This paper acquaints the adopted parabolic with the nonlocal law of self-phase modulation form of the complex Ginzburg–Landau model, which regularizes the evolution of specific amplitudes of instability pulses in diverse dissipative systems. We employ the new Kudryashov and Sinh-Gordon equation expansion schemes to obtain bright and dark soliton families under particular conditions on the parameters of the physical model. Furthermore, the effect of diverse model parameters such as the chromatic dispersion, the parabolic law, and the nonlocal nonlinearity terms on the behaviors of bright and dark soliton solutions is also explored. We also search for modulation instability analysis for the model. The primary contribution of this study is the examination of a different version of the complex Ginzburg–Landau model, which is not yet available in the literature, along with the first comprehensive analysis of its modulation instability. Thus, this study highlights the practical and prompt results received by the Sinh-Gordon equation expansion scheme. Thus, this study is expected to offer meaningful implications for ongoing and future research within the framework of this model.

This research investigated an optimal control problem formulated using a Hilfer fractional-order differential model to describe the dynamics between tumor cells and the immune system during immuno-chemotherapy. The model emphasized the activation timing of effector cells and their capacity to generate a potent immune response against the tumor. The primary objective was to minimize the costs associated with immuno-chemotherapy while simultaneously reducing the tumor cell population through optimal control strategies. By solving both the state and adjoint equations, the optimal control problem was addressed numerically. Simulation results demonstrated that the proposed immuno-chemotherapy protocol significantly reduced the tumor burden.

This study examines the variable thermal conductivity and electroosmotic performance of Sutterby hybrid nanofluid (SBHNF) thin film flow over a stretched inclined sheet using an artificial neural network (ANN)-based on NARX (Multilayer Nonlinear Autoregressive Networks with Exogenous Inputs) multiple-layer backpropagation simulation with the Levenberg-Marquardt algorithm (LMA). AA7075 and AA7072 nanoparticles suspended in sodium alginate (SA) base fluid make up the hybrid nanofluid (HNF), which was selected due to its improved heat transfer properties and superior thermal conductivity. The model’s practical applicability is enhanced by melting heat, nonlinear thermal radiation, boundary slip, and Newtonian heating effects, which are considered for surface heat flow. A dataset spanning three cases and seven scenarios of SBHNF is generated by solving the simplified governing equations using the built-in MATLAB bvp4c numerical methods. The dataset comprises three divisions: 80% allocated for training, 10% for validation, and 10% for testing. The proposed system is employed for the analysis of stream and thermal transmission, with conclusions validated by several approaches, including error histograms, regression plots, time series analysis, mean square error (MSE) of the loss function, autocorrelation, and cross-correlation. Findings from the AI-based LMA validate the suggested method for solving the SBHNF accurately. Joule heating, variable thermal conductivity, and other external sources elevate fluid temperature, whereas radiation heating markedly amplifies surface heat energy by accumulating, hence improving heat transfer. The opposing forces produced by magnetic fields, Darcy’s law, and electro-osmosis reduce fluid velocity, which is effective for wellbore stability and hydraulic efficiency. The MSE and coefficient of determination (R²) are used to assess the correctness and robustness of the suggested computational framework. The trained network indicated outstanding predictive accuracy with $R^2=0.999$ for all scenarios. The error histogram for the proposed model is ${10}^{-6}$ to ${10}^{-7}$. The seven scenarios of SBHNF fall within the range of ${10}^{-8}$ to ${10}^{-13}$ for the attained high MSE (loss function) convergence levels.

Despite the availability of continuous distributions, few utilize the modeling potential of trigonometric functions, and none are based on the hyperbolic secant function. To fill this gap, we introduce the hyperbolic Sec-B family of trigonometric distributions. As a specific application, we introduce an adapted half-power logistic distribution (HS-PHLD) that retains a simple two-parameter form while offering greater versatility, particularly in tail behavior and skewness. Our research comprehensively explores this, establishing its fundamental mathematical properties, providing series expansions for its functions, and using both non-Bayesian and Bayesian estimation techniques. Monte Carlo simulations are used to validate the effectiveness of these estimators. Practically speaking, the HS-PHLD outperforms well-established models on three real-world datasets from the engineering and survival domains.

This investigation gives a comprehensive dynamical and analytical analysis of the first extended (3+1)-dimensional Kadomtsev-Petviashvili (eKP) equation, which is featured in the fields of non-linear optics and wave propagation. By the method of traveling wave transformation, the nonlinear partial differential equation is transformed into a planar dynamical system, and detailed phase plane and bifurcation analysis can be done. We analyze the qualitative characteristics of equilibrium points, i.e., centers, saddles, and cusps, for different physical conditions. An external periodic disturbance is introduced to study some complicated dynamics, and we observe chaos using a number of diagnostic tools, such as phase portraits, time series, return maps, Lyapunov exponents, and multistability analysis. A sensitivity study indicates that the dynamics of the waves depend greatly upon the initial condition, and this reveals the non-linear, unpredictable nature of the system. In parallel with the dynamical study, we obtained exact analytical solutions to the eKP equation with the help of a bilinear form. We applied various newly developed analytical techniques to obtain exact solutions, such as homoclinic solutions, multiwave solutions, and M-type rational solutions. We obtained homoclinic breather waves, M-type and rational waves, and multi-wave interactions, which exhibit localized oscillating states, stable rogue-wave solutions, and wave-number coupling. Plots show the robustness and toughness of these solutions. Merging dynamic insights and precise solutions of the extended KP model allows a better understanding of the complex nonlinear behavior, opening new horizons in soliton theory as well as applications in the fields of nonlinear optics, fluid mechanics, and complex wave systems.

This paper developed a novel nonlinear Susceptible–Vaccinated–Exposed–Infectious–Treated–Recovered–Susceptible (SVEITRS) compartmental model to investigate the transmission dynamics of tuberculosis (TB) in Algeria over the period between 1990–2024, explicitly accounting for partial Bacillus Calmette-Guérin (BCG) vaccine efficacy, endogenous reactivation of latent infection, and exogenous reinfection. The basic reproduction number ${\mathcal{R}}_0$ was derived using the next-generation matrix approach, and rigorous analytical results were established for the local and global stability of both the disease-free and endemic equilibrium points. Model parameters were estimated by fitting the model to Algerian TB surveillance data, achieving a coefficient of determination of $R^2=0.84$, which indicates a strong agreement between the model predictions and reported epidemiological trends. A global sensitivity analysis based on partial rank correlation coefficients (PRCCs) reveals that demographic recruitment and transmission-related parameters exert the greatest influence on TB dynamics, whereas intervention-related parameters such as vaccination and treatment rates have a comparatively weaker impact. These findings indicate that vaccination alone is insufficient to eliminate TB and highlight the need for integrated control strategies that combine sustained vaccination coverage, early case detection, and effective management of latent infections. The proposed modeling framework provides a quantitative basis for evaluating TB control policies in high-burden settings such as Algeria.

Bipolar fuzzy sets (BPFs) provide a suitable framework for knowledge representation if some data contains imprecise and ambiguous information. In this manuscript, the lower and upper bounds of the Seidel Laplacian energy of a bipolar fuzzy graph were examined with suitable illustrative examples. Moreover, the energy of a bipolar fuzzy graph, the Laplacian energy of a bipolar fuzzy graph, and the Seidel Laplacian energy of a bipolar fuzzy graph were examined. Furthermore, to address complex multi-criteria decision-making (MCDM) problems involving uncertainty and bipolar information, we proposed novel score functions: The score function, improved score function, and double improved score function. These functions were demonstrated through examples to effectively handle ambiguity and duality in decision-makers' inputs represented via bipolar fuzzy sets.

This research extensively investigates the solitary wave solutions of the fractional coupled Konno–Oono model, a prevalent framework in diverse scientific and engineering disciplines. Three mathematical methods called the simple equation method, the modified extended auxiliary equation mapping method, and the exponential-expansion method are gradually employed to derive the analytical solutions. Moreover, Mathematica 13.0 software is used to perform the analytical computations and graphical simulations. The explored outcomes have significant applications in the realm of magnetic fields. After the careful selection of parametric values under constrained conditions, some solutions are plotted in 2-dimensional and 3-dimensional spaces to understand the physical phenomena of the concerned model. Importantly, our findings affirm that the employed methods not only yield complete and uniform responses but also showcase simplicity, effectiveness, and remarkable computational efficiency. Hence, our research contributes valuable insights into the behavior of the fractional coupled Konno–Oono model, paving the way for enhanced comprehension and potential applications in magnetic field studies.

In this paper, we analyze and provide innovative soliton solutions for a (2+1)-dimensional generalized Korteweg-de Vries (gKdV) problem. We obtain phase shifts and dispersion relations by using the generalized Arnous technique and the Riccati equation approach, thus allowing different soliton solutions to be developed. Several precise solutions with special structural properties, including kink and solitary soliton solutions, are included in our study. This detailed examination demonstrates the complex behavior of the model and its capability to explain a large scale of nonlinear wave occurrences in many physical settings. Thus, in scientific domains such as fluid mechanics, plasma physics, and wave propagation in media ranging from ocean surfaces to optical fibers, our results are crucial to comprehend the principles behind the production and propagation of many complicated phenomena. Finally, we provide 2D and 3D graphs for various solutions that have been obtained using Maple.