclasses as ${\mathcal{I}}_1=[0,t_1]$, ${\mathcal{I}}_2=[t_1,t_2]$, for $t_1,t_2\in \mathbb{R}$ with $t_1<t_2$. In ${\mathcal{I}}_1$, the classical derivative is considered for the study of the leukemia growth while in ${\mathcal{I}}_2$ we presume modified ABC fractional differential operator. As a result, the study is initiated in the piecewise modified ABC sense of derivative for the dynamical systems. The novel constructed model is then studied for the solution existence and stability as well computational results. The symmetry in dynamics for all the three classes can be graphically observed in the presented six plots.

Vaccination against any infectious disease is considered to be an effective way of controlling it. This paper studies a fractional order model with vaccine efficacy and waning immunity. We present the model’s dynamics under vaccine efficacy, the impact of immunization, and the waning of the vaccine on coronavirus infection disease. We analyze the model under their equilibrium points. The model under the equilibrium points is discussed and proven that it is locally asymptotically stable if ${\mathcal{R}}_v$ is lesser than unity. We present the backward bifurcation hypothesis of the model and show that there is a parameter ${\beta }_2$ that causes the backward bifurcation in the imperfect vaccine model. We show certain assumptions when $\psi =1$ for the imperfect vaccine case; the model is both stable globally asymptotically at the disease-free (${\mathcal{R}}_0\le 1$) and endemic cases (${\mathcal{R}}_0>1$). By using infected cases from the recent wave throughout Pakistan, we shall estimate the model parameters and calculate the numerical value of the basic reproductive number ${\mathcal{R}}_0\approx 1.2591$. We present the comprehensive graphical results for the realistic parameter values and show many useful suggestions regarding the elimination of the infection from society. The vaccination efficacy that provides an important role in disease elimination is discussed in detail.

In this paper, a synthesized zeolite (ZSM-5) is used as an adsorbent to analyze the adsorption phenomenon of carbon dioxide. This investigation, based on the statistical physics treatment, applied the multilayer model with saturation to understand the CO₂ adsorption on four samples, namely M-ZSM-5 (M = Na⁺, Mg²⁺, Zn²⁺, La³⁺), at various temperatures T = 0 °C, 30 °C and 60 °C. The modeling results indicated that CO₂ adsorption occurred via a non-parallel orientation on the ZSM-5 surface. The CO₂ adsorption capacities varied from 26.14 to 28.65 cm³/g for Na-ZSM-5, from 25.82 to 27.97 cm³/g for Mg-ZSM-5, from 54.82 to 68.63 cm³/g for La-ZSM-5 and from 56.53 to 74.72 cm³/g for Zn-ZSM-5. Thus, Zn-ZSM-5 exhibits the highest adsorption amount. The analysis of the adsorption energies shows that the adsorption of CO₂ on ZSM-5 zeolite is a physisorption phenomenon that could be controlled thanks to the energy parameters obtained via the numerical findings using the multilayer statistical model. Finally, the distribution of site energy was determined to confirm the physical character of the interactions between adsorbate/adsorbent and the heterogeneity of the zeolite surface.

In the current analysis, the steady and incompressible magnetohydrodynamics hybrid nanofluid (hnf) flow across two spinning permeable surfaces is studied. The hybrid nanoliquid has been examined under the additional effects of heat source, magnetic field, and Arrhenius activation energy. The hnf is synthesized by the dispersion of silicon dioxide and molybdenum disulfide nanoparticles in ethylene glycol. The flow scenario has been communicated in the form of a system of nonlinear Partial Differential Equations (PDEs), which are degraded and dimensionless to a set of nonlinear ordinary differential equations through appropriate similarity replacement. For the solution of reduced first-order differential equations, a numerical technique is employed. The consequences of physical flow parameters on energy, mass, and velocity profiles are shown through figures. It has been observed that the radial velocity profile increases with the influence of the suction factor and decreases with the effect of surface stretching and the rotation factor. The heat transmission rate increases with the impact of the Reynolds number.

The Levenberg–Marquardt (LM) backpropagation optimization algorithm, an artificial neural network algorithm, is used in this study to perform integrated numerical computing to evaluate the electromagnetohydrodynamic bioconvection flow of micropolar nanofluid with thermal radiation and stratification. The model is then reduced to a collection of boundary value problems, which are solved with the help of a numerical technique and the proposed scheme, i.e., the LM algorithm, which is an iterative approach to determine the minimum of a nonlinear function defined as the sum of squares. As a blend of the steepest descent and the Gauss–Newton method, it has become a typical approach for nonlinear least-squares problems. Furthermore, the stability and consistency of the algorithm are ensured. For validation purposes, the results are also compared with those of previous research and the MATLAB bvp4c solver. Neural networking is also utilized for velocity, temperature, and concentration profile mapping from input to output. These findings demonstrate the accuracy of forecasts and optimizations produced by artificial neural networks. The performance of the bvp4c solver, which is used to reduce the mean square error, is used to generalize a dataset. The artificial neural network-based LM backpropagation optimization algorithm operates using data based on the ratio of testing (13%), validation (17%), and training (70%). This stochastic computing work presents an activation log-sigmoid function based LM backpropagation optimization algorithm, in which tens of neurons and hidden and output layers are used for solving the learning language model. The overlapping of the results and the small computed absolute errors, which range from 10−3 to 10−10 and from 106 to 108 for each model class, indicate the accuracy of the artificial neural network-based LM backpropagation optimization algorithm. Furthermore, each model case’s regression performance is evaluated as if it were an ideal model. In addition, function fitness and histogram are used to validate the dependability of the algorithm. Numerical approaches and artificial neural networks are an excellent combination for fluid dynamics, and this could lead to new advancements in many domains. The findings of this research could contribute to the optimization of fluid systems, resulting in increased efficiency and production across various technical domains.