Higher Institute of Engineering

The stochastic Gardner equation with Itô-type advection Brownian motion provides an effective mathematical framework for modeling the propagation of nonlinear waves in random environments. It plays a key role in understanding the transition from coherent wave structures to irregular dynamics by elucidating the interplay among randomness, nonlinearity, and dispersion in complex physical systems. Consequently, obtaining exact analytical solutions of the stochastic Gardner equation is of significant theoretical and practical importance. By employing appropriate transformation techniques together with Itô calculus, the stochastic Gardner equation is decomposed into two coupled components: a deterministic Gardner equation with an additional diffusion term and a stochastic ordinary differential equation. The extended tanh-function method is applied to derive exact traveling wave solutions of the deterministic Gardner equation. These solutions are then combined with the analytical solution of the stochastic ordinary differential equation to construct exact solutions of the original stochastic Gardner equation. The effectiveness of the proposed framework can be demonstrated by deriving various stochastic wave solutions and graphing them, thus proving its ability to investigate nonlinear wave propagation in the presence of stochastic effects. A variety of exact analytical solutions for the stochastic Gardner equation are successfully obtained, including solitary wave structures influenced by stochastic effects. The impact of advection Brownian motion on the wave dynamics is systematically investigated. Three-dimensional graphical simulations, generated using MATLAB, illustrate how stochastic advection modifies the shape, amplitude, and evolution of the solutions compared to their deterministic counterparts. Additionally, these findings shed light on how stochastic perturbations affect the amplitude and propagation characteristics of nonlinear waves in the Gardner model. The proposed analytical framework provides explicit exact solutions for the stochastic Gardner equation and reveals the significant role of advection Brownian motion in altering nonlinear wave behavior. These results enhance the understanding of stochastic nonlinear wave propagation and may be useful for modeling realistic physical systems subject to random perturbations.

The contamination of aquatic systems with cationic dyes such as Basic Fuchsine (BF) poses substantial ecological and public-health risks, highlighting the need for efficient, sustainable, and regenerable adsorbents. In this work, Ca-Alginate/Carboxymethyl Cellulose (Ca-A/CMC) hydrogel beads were fabricated via Ca2+-induced ionic crosslinking of sodium alginate and CMC, yielding a biopolymeric network enriched with -COO- and -OH functional groups and a well-defined mesoporous structure. To the best of our knowledge, the application of Ca-A/CMC hydrogel beads for BF adsorption has not yet been systematically investigated. Extensive characterization (FTIR, XRD, SEM, TGA, XRF, and N-2 adsorption-desorption) verified successful polymer integration, Ca2+ coordination, and hierarchical mesoporosity. Under optimised conditions (100 mg L-1 BF, pH 7, 25 +/- 1 degrees C, 180 min), the beads exhibited outstanding performance, achieving a monolayer adsorption capacity of 222.5 +/- 2.1 mg g(-1) with 97.14 +/- 0.8% removal efficiency. Kinetic modeling showed excellent agreement with the PSO model (R-2 = 0.998), suggesting that the adsorption rate is governed by the availability of active surface sites rather than simple mass-transfer control. Equilibrium data were best described by the Langmuir isotherm (R-2 = 0.996), indicating predominantly monolayer-like adsorption behaviour under the investigated conditions. Thermodynamic analysis showed negative Delta G degrees, a positive enthalpy change (Delta H degrees = +86.4 +/- 2.8 kJ mol(-1)), and an increase in entropy (Delta S degrees > 0), demonstrating that BF adsorption is spontaneous, endothermic, and entropy-driven. The beads further displayed good reusability, maintaining 90.24 +/- 1.4% removal efficiency after five adsorption-desorption cycles. Overall, the Ca-A/CMC hydrogel beads offer a structurally stable and environmentally benign platform for cationic dye removal in wastewater treatment applications.

This paper focuses on analyzing how initial stress influences wave propagation phenomena in a microelongated thermoelastic medium described within the framework of fractional conformable derivative, considering both the dual phase lag (DPL) and refined dual phase lag (RDPL) theories. The fundamental governing equations for heat transfer, mechanical motion, and microelongation are established to incorporate finite thermal wave speed and microelongation effects. Through an appropriate non-dimensionalization procedure and the application of the normal mode analysis technique, the coupled partial differential system is transformed into a form that admits explicit analytical solutions. These solutions provide expressions for displacement, microelongation, temperature distribution, and stress components, allowing a comprehensive examination of the thermomechanical wave behavior within the medium. To better comprehend the theoretical results, numerical evaluations are performed to emphasize the comparison of DPL and RDPL in the presence and absence of initial stress, as well as the influence of the fractional-order parameter and different times on wave properties. The results show that initial stress has a considerable effect on wave propagation characteristics such as amplitude modulation, propagation speed, and attenuation rate. Furthermore, the use of fractional conformable derivatives and the RDPL formulation allows for more precise modeling and control of the thermal relaxation dynamics. The current study contributes to a better understanding of the linked microelongated and thermal effects in thermoelastic media, as well as significant insights for designing and modeling advanced microscale thermoelastic systems.

With the use of the modified extended direct algebraic scheme (MEDAS), this work investigates all possible envelope solitons and their propagation properties in a birefringent fiber where light propagation is controlled by two coupled nonlinear Schrödinger equations (CNLSEs) via coherent and incoherent nonlinear couplings. With regard to elliptical core optical fibers, we investigate the physical significance of different optical solitons in the presence of third-order nonlinearity and group-velocity dispersion. This suggests that a birefringent fiber medium may produce a potentially plentiful array of nonlinear periodic waves and localized pulses. The study’s conclusions have significant implications for the propagation of solitons in nonlinear optics. Possible formulations for the resultant solutions include single periodic solutions, Weierstrass elliptic doubly periodic solutions, singular, dark, and bright solitons, exponential solutions, and Jacobi elliptic functions (JEFs). The wave solutions we found are contrasted with existing literature to illustrate their originality and significant contribution to contemporary study. Given the numerous applications of the studied model, the efficacy of our approach implies that it might be used to a wide variety of nonlinear problems in a broad range of areas, including soliton theory. To further visualize their behaviors, we also displayed the contours of a few of these solutions in 2D and 3D contour plots.

This work presents a comprehensive investigation into the thermoelastic response of a porous micro-stretch elastic half-space medium subjected to the effect of rotation. The analysis is carried out within the frameworks of the Lord-Shulman (L-S) theory, the Dual-Phase-Lag (DPL) model, and Refined Dual-Phase-Lag (RDPL) extension. The novelty of this work lies in integrating rotational effects into the RDPL model for porous micro-stretch materials, deriving the fully coupled field equations using the Normal-Mode Analysis (NMA) technique, and offering a comprehensive comparison among the three thermoelastic theories. The inclusion of rotation introduces significant modifications to the thermo-mechanical fields due to Coriolis and centrifugal contributions, which are effectively captured by the refined phase-lag structure of the RDPL model. The results demonstrate that rotation strongly alters the temperature distribution, displacement field, micro-stretch response, and stress components. Specifically, the RDPL model predicts reduced displacement amplitudes, more damped thermal waves, and smoother stress distributions compared with the L-S and DPL theories, highlighting the influence of refined phase-lag parameters on wave attenuation and energy transport. Furthermore, rotation enhances the coupling between thermal and mechanical fields, leading to noticeable changes in wave speed, amplitude, and stability. These findings provide deeper insight into the dynamic behavior of rotating porous micro-structured materials and offer a more realistic framework for analyzing advanced thermoelastic systems.