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This study investigates the nonlinear fractional-order Pochhammer-Chree equation, featuring a power-law nonlinearity of order τ, a model that describes how nonlinear longitudinal waves travel in elastic materials with memory of prior deformations. Understanding nonlinear wave propagation in elastic materials with memory effects is important for accurately modeling complex physical phenomena in nonlinear elasticity, geophysics, and material science. To incorporate these memory effects, conformable fractional derivative is utilized that allowing us to examine the fractional-order spatial and temporal changes. Kumar-Malik method is used to extract analytical solutions like bright soliton, dark soliton, kink soliton and solitary wave forms, that demonstrates the dynamics of the model. To better understand these solutions behaviours, visual analysis is considered. In this analysis mathematica 14.2 is used to construct the two-dimensional, three-dimensional, and contour plots, which illustrate the influence of the fractional parameter χ on the waveforms. The shape, width and amplitude of the solitons also vary with the change in the value of the parameter, χ. These diagrams are able to make it clear how the system would respond to various physical situations. These findings indicate the efficiency of the Kumar-Malik method which guarantees accurate solutions of the fractional Pochhammer-Chree equation having power laws nonlinearity. The work enhances the theoretical knowledge on the nonlinear waves of the fractional-order systems and brings forth new uses in the field of mathematical physics and nonlinear elasticity.

In this paper, we investigate exact analytical solutions of the (2+1)-dimensional complex modified Korteweg-de Vries (CmKdV) system using the truncated M-fractional derivative together with the Jacobi elliptic function expansion method. The CmKdV system plays a central role in modeling nonlinear wave propagation in optics, plasma physics, and fluid dynamics. By applying the truncated M-fractional derivative, the system is transformed into a more tractable form, enabling the effective use of the Jacobi elliptic function expansion method to construct exact soliton and periodic wave solutions. The results offer deeper insight into the system's nonlinear dynamics and highlight the robustness of the proposed method. Graphical simulations generated in Mathematica illustrate the physical behavior of the obtained solutions across multiple dimensions, such as two-dimensional, three-dimensional, and contour, and the influence of time on the wave propagations. Overall, combining the Jacobi elliptic function expansion method and truncated M-fractional derivatives creates a strong framework for solving complex differential equations, leading to new possibilities. Opportunities for research and development exist. This research adds to our understanding of the (2+1)-dimensional complex modified Korteweg-de Vries (CmKdV) system and demonstrates how theoretical mathematics can be applied to real concerns. Mathematical modeling and computational visualization can significantly impact engineering and science, and our findings promote a multidisciplinary approach to research.

Peristaltic or undulating flow plays a significant role in various biomedical and industrial processes, where it provides an efficient mechanism for transporting fluids through flexible conduits such as catheters and endoscopic channels. Understanding such flow behavior is essential for improving medical devices and industrial applications involving non-Newtonian fluids. This study investigates the peristaltic motion of a Carreau fluid whose viscosity varies with both temperature and concentration within a flexible, axisymmetric channel composed of two overlapping cylindrical tubes. The outer wall of the channel exhibits a sinusoidal wave pattern, simulating a realistic endoscopic configuration. The governing nonlinear, nonhomogeneous partial differential equations were formulated in cylindrical coordinates under the assumption of a long wavelength and low Reynolds number. The equations were transformed into a dimensionless form and solved using the uniform perturbation method. Graphical analyses were performed using Mathematica software. The results illustrate the combined effects of temperature-dependent and concentration-dependent viscosity on the velocity distribution and pressure gradient within the channel. Increasing temperature and solute concentration were found to enhance fluid velocity and reduce flow resistance. The study provides a comprehensive understanding of peristaltic transport in variable-viscosity Carreau fluids under realistic physiological conditions. These findings may contribute to optimizing the design and performance of endoscopic and biomedical fluid transport systems.

The objective of this study is to examine the influence of nonlocal thermoelasticity parameters on an orthotropic medium subjected to magnetic fields and rotational effects, within the framework of Green-Naghdi thermoelasticity theory (Type III). A time-dependent thermal load is applied to the free surface of the medium, and analytical solutions for the resulting thermal stresses, displacement, and temperature fields are derived using the normal mode analysis and eigenvalue approach techniques. Numerical simulations, implemented through MATHEMATICA programming, are conducted for a representative material to validate the theoretical model. The results are presented graphically to highlight the effects of various parameters, including time, non-locality, magnetic intensity, and rotational speed, on the thermoelastic response. These findings offer significant insights for advanced engineering and scientific applications, especially in geophysics, aerospace, and biomedical engineering, where complex multiphysical interactions and nonlocal effects play a critical role. The study also contributes to the ongoing development of generalized thermoelastic models capable of accurately capturing wave propagation and heat conduction behaviours in anisotropic and heterogeneous materials.

Frictional moving load-induced dynamic response of a porous piezoelectric micro/nano plate with superficial parabolic discontinuity. The present paper aims to analyze the complex dynamic response of micro/nano-scale components through investigating the stress distribution within a Nonlocal Porous Piezoelectric Layer (NPPEL) of finite thickness. The study specifically focuses on quantifying the combined effects of material porosity, size-dependent elasticity, and geometrical surface imperfections when the layer is subjected to a load moving across its upper boundary. This comprehensive model provides a more realistic assessment of reliability for small-scale smart devices. The layer's constitutive behavior is modeled using Eringen's nonlocal elasticity theory to account for the essential size effects present at the micro/nano scale. The governing equations for the coupled porous and piezoelectric medium are derived, incorporating appropriate boundary conditions for a moving load. Crucially, the superficial parabolic discontinuity on the upper surface is handled analytically through a robust perturbation technique, allowing for the derivation of closed analytical forms for the resulting shear and normal stresses. The final solutions are then computed using Mathematica to illustrate the transient stress fields. Numerical results demonstrate that the nonlocal parameter is highly effective at amplifying the magnitude of the stresses, which is characteristic of the stiffening effect in nonlocal models. The depth and factor of the parabolic irregularity significantly amplify the stress concentrations at the interface, indicating a critical pathway for potential failure. Furthermore, the frictional coefficient of the moving load plays a non-linear role in dictating the shear stress distribution, providing crucial insight into contact mechanics at the nanoscale. The core novelty lies in the simultaneous analytical incorporation of nonlocal effects, porosity, and an arbitrary surface irregularity under dynamic moving load conditions-a combination highly relevant to microfabrication. The model is directly applicable to enhancing the design and performance assessment of MEMS/NEMS pressure sensors, ultra-thin piezoelectric energy harvesters, and other micro-electromechanical devices where surface quality and size effects dictate device lifespan and reliability.

Ternary hybrid nanofluids(THNFs) are a revolutionary advancement in thermal management, offering remarkable thermal conductivity that significantly boosts heat transfer efficiency. THNFs are ideal for cooling systems, solar energy applications, and electronic device regulation, where effective heat management is crucial. By fine tuning their composition, researchers can tailor these nanofluids to meet specific industrial needs, leading to improved efficiency and energy savings. The goal of current study is twofold: first, to examine the improvement of heat, mass, and motile density of THNF, and secondly, to investigate the irreversibilities in bioconvective dihybrid and trihybrid nanofluids. The THNF is formulated by suspension of nanoparticles of cobalt ferrite [Formula: see text], disulfide (dithioxo) molybdenum [Formula: see text], and copper (Cu) into pure engine oil [Formula: see text]. For dihybrid nanofluid the volume fraction of copper is taken as zero. Maxwell fluid model is utilized to analyze the performance of THNF and dihybrid nanofluid(DHNF). The flow in THNF and DHNF is induced due to an impermeable stretched sheet. Flow governing equations for DHNF and THNF are obtained considering diverse assumptions like electro-magnetohydrodynamic(EMHD), Dufour, Soret, chemical reaction, thermal radiation, and activation energy. Irreversibilities are modeled with the help of thermodynamics second law. The model equations are altered into ordinary system via transformation procedure. Numerical simulations are carried out through built-in function(NDSolve) of Mathematica. Impact of generated variables on DHNF and THNF velocity, thermal field, motile density profile and mass concentration are examined. Comparative analysis for THNF and DHNF is performed. Furthermore, local heat, mass, density, and skin friction coefficient for THNF and DHNF are numerically investigated. Numerical results show that the skin friction coefficient for THNF is 3.2% more than that of DHNF, and the heat transfer rate of THNF is up to 16% higher than that of DHNF. The density number of DHNF is about 1.08% less than that of THNF.

This research explores wave phenomena in the [Formula: see text]-dimensional integro-differential Jaulent-Miodek equation associated with dispersive nonlinear systems. The primary objective of this study is to derive and analyze exact wave solutions of the governing model using the generalized Arnous approach and the modified simple equation method. By applying a suitable travelling-wave transformation, the nonlinear partial differential equation is reduced to an ordinary differential equation, which is then solved analytically using the proposed techniques. The results reveal a rich variety of nonlinear wave structures such as kink, anti-kink, bright, dark, bell-shaped, anti-bell, periodic, singular periodic, V-shaped, W-shaped, and singular wave solutions. The physical characteristics of the obtained solutions are illustrated through two-dimensional, three-dimensional, and contour plots generated using Mathematica. Furthermore, the dynamical behavior of the system is investigated through bifurcation analysis using two-dimensional phase portraits, while chaotic dynamics are examined via corresponding three-dimensional phase portraits together with time-series plots to identify transitions from periodic to quasi-periodic and chaotic behavior. In addition, sensitivity analysis is carried out using two-dimensional phase portraits to examine the dependence of the system response on initial conditions. The findings confirm the effectiveness and applicability of the proposed analytical methods for constructing exact solutions of higher-dimensional nonlinear evolution equations and provide deeper insight into nonlinear wave propagation phenomena arising in optics, plasma physics, and fluid flows.

In this work, we present a complete dynamical analysis of lump, breather, M-shaped, and other waveforms propagating in a nonlinear PDE governing nonlinear low-pass electrical transmission lines. We utilize the Hirota bilinear transformation approach with the help of Mathematica to report a number of wave solutions, including bright and dark lumps, solitons, breathers, and kink waves, along with their periodic and aperiodic forms. Energy distribution, wave interactions, and changes are presented in the form of 3D, contour, and 2D plots, which demonstrate the nonlinear characteristics that govern the dynamics. These results provide a better understanding of the propagation, stability, and interaction of waveforms which are useful in signal and energy transport and also in the construction of complex nonlinear electric circuits.

In this research, the impact of thermal effects, viscous energy loss, and magnetic-field interaction on a Williamson ternary hybrid nanofluid (Ag, SWCNT, MWCNT) that simulates blood flow (in 2D) over an elastically moving surface is assessed. An analytical technique is developed to provide a framework for enhancing convective heat transfer and reducing streamwise resistance in high-energy systems. Obtaining semi-analytical solutions to the governing nonlinear partial differential equations within the BVPh 1.0 and BVPh 2.0 packages for Mathematica involves transforming them into ordinary differential equations via special similarity variables, applying the homotopy analysis approximating method, and achieving residuals below 10-5 in fewer than 20 steps. The analysis reveals that the resultant average heat transfer (Nu) is over 21% due to the surface cooling heat flux, the thickening of the opaque thermal layer from thermal effects, the extension of the Eckert number, the nanofluid volumetric concentration, and the magnetic Williamson number (slowing rates). And accurately including these ternary hybrid nanofluids in biomedical wearables, blood cooling, polymer extrusion cooling, and highly oriented micro- and electronic heat exchangers for efficient simultaneous temperature and shear stress, is revealing.

This study investigates elastic wave propagation in a generalized magneto-micropolar thermoelastic medium under the combined effects of gravity, initial stress, and a magnetic field. The governing equations, based on the micropolar thermoelastic framework, account for micro-rotation, thermal conduction, electromagnetic interactions, and initial stress, providing a comprehensive description of the medium's behavior. Analytical expressions for displacement, stress components, and microrotation fields are obtained using the normal mode method, while numerical simulations in Mathematica illustrate how time, magnetic field, gravity, and initial stress influence wave speed, dispersion, and attenuation. The results show that magnetic and micropolar properties reshape wave behavior and introduce distinct dispersive and damping patterns. These findings highlight the complex interplay of multi-physical effects and offer insights for optimizing wave control in microstructured and multifunctional materials, with potential applications in materials science, geophysics, and advanced engineering systems.

In this work, a time-fractional form of Richards' equation is considered to study the infiltration phenomenon in unsaturated porous media. The memory effects in the flow process are described using the Caputo-Fabrizio fractional operator. To obtain an approximate analytical solution of the governing nonlinear fractional partial differential equation, a hybrid analytical technique combining the Natural transform method and the variational iteration method is employed. The proposed Natural transform variational iteration method (NVIM) provides a rapidly convergent series solution and avoids complicated discretization or linearization procedures. A rigorous convergence and uniqueness analysis is carried out, which confirms the validity and accuracy of the proposed approach. The effectiveness and reliability of the method are demonstrated through the solution of the considered problem. The obtained solutions are illustrated through graphical representations using MATHEMATICA, where a comparison between the approximate analytical solution and the exact solution is carried out to validate the accuracy and effectiveness of the proposed method. In addition, the influence of the fractional parameter and other model parameters on the infiltration process is analyzed through graphical illustrations. The results demonstrate that the proposed approach provides accurate and efficient solutions and can serve as a useful analytical tool for solving nonlinear fractional differential equations arising in groundwater hydrology and related physical processes.

Complex-valued chemometrics offers a promising extension of classical regression methods by exploiting both real and imaginary spectral components. Here, we show that conventional absorbance (χ(1)) and Raman (χ(3)) spectra can be transformed into complex-valued forms by combining the measured intensities as imaginary parts with their Kramers-Kronig-derived real parts. We benchmark four regression methods─classical least squares (CLS), inverse least squares (ILS), principal component regression (PCR), and partial least-squares regression (PLSR)─across four representative systems: the quasi-ideal benzene-toluene and benzene-cyclohexane mixtures, the nonideal acetone-chloroform mixture, and blood plasma spiked with glucose and urea. Compared to conventional chemometrics, complex-valued approaches consistently reduce prediction errors (MAE, RMSE, and R2). Implementation is computationally inexpensive, since the Kramers-Kronig transform of absorbance or Raman spectra can be obtained within seconds using FFT-based routines, even for large data sets. Software implementation is straightforward, and programs can be adapted within minutes using standard environments such as Mathematica. Surprisingly, complex-valued ILS matches or surpasses complex-valued PLSR, echoing earlier results in infrared spectroscopy, using the complex refractive index function, and suggesting a re-evaluation of regression hierarchies when complex spectra are available. These findings demonstrate that complex-valued chemometrics is broadly applicable, physically grounded, and capable of enhancing both classical and modern regression strategies in analytical spectroscopy.

Electrokinetically driven squeezing flows are of increasing relevance in biomedical transport and microscale pumping systems, particularly in understanding complex blood dynamics in constricted arteries. This study explores the electroosmotic bi-layered hybrid nano-blood flow through a cardiovascular squeezing channel, incorporating nanolayer and thermal slip effects. The lower layer consists of Casson-type non-Newtonian blood with SWCNTs and gold nanoparticles, while the upper layer is modeled as a Newtonian fluid. The controlling equations for various flow quantities are presented using nonlinear partial differential equations and subsequently converted to a scale-invariant form through scale-invariant transformations. The coupled nonlinear system is numerically resolved through the Runge-Kutta-Fehlberg (RKF45) approach in conjunction with a shooting scheme, executed in Mathematica to achieve stable and precise computational results. Results indicate that temperature enhances with increasing Hartmann number due to magnetic heating but diminishes with stronger interfacial ratio parameter. The Casson region exhibits more pronounced thermal and velocity gradients compared to the Newtonian region, reflecting physiological shear-thinning characteristics. This study employs an artificial neural network for rapid and precise evaluation of the skin friction coefficient demonstrating strong predictive accuracy with minimal error rates of 0.01%. These findings provide new insights into the interplay between electromagnetic and electroosmotic forces in nanofluidic blood transport. The model offers potential applications in optimizing targeted drug delivery, hyperthermia treatments and microvascular flow control in cardiovascular systems. Blood flow inside the human body becomes more complicated when it moves through narrow or partially blocked vessels, such as those affected by cardiovascular disease. In these tight spaces, forces from the vessel walls, electric fields, magnetic fields and tiny particles inside the blood can all influence how the blood moves and how heat is transferred. Understanding these effects is important for improving medical devices, treatment techniques and drug delivery methods. In this study, we examine how blood behaves when it flows through a channel that repeatedly squeezes in and out, similar to how some sections of the cardiovascular system contract. The model considers two layers of blood: a lower layer that behaves like real, thickened blood (called a non-Newtonian Casson fluid) mixed with two types of nanoparticles, and an upper layer that behaves more like a simple fluid. We also include the effects of an electric field, a magnetic field, and heat transfer at the walls. We solve the governing equations using well-established numerical methods to predict how the blood flow speed and temperature change under different conditions. The results show that stronger magnetic fields increase the temperature because they create heating inside the fluid. However, changes at the interface between the two layers can reduce this temperature rise. The complex upper layer shows sharper changes in speed and temperature, which matches the natural behavior of real blood that becomes thinner when it flows faster. Overall, this research improves our understanding of how electric and magnetic forces interact with blood containing nanoparticles. These insights may help advance targeted drug delivery, magnetic-based heating therapies and better control of blood flow in small vessels.

This paper explores the nonlinear Zoomeron equation that is much used in the modeling of nonlinear wave propagation processes in several physical systems including fluid dynamics and nonlinear optics. The great nonlinearity of this equation renders search of analytical solutions to this equation to be a hard and significant problem. In order to solve this, the governing nonlinear partial differential equation is converted into an ordinary differential equation through the use of the modified Khater method (MKM) and its exact solutions are then built. In this method, a vast variety of new analytical solutions are discovered, including kink, anti-kink, singular periodic and dark solitary wave solutions. These solutions are further illustrated using the dynamical behavior by two-dimensional, three-dimensional and contour plots in mathematica. The results obtained prove that the Modified Khater method is a good and valid tool of analysis in solving nonlinear evolution equations. Additionally, the derived solutions add to the solution space of the Zoomeron model and give further insight into the complex nonlinear wave dynamics, which can be helpful in future theoretical and practical research.

In several engineering systems, wavy surfaces are utilized to enhance thermal distribution such as heat exchangers and aerodynamics and drag control. However, when radiative heat transfer, heat generation and magnetic fields are considered, velocity and thermal distribution become more difficult, making it significant to understand their combined influences for improved heat transfer. Therefore, this problem focuses on the flow rate, isotherms, streamlines and thermal distribution of the hydromagnetic fluid flow over a wavy surface subjected to a constant heat flux with radiative heat transfer and magnetic fields influence. The impermeable wavy texture is considered to be heated via a constant heat flux, which supplies the fluid a consistent source of heating energy. The transformed nonlinear governing equations are solved numerically using a robust and efficient computational scheme known as the Spectral Quasi-Linearization Method (SQLM), implemented in Wolfram Mathematica to ensure precise and reliable solutions to the physical problem. The numerical outcomes obtained in the present study are validated through comparison with previously published results, showing excellent agreement and good concordance. This consistency confirms the reliability, effectiveness, and accuracy of the adopted numerical methodology. The findings display that radiation-conduction parameter enhance the thermal distribution within the boundary layer, whereas the amplitude of waviness tends to reduce the fluid velocity.

Radioactive seed localization (RSL) using 125I is widely employed for the surgical resection of nonpalpable breast lesions due to its precision and scheduling flexibility. Although clinical experience supports its safety, quantifying radiation dose to surrounding healthy breast tissue remains important for establishing conservative operational guidelines, including acceptable implant duration prior to surgery. In this study, cumulative dose to healthy breast tissue from an implanted 125I localization seed was evaluated using two independent computational approaches: a simplified hemispherical breast model implemented in MATLAB and a Monte Carlo simulation developed in Mathematica. Both models incorporated dosimetric parameters from AAPM Task Group 43, including the dose-rate constant, radial dose function, and two-dimensional anisotropy function, along with tissue properties consistent with International Commission on Radiological Protection recommendations. Results from both models demonstrated that cumulative dose decreases rapidly with increasing distance from the seed and increases slowly with implantation time due to the low dose rate and long half-life of 125I. At a distance of 1 cm, cumulative dose reached approximately 3-4 cGy after 5 d and approximately 50 cGy only after extended implantation times exceeding 200 d. Agreement between the MATLAB and Monte Carlo models was within expected limits given differences in deterministic and stochastic modeling approaches. These findings indicate that radiation dose to healthy breast tissue from RSL remains well below thresholds associated with deterministic effects for clinically relevant implantation times. The results support the use of a conservative "no later than" removal date while reaffirming the radiological safety and practical advantages of 125I seed localization in breast cancer surgery.

After being motivated by the diverse applications of blood rheology, nanotechnology, magnetic field, chemical reaction, solar radiation, and non-Darcy porous media in nano-industrial, medical, and chemical engineering domains. The current computational study aims to numerically examine the influences of velocity slip, internal thermal generation or absorption, chemical reactions, and thermal radiation on magneto-hydrodynamic blood-based nanofluid flow with thermo-Brownian motion through an extending interface within a high-permeability medium. Furthermore, the sensitive analysis of flow features with respect to the independent flow parameters is considered. Suitable similarity equations are employed to convert the partial differential equations into ordinary differential equations together with their boundary constraints. The NDSolve method in Mathematica 11.0 is employed to numerically analyze the flow model, yielding data for the stream function, velocity profile, frictional force coefficient, temperature profile, concentration profile, local Nusselt number, and Sherwood number across several rheological parameters. A boundary slip diminishes momentum transmission from the fluid to the surface; when velocity slip escalates, the velocity profile declines. The intensity of the thermal boundary layer escalates with the thermal Grashof number. The temperature distribution is exacerbated by the influence of radiation. As the Brownian parameter grows, the nanofluid temperature intensifies. The chemical reaction parameter substantially affects the enhancement of both skin friction and the Sherwood number. The Nusselt number is enhanced by increasing the thermal Grashof number. The sensitivity analysis indicates that the chemical reaction and concentration Grashof number significantly influence the improvement of rheological properties. The results of this work are relevant for regulating film thickness, chemical vapour deposition, drug delivery systems, and process optimization.

Nonlinear oscillators with two degrees of freedom (2DOF) serve as fundamental models for describing complex dynamical behavior in engineering and applied mechanics. Accurate prediction of their responses is crucial for stability enhancement, vibration suppression, and optimal design of coupled mechanical systems. In this study, three distinct 2DOF coupled oscillator models are examined, encompassing both linear and strongly nonlinear restoring forces that govern free and damped vibration regimes. These models provide realistic frameworks for analyzing nonlinear interactions, resonance phenomena, and stability boundaries in coupled dynamical systems. The primary objective is to develop and apply a robust non-perturbative approach (NPA) for deriving periodic solutions of conservative and damped coupled oscillators. The proposed approach, rooted in He’s Frequency Formula (HFF), fundamentally differs from classical perturbation techniques as it avoids Taylor-series expansions, linearization assumptions, and small-parameter constraints. Instead, the nonlinear governing equations are transformed into analytically tractable linear forms, enabling efficient treatment of strongly nonlinear performance. The analytical solutions are validated through comprehensive numerical simulations implemented in Mathematica Software (MS), and are systematically compared with direct numerical integrations, demonstrating excellent accuracy and computational efficiency. Furthermore, bifurcation diagrams and Poincaré maps (PMs) are employed to characterize the qualitative dynamical transitions and classify the complex response patterns exhibited by each coupled model.

This article examines the significance of entropy production of two-phase Casson nanomaterial considering microbes cell, compelled by the Homann stagnant point past a vertical deformable sheet. The flow is modeled via porous medium laden by Darcy-Forchheimer drag effects, influenced by heated radiation and buoyancy effect. The suction/injection and convective flow have been applied on the boundary conditions. The double diffusive heat/mass theory model has been considered to signify the influences of thermal/solutal relaxation time in contrast to traditional Fourier and Fick's equations. Employing similarity alteration, the constitutive equations are dimensionalized and rehabilitated into a system of ODE's. These are then semi-analytically considered through a robust Homotopy analysis method (HAM) along convergence analysis on MATHEMATICA 12.0. The various profiles have been examined against the various physical variables. The findings clarify the combined effects on entropy production along transfer mechanism of the fluid factor, inertia coefficient, buoyancy factor, heated radiation factor, and thermal/mass relaxation time. Moreover, Casson fluid parameter is declined the velocity filed. Moreover, thermophoresis factor is greatly enhanced for the temperature and concentration fields. Additionally, the entropy production is declined via the Brickman number and porosity factor but boosted through the Bejan number. This study sheds new light on the thermal efficiency of biological-nanofluidic structures, which may find usage in solar and thermal collectors, medicinal tools, including innovative energy technologies.