Search Results (433)

In this article, we propose a novel nonlinear cross-diffusion framework to model the distribution of susceptible and infected individuals within their habitat using a reduced SIR model that incorporates saturated incidence and treatment rates. The study investigates solution boundedness through the theory of parabolic partial differential equations, thereby validating the proposed spatio-temporal model. Through the implementation of the suggested cross-diffusion mechanism, the model reveals at least one non-constant positive equilibrium state within the susceptible–infected (SI) system. This work demonstrates the potential coexistence of susceptible and infected populations through cross-diffusion and unveils Turing instability within the system. By analyzing codimension-2 Turing–Hopf bifurcation, the study identifies the Turing space within the spatial context. In addition, we explore the results for Turing–Bogdanov–Takens bifurcation. To account for seasonal disease variations, novel perturbations are introduced. Comprehensive numerical simulations illustrate diverse emerging patterns in the Turing space, including holes, strips, and their mixtures. Additionally, the study identifies non-Turing and Turing–Bogdanov–Takens patterns for specific parameter selections. Spatial series and surfaces are graphed to enhance the clarity of the pattern results. This research provides theoretical insights into the implications of cross-diffusion in epidemic modeling, particularly in contexts characterized by localized mobility, clinically evident infections, and community-driven isolation behaviors. Full article

As power grids scale and aging assets edge toward obsolescence, grounding grid corrosion has become a critical vulnerability. Conventional diagnosis must fit high-dimensional electrical data to a physical model, typically yielding a nonlinear under-determined system fraught with computational burden and uncertainty. We propose the Enhanced Biomimetic Hippopotamus Optimization (EBOHO) algorithm, which distills the river-dwelling hippo’s ecological wisdom into three synergistic strategies: a beta-function herd seeding that replicates the genetic diversity of juvenile hippos diffusing through wetlands, an elite–mean cooperative foraging rule that echoes the way dominant bulls steer the herd toward nutrient-rich pastures, and a lens imaging opposition maneuver inspired by moonlit water reflections that spawn mirror candidates to avert premature convergence. Benchmarks on the CEC 2017 suite and four classical design problems show EBOHO’s superior global search, robustness, and convergence speed over numerous state-of-the-art meta-heuristics, including prior hippo variants. An industrial case study on grounding grid corrosion further confirms that EBOHO swiftly resolves the under-determined equations and pinpoints corrosion sites with high precision, underscoring its promise as a nature-inspired diagnostic engine for aging power system infrastructure. Full article

The non-monotonic behavior of amperometric enzyme-based biosensors under uncompetitive and noncompetitive (mixed) substrate inhibition is investigated computationally using a two-compartment model consisting of an enzyme layer and an outer diffusion layer. The model is based on a system of reaction–diffusion equations that includes a nonlinear term associated with non-Michaelis–Menten kinetics of the enzymatic reaction and accounts for the partitioning between layers. In addition to the known effect of substrate inhibition, where the maximum biosensor current differs from the steady-state output, it has been determined that external diffusion limitations can also cause the appearance of a local minimum in the current. At substrate concentrations greater than both the Michaelis–Menten constant and the uncompetitive substrate inhibition constant, and in the presence of external diffusion limitation, the transient response of the biosensor, after immersion in the substrate solution, may follow a five-phase pattern depending on the model parameter values: it starts from zero, reaches a global or local maximum, decreases to a local minimum, increases again, and finally decreases to a steady intermediate value. The biosensor performance is analyzed numerically using the finite difference method. Full article

We propose a hybrid numerical framework for solving time-fractional Navier–Stokes equations with nonlinear damping. The method combines the finite difference L1 scheme for time discretization of the Caputo derivative ($0<\alpha <1$) with mixed finite element methods (P1b–P1 and $P_2$–$P_1$) for spatial discretization of velocity and pressure. This approach addresses the key challenges of fractional models, including nonlocality and memory effects, while maintaining stability in the presence of the nonlinear damping term ${\gamma \left|\mathbf{u}\right|}^{r-2}\mathbf{u}$, for $r\ge 2$. We prove unconditional stability for both semi-discrete and fully discrete schemes and derive optimal error estimates for the velocity and pressure components. Numerical experiments validate the theoretical results. Convergence tests using exact solutions, along with benchmark problems such as backward-facing channel flow and lid-driven cavity flow, confirm the accuracy and reliability of the method. The computed velocity contours and streamlines show close agreement with analytical expectations. This scheme is particularly effective for capturing anomalous diffusion in Newtonian and turbulent flows, and it offers a strong foundation for future extensions to viscoelastic and biological fluid models. Full article

This paper presents a spectral method to solve nonlinear distributed-order diffusion equations with both time-distributed-order and two-sided space-fractional terms. These are highly challenging to solve analytically due to the interplay between nonlinearity and the fractional distributed-order nature of the time and space derivatives. For this purpose, Hexic-kind Chebyshev polynomials (HCPs) are used as the backbone of the method to transform the primary problem into a set of nonlinear algebraic equations, which can be efficiently solved using numerical solvers, such as the Newton–Raphson method. The primary reason of choosing HCPs is due to their remarkable recurrence relations, facilitating their efficient computation and manipulation in mathematical analyses. A comprehensive convergence analysis was conducted to validate the robustness of the proposed method, with an error bound derived to provide theoretical guarantees for the solution’s accuracy. The method’s effectiveness is further demonstrated through two test examples, where the numerical results are compared with existing solutions, confirming the approach’s accuracy and reliability. Full article

This study analyzes the family of one of the most essential fractional differential equations due to its wide applications in physics and engineering: the multidimensional fractional linear and nonlinear diffusion equations. The Caputo fractional derivative operator is used to treat the time-fractional derivative. To complete the analysis and generate more stable and highly accurate approximations of the proposed models, three extremely effective techniques, known as the direct Tantawy technique, the new iterative transform technique (NITM), and the homotopy perturbation transform method (HPTM), which combine the Elzaki transform (ET) with the new iterative method (NIM), and the homotopy perturbation method (HPM), are employed. These reliable approaches produce more stable and highly accurate analytical approximations in series form, which converge to the exact solutions after a few iterations. As the number of terms/iterations in the problems series solution rises, it is found that the derived approximations are closely related to each problem’s exact solutions. The two- and three-dimensional graphical representations are considered to understand the mechanism and dynamics of the nonlinear phenomena described by the derived approximations. Moreover, both the absolute and residual errors for all generated approximations are estimated to demonstrate the high accuracy of all derived approximations. The obtained results are encouraging and appropriate for investigating diffusion problems. The primary benefit lies in the fact that our proposed plan does not necessitate any presumptions or limitations on variables that might affect the real problems. One of the most essential features of the proposed methods is the low computational cost and fast computations, especially for the Tantawy technique. The findings of the present study will be valuable as a tool for handling fractional partial differential equation solutions. These approaches are essential in solving the problem and moving beyond the restrictions on variables that could make modeling the problem challenging. Full article

This work examines the fractional Sawada–Kotera and Korteweg–de Vries (KdV)–Burgers equations, which are essential models of nonlinear wave phenomena in many scientific domains. The homotopy perturbation transform method (HPTM) and the Yang transform decomposition method (YTDM) are two sophisticated techniques employed to derive analytical solutions. The proposed methods are novel and remarkable hybrid integral transform schemes that effectively incorporate the Adomian decomposition method, homotopy perturbation method, and Yang transform method. They efficiently yield rapidly convergent series-type solutions through an iterative process that requires fewer computations. The Caputo operator, used to express the fractional derivatives in the equations, provides a robust framework for analyzing the behavior of non-integer-order systems. To validate the accuracy and reliability of the obtained solutions, numerical simulations and graphical representations are presented. Furthermore, the results are compared with exact solutions using various tables and graphs, illustrating the effectiveness and ease of implementation of the proposed approaches for various fractional partial differential equations. The influence of the non-integer parameter on the solutions behavior is specifically examined, highlighting its function in regulating wave propagation and diffusion. In addition, a comparison with the natural transform iterative method and optimal auxiliary function method demonstrates that the proposed methods are more accurate than these alternative approaches. The results highlight the potential of YTDM and HPTM as reliable tools for solving nonlinear fractional differential equations and affirm their relevance in wave mechanics, fluid dynamics, and other fields where fractional-order models are applied. Full article

A mathematical model of the anomalous diffusion of surfactant and the process of adsorption–desorption on an interface is analyzed using a fractional calculus approach. The model is based on time-fractional partial differential equations in the bulk phases and the corresponding time-fractional description of the flux bulk–interface. The general case, when the surfactant is soluble in both phases, is considered under the assumption that the adsorption–desorption process is diffusion-controlled. Some of the most popular kinetic models of Henry, Langmuir, and Volmer are considered. Applying the Laplace transform, the partial differential model is transformed into a single multi-term time-fractional nonlinear ordinary differential equation for the surfactant concentration on the interface. Based on existing analytical solutions of linear time-fractional differential equations, the exact solution in the case of the Henry model is derived in terms of multinomial Mittag–Leffler functions, and its asymptotic behavior is studied. Further, the fractional differential model in the general nonlinear case is rewritten as an integral equation, which is a generalization of the well-known Ward–Tordai equation. For computer simulations, based on the obtained integral equation, a predictor–corrector numerical technique is developed. Numerical results are presented and analyzed. Full article

Aimed at dealing with the problems of high reliability solution cost and low solution accuracy under random excitation, especially Gaussian white noise excitation, this paper proposes a probability density evolution and reliability analysis method for nonlinear gear transmission systems under Gaussian white noise excitation based on the path integration method. This method constructs an efficient probability density evolution framework by combining the path integration method, the Chapman–Kolmogorov equation, and the Laplace asymptotic expansion method. Based on Rice’s theory and combined with the adaptive Gauss–Legendre integration method, the transient and cumulative reliability of the system are path integration method calculated. The research results show that in the periodic response state, Gaussian white noise leads to the diffusion of probability density and peak attenuation, and the system reliability presents a two-stage attenuation characteristic. In the chaotic response state, the intrinsic dynamic instability of the system dominates the evolution of the probability density, and the reliability decreases more sharply. Verified by Monte Carlo simulation, the method proposed in this paper significantly outperforms the traditional methods in both computational efficiency and accuracy. The research reveals the coupling effect of Gaussian white noise random excitation and nonlinear dynamics, clarifies the differences in failure mechanisms of gear systems in periodic and chaotic states, and provides a theoretical basis for the dynamic reliability design and life prediction of nonlinear gear transmission systems. Full article

4H-silicon carbide (4H-SiC) is a cornerstone for next-generation optoelectronic and power devices owing to its unparalleled thermal, electrical, and optical properties. However, its chemical inertness and low dopant diffusivity for most dopants have historically impeded effective doping. This study unveils a transformative laser-assisted boron doping technique for n-type 4H-SiC, employing a pulsed Nd:YAG laser (λ = 1064 nm) with a liquid-phase boron precursor. By leveraging a heat-transfer model to optimize laser process parameters, we achieved dopant incorporation while preserving the crystalline integrity of the substrate. A novel optical characterization framework was developed to probe laser-induced alterations in the optical constants—refraction index ($\mathrm{n}$) and attenuation index ($\mathrm{k}$)—across the MIDIR spectrum (λ = 3–5 µm). The optical properties pre- and post-laser doping were measured using Fourier-transform infrared spectrometry, and the corresponding complex refraction indices were extracted by solving a coupled system of nonlinear equations derived from single- and multi-layer absorption models. These models accounted for the angular dependence in the incident beam, enabling a more accurate determination of $\mathrm{n}$ and $\mathrm{k}$ values than conventional normal-incidence methods. Our findings indicate the formation of a boron-acceptor energy level at 0.29 eV above the 4H-SiC valence band, which corresponds to λ = 4.3 µm. This impurity level modulated the optical response of 4H-SiC, revealing a reduction in the refraction index from 2.857 (as-received) to 2.485 (doped) at λ = 4.3 µm. Structural characterization using Raman spectroscopy confirmed the retention of crystalline integrity post-doping, while secondary ion mass spectrometry exhibited a peak boron concentration of 1.29 × 10¹⁹ cm⁻³ and a junction depth of 450 nm. The laser-fabricated p–n junction diode demonstrated a reverse-breakdown voltage of 1668 V. These results validate the efficacy of laser doping in enabling MIDIR tunability through optical modulation and functional device fabrication in 4H-SiC. The absorption models and doping methodology together offer a comprehensive platform for paving the way for transformative advances in optoelectronics and infrared materials engineering. Full article

This paper introduces a functional framework for modeling cyclical and feedback-driven information flow using a generalized family of derangetropy operators. In contrast to scalar entropy measures such as Shannon entropy, these operators act directly on probability densities, providing a topographical representation of information across the support of the distribution. The proposed framework captures periodic and self-referential aspects of information evolution through functional transformations governed by nonlinear differential equations. When applied recursively, these operators induce a spectral diffusion process governed by the heat equation, with convergence toward a Gaussian characteristic function. This convergence result establishes an analytical foundation for describing the long-term dynamics of information under cyclic modulation. The framework thus offers new tools for analyzing the temporal evolution of information in systems characterized by periodic structure, stochastic feedback, and delayed interaction, with potential applications in artificial neural networks, communication theory, and non-equilibrium statistical mechanics. Full article

Viscosimetric experiments and microscopy measurements on microdispersions of polycaprolactone (PCL) plastics showed an unexpected exponential decrease in viscosity over the first 3 months and a plateau for a further 4 months of observations. This behavior is due to the release of nanoplastics from semicrystalline particles that reduce the viscosity of the dispersion, and leave stable and fine crystalline microplastics ranging in size from 30 to 180 μm. The development of nonlinear kinetic models for the fragmentation process from macro- to meso-, micro-, and nanoplastics reveals complex behavior that we call a cracking–leaching mechanism. The autocatalytic mechanical cracking of macroplastics larger than 5 mm is followed by a logistic-type mechanical cracking of mesoplastics between 5 and 1 mm. Therefore, microplastics smaller than 1 mm experience the leaching diffusion modeled via nonlinear coupled kinetic differential equations: semicrystalline microplastics quickly release nanoplastics from the amorphous fraction, followed by fine and stable crystalline microplastics. This proposed mechanism explains the size distribution of floating plastic debris in the oceans, with an unexpected gap of microplastics. Considering the outcome, a general reflection is made on the critical issues that currently appear unsolvable regarding plastic pollution. Full article

This paper develops a novel probabilistic theory of belief formation in social networks, departing from classical opinion dynamics models in both interpretation and structure. Rather than treating agent states as abstract scalar opinions, we model them as belief-adoption probabilities with clear decision-theoretic meaning. Our approach replaces iterative update rules with a fixed-point formulation that reflects rapid local convergence within social neighborhoods, followed by slower global diffusion. We derive a matrix logistic equation describing uncorrelated belief propagation and analyze its solutions in terms of mean learning time (MLT), enabling us to distinguish between fast local consensus and structurally delayed global agreement. In contrast to memory-driven models, where convergence is slow and unbounded, uncorrelated influence produces finite, quantifiable belief shifts. Our results yield closed-form theorems on propaganda efficiency, saturation depth in hierarchical trees, and structural limits of ideological manipulation. By combining probabilistic semantics, nonlinear dynamics, and network topology, this framework provides a rigorous and expressive model for understanding belief diffusion, opinion cascades, and the temporal structure of social conformity under modern influence regimes. Full article

Various numerical techniques have been developed to address multiple problems in computational fluid dynamics (CFD). The finite volume method (FVM) is a numerical technique used for solving partial differential equations that represent conservation laws by dividing the domain into control volumes and ensuring flux balance at their boundaries. Its conservative characteristics and capability to work with both structured and unstructured grids make it suitable for addressing issues related to fluid flow, heat transfer, and diffusion. This article introduces an FVM for the linear advection and nonlinear Burgers’ equations through a fifth-order targeted essentially non-oscillatory (TENO5) scheme. Numerical experiments showcase the precision and effectiveness of TENO5, emphasizing its benefits for computational fluid dynamics (CFD) simulations. Full article

Reaction–diffusion equations can model complex systems where randomness plays a role, capturing the interaction between diffusion processes and random fluctuations. The Kolmogorov equations associated with these systems play an important role in understanding the long-term behavior, stability, and control of such complex systems. In this paper, we investigate the existence of a classical solution for the Kolmogorov equation associated with a stochastic reaction–diffusion equation driven by nonlinear multiplicative trace-class noise. We also establish the existence of an invariant measure $\nu$ for the corresponding transition semigroup $P_t$, where the infinitesimal generator in $L^2(H,\nu )$ is identified as the closure of the Kolmogorov operator $K_0$. Full article