In this article, we proposed a comprehensive pest control model that integrated both delay differential equations and stochastic processes to mitigate the spread of whiteflies in coconut plantations. The delay model introduced a time lag in the implementation of awareness programs and investigated its impact on the system’s equilibrium stability. Numerical simulations validated the model's effectiveness in enhancing pest management. A stochastic component, formulated using a Wiener process within a system of non-linear ordinary differential equations (ODEs), was used to estimate the probability of disease elimination under environmental fluctuations. The study was novel in combining both deterministic delays and stochastic effects in a unified framework, offering deeper insights into timing and control efficiency. Although focused on coconut farming, the modeling approach and techniques had broader applicability in agricultural pest control scenarios. The findings enhanced our understanding of pest dynamics under uncertainty and delay, providing a foundation for more informed and effective control interventions in agricultural ecosystems.

This paper presents a novel application of the improved modified extended tanh-function method to derive exact traveling wave solutions for the space-time fractional modified third- order Korteweg-de Vries (KdV) equation. The fractional derivatives are considered in the con- formable fractional derivative form. The proposed method systematically reduces the nonlinear fractional partial differential equation to an ordinary differential equation, which is then solved using a generalized ansatz. A specific kink-type solitary wave solution is obtained in closed form. The results demonstrate the robustness and efficiency of the method for handling com- plex nonlinear fractional differential equations, providing valuable new solutions that enhance our understanding of wave propagation in fractional media.

Abstract Enhancing the energy efficiency in controlling an overhead crane during repeated rest-to-rest movements can result in significant energy savings. Moreover, it influences both maintenance expenditures and safety. This study focuses on optimizing a smooth wave command shaping profile to reduce energy usage and enhance the motion efficiency of an overhead crane. A smooth single-mode command shaper is modified to reduce the energy required during the trolley maneuver. The nonlinear equation of motion for a simple crane is derived, linearized, and then solved to determine the optimal controller performance. An extra constant is added to a smooth waveform command shaper and then optimized to enhance the required energy. Furthermore, the selectable maneuvering time feature of the smooth command shaper is utilized to further enhance the maneuver’s efficiency. The results obtained are compared with several well-known input/command shapers. The optimized command shaper profile can eliminate all residual vibrations induced and reduce energy consumption by 30% compared to the most effective unoptimized input shaper and 45% compared to the classical smooth command shaper. The performance of all shapers is numerically and experimentally validated on an experimental overhead crane.

In this paper, we present a new iterative method for solving nonlinear equations. The method is a combination of the method of Chun and the method of Hu et.al. The new method requires six function evaluations and has the order of convergence sixteenth. Numerical experiments are made to demonstrate the convergence and validation of the iterative method.

This paper considers the stochastic Korteweg-de--de Vries (SKdV) equation perturbed by multiplicative Brownian motion, which is an important model reflecting the nonlinear science. After a systematic change and a rescaling, the SKdV equation is exactly recast into a deterministic KdV equation with random variable coefficients (KdV-RVCs). By using the Jacobi elliptic equation method and the generalized Riccati equation mapping approach, we obtain new exact solutions (rational, hyperbolic, trigonometric, and elliptic) for the KdV-RVCs. The latter are then used to form stochastic solutions for the SKdV equation. Of practical interest, these results are related to specific physical systems: magnetized plasmas in astrophysics and in 1D/2D fusion, soliton propagation in fiber optical communication, and surface-wave dynamics in fluid mechanics. For example, the resulting solutions explain how noise-induced perturbations change soliton propagation in optical fibers and stabilize wave patterns in Turbulence. To give some (visual) impression of how multiplicative noise influences the solution behavior, we use pictures of the probability density distributions and ensemble-averaged trajectories as examples. These findings show that multiplicative Brownian motion has a stabilizing effect on the SKdV solutions by keeping their variations more or less near zero. This connection between stochastic modeling and experimental observations in plasma turbulence, nonlinear optical signal processing, and fluid wave dynamics has implications for the development of predictive theories for noise-driven nonlinear systems.

Abstract This paper presents a robust computational technique to tackle the intricate nonlinear partial differential equations (PDEs) encountered in mathematical physics. The method is applied to the time-fractional Burgers-Huxley equation, where the time derivative is considered in the Liouville-Caputo sense. This equation, which combines the well-known Burgers and Huxley equations, describes the interplay of reaction, convection, and diffusion in transport phenomena and finds application in acoustics, turbulence theory, traffic flow, and hydrodynamics. The proposed method transforms this complex non-linear fractional PDE into a simple algebraic system. Its ability to handle the non-linear terms without perturbation, discretization, or the calculation of extraneous terms is a major advantage over available analytical approaches. Five different cases of the equation with diverse initial and boundary conditions are discussed. To demonstrate the accuracy and reliability of the semi-analytic approach, the obtained outcomes are compared with existing exact and analytical solutions in the literature, showing a strong level of agreement. Error analysis and the convergence criterion are also discussed.

Stability analysis of multi-spatial Riesz and multi-fractional non-linear damped wave equations involving Caputo-Fabrizio derivative

Abstract. The flow of fluids within porous rocks is an important process with numerous applications in Earth sciences. Modeling the compaction-driven fluid flow requires the solution of coupled nonlinear partial differential equations that account for the fluid flow and the solid deformation within the porous medium. Despite the nonlinear relation of porosity and permeability that is commonly encountered, natural data show evidence of channelized fluid flow in rocks that have an overall layered structure. Layers of different rock types have discontinuous hydraulic and mechanical properties. We present numerical results obtained by a novel space-time method, which can handle discontinuous initial porosity (and permeability) distributions efficiently. The space-time method enables straightforward coupling to models of mass transport for trace elements. Our results indicate that, under certain conditions, the discontinuity of the initial porosity influences the distribution of incompatible trace elements, leading to sharp concentration gradients and large degrees of elemental enrichment. Finally, our results indicate that the enrichment of trace elements depends not only on the channelization of the flow but also on the interaction of fluid-filled channels with layers of different porosity and permeability.

This paper investigates a new class of two-dimensional fuzzy difference equations that integrate logarithmic nonlinearities with interaction effects between system variables. Motivated by the need to model complex dynamical systems influenced by uncertainty and interdependencies, we propose a system that extends existing one-dimensional models to capture more realistic interactions within a discrete-time framework. Our approach employs the characterization theory to transform the fuzzy system into an equivalent family of classical difference equations, thereby facilitating a rigorous analysis of the existence, uniqueness, and boundedness of positive solutions. To support the theoretical findings, two numerical examples are provided, illustrating the model’s capacity to capture complex dynamical patterns under fuzzy conditions. An application to a fuzzy population growth model illustrates how the model captures both interaction effects and uncertainty while ensuring well-defined and stable solutions. Numerical simulations show that, for instance, with α=0.10, β=δ=1.0, γ=0.08, and ρx=ρy=0.10, the trajectories of (xt,yt) rapidly converge toward a stable fuzzy equilibrium, with uncertainty bands confirming the positivity and boundedness of the solutions.

This study conducts a systematic analysis of the secondary buckling behavior of composite laminated plates under thermal loads based on Reddy’s Higher-Order Shear Deformation Theory (HSDT) and the Isogeometric Analysis (IGA) method. By introducing the von Kármán large-deformation theory and initial geometric imperfections, a mechanical model considering the higher-order shear effects and geometric nonlinearity is established. The Non-Uniform Rational B-Spline (NURBS) basis functions are employed for spatial discretization, effectively achieving the unification of geometrically exact description and high-order continuity. In the model, the in-plane, bending, and initial imperfection stiffness matrices are derived in detail, and the nonlinear equilibrium equations are solved using the Newton-Raphson iterative method. This research proposes a critical load prediction method based on the minimum eigenvalue of the tangent stiffness matrix. By combining the linear interpolation technique, the secondary buckling loads and corresponding modes are accurately captured. Numerical examples verify the effectiveness and accuracy of this method in analyzing the thermal buckling paths, initial imperfection sensitivity, and higher-order mode evolution of composite laminated plates. The results show that initial imperfections significantly reduce the critical buckling load of the structure. Moreover, the combination of IGA and HSDT can accurately characterize the complex deformation behavior of thick plates, providing a theoretical basis for the thermal stability design of composite structures in engineering.

Finite element analysis of a multi-term nonlinear time-fractional convection-diffusion equation with Caputo-Fabrizio derivative

This study aims to explore the Peyrard-Bishop Deoxyribonucleic Acid (PB-DNA) dynamic model utilizing He’s Variational Iteration (HVI) method, supported by the Khater III (Khat III) and improved Kudryashov (IKud) mathematical techniques. The PB-DNA model is essential for illustrating the nonlinear dynamics of DNA, a process vital for gene expression and DNA replication. This model shares similarities with other nonlinear evolution equations, such as the sine-Gordon and Klein-Gordon equations, which depict wave propagation in nonlinear media. Our approach involves employing the HVI method to obtain numerical solutions and leveraging the Khat III and IKud schemes to formulate exact solutions. This integrated strategy provides a thorough methodology for addressing the PB-DNA model, offering dual numerical and analytical insights.

Nowadays a vast literature is available on the Hele-Shaw or incompressible limit for nonlinear degenerate diffusion equations. This problem has attracted a lot of attention due to its applications to tissue growth and crowd motion modeling as it constitutes a way to link soft congestion (or compressible) models to hard congestion (or incompressible) descriptions. In this paper, we address the question of estimating the rate of this asymptotics in the presence of external drifts. In particular, we provide improved results in the 2-Wasserstein distance which are global in time thanks to the contractivity property that holds for strictly convex potentials.

Fractional differential equations offer a natural framework for describing systems in which present states are influenced by the past. This work presents a nonlinear Caputo-type fractional differential equation (FDE) with a nonlocal initial condition and attempts to describe a model of memory-dependent behavioral adaptation. The proposed framework uses a fractional-order derivative η∈(0,1) to discuss the long-term memory effects. The existence and uniqueness of solutions are demonstrated by Banach’s and Krasnoselskii’s fixed-point theorems. Stability is analyzed through Ulam–Hyers and Ulam–Hyers–Rassias benchmarks, supported by sensitivity results on the kernel structure and fractional order. The model is further employed for behavioral despair and learned helplessness, capturing the role of delayed stimulus feedback in shaping cognitive adaptation. Numerical simulations based on the convolution-based fractional linear multistep (FVI–CQ) and Adams–Bashforth–Moulton (ABM) schemes confirm convergence and accuracy. The proposed setup provides a compact computational and mathematical paradigm for analyzing systems characterized by nonlocal feedback and persistent memory.

This paper studies the existence of solution of hybride fractional differential equation with hybride boundary conditios were ivestigation by the Dhage’s fixed theorem. Additionally, the U.H technique is employed to verify the stabiliy of this solution. The stady is concluded whit two examples that illustrate the pratical application of the theorecal results.

Background: Estimation of forest biomass has become critical as afforestation has been proposed to sequester carbon from the atmosphere in order to mitigate climate change. New Zealand Dryland Forestry Innovation (NZDFI), in collaboration with the University of Canterbury’s School of Forestry and the Marlborough Research Centre, has initiated a research and development programme to gather seed, breed, propagate, identify site limitations, model growth, investigate silviculture, and develop wood products from a suite of eucalypts that grow durable heartwood. The aim is to supply naturally durable wood for uses that formerly required either imports of durable wood or copper-chrome-arsenate treated pine. Methods: As part of a project examining land-use and greenhouse gas budget case studies in Marlborough, New Zealand, we collected and summarised data describing above-ground biomass (AGB) of Eucalyptus bosistoana F.Meull., and Eucalyptus globoidea Blakely trees across a wide range of combinations of height (h) and diameter at breast height (dbh). One hundred and eleven trees were felled, separated into stems, branches and foliage, and the components were weighed in the field. Subsamples of these tree parts were collected and weighed in the field after separating bark from stem discs. The subsamples were dried in an oven at 105°C, and then weighed. Ratios of dry to wet weights for samples were applied to total green weights from the field in order to calculate AGBs of tree components. Systems of non-linear equations were simultaneously fitted to the data to ensure additivity; that sums of estimates of tree part AGBs versus dhb, h and slenderness (h/dbh) equalled estimates from a model of total tree AGB versus the same independent variables. The study also included the development of a plot-level estimation model of above-ground CO2-e/ha for E. globoidea and its incorporation in an on-line growth and yield simulator. Moreover, a comparison of two pathways to estimating AGB by aerial LiDAR was made: One including estimates of dbh and h from LiDAR and applying the tree-level equations developed in this study, and one going directly from LiDAR metrics to estimates of AGB. Results: A system of models created for both species with a dummy variable denoting species yielded the least biased residuals, with 22 coefficients estimated in one simultaneous fit. Standard errors varied with plant part and with the size of the prediction, requiring transformations prior to fitting. R2 values also varied with part, but were typically between 0.96 and 0.98. An exception was foliage and seeds which were influenced by one tree with an unusually high loading of seeds. The standard error for plot level estimates of CO2-e was 1.9 tonnes CO2-e /ha and residuals were relatively unbiased. Directly predicting individual tree AGB from LiDAR metrics yielded less biased estimates than predicting dbh and h and then using those estimates to predict AGB. Conclusions: A system of related, additive equations with a dummy variable denoting species represented the above-ground biomass of Eucalyptus globoidea and Eucalyptus bosistoana with precision adequate for prediction of biomass for fuel and carbon storage to mitigate climate change. Direct predictions of biomass from LiDAR metrics were less biased than predictions of biomass from tree height and diameter at breast height that were in turn predicted from LiDAR metrics.

On compact manifolds (M,g) , we derive the existence of metrics in a given conformal class [g] with prescribed negative partial curvature. This curvature corresponds to a fully nonlinear equation derived from conformal geometry. For manifolds with boundary, we demonstrate the solvability of equations involving prescribed negative partial curvature within M , coupled with mean curvature along \partial M .

Many problems in science, engineering, and economics require solving of nonlinear equations, often arising from attempts to model natural systems and predict their behavior. In this context, iterative methods provide an effective approach to approximate the roots of nonlinear functions. This work introduces five new parametric families of multipoint iterative methods specifically designed for solving nonlinear equations. Each family is built upon a two-step scheme: the first step applies the classical Newton method, while the second incorporates a convex mean, a weight function, and a frozen derivative (i.e., the same derivative from the previous step). The careful design of the weight function was essential to ensure fourth-order convergence while allowing arbitrary parameter values. The proposed methods are theoretically analyzed and dynamically characterized using tools such as stability surfaces, parameter planes, and dynamical planes on the Riemann sphere. These analyses reveal regions of stability and divergence, helping identify suitable parameter values that guarantee convergence to the root. Moreover, a general result proves that all the proposed optimal parametric families of iterative methods are topologically equivalent, under conjugation. Numerical experiments confirm the robustness and efficiency of the methods, often surpassing classical approaches in terms of convergence speed and accuracy. Overall, the results demonstrate that convex-mean-based parametric methods offer a flexible and stable framework for the reliable numerical solution of nonlinear equations.

ABSTRACT The Jeffery–Hamel flow, a classic benchmark in fluid dynamics, describes the motion of an incompressible viscous fluid within convergent or divergent channels. Although extensively studied for Newtonian fluids, the dynamics of such flows in channels with stretching or shrinking walls, especially for couple‐stress fluids, remain largely unexplored. In this study, we pioneer the use of artificial neural networks (ANNs) to solve a fifth‐order nonlinear differential equation arising from the two‐dimensional Jeffery–Hamel flow of couple‐stress fluids within stretching/shrinking channels, addressing a complex, nonlinear fluid dynamics problem. By capturing microstructural effects and the unique rheology of couple‐stress fluids, our approach enables high‐accuracy solutions for complex flow behaviour influenced by wall deformation. We focus exclusively on fluid flow behaviour, analysing the influence of key parameters such as Reynolds number, magnetic parameter, channel angle, stretching parameter, and couple stress parameter on velocity distribution and flow structure. Our results reveal new flow topologies and response patterns that are unattainable with traditional analytical or numerical methods. The proposed ANN‐based methodology bridges significant gaps in the literature and provides a powerful tool for modelling biological, industrial, and microfluidic flows in adaptive geometries. This work advances the understanding of the dynamics of Jeffery–Hamel flow in couple‐stress fluids within magnetically influenced stretching/shrinking channels, demonstrating unprecedented microstructural interactions absent in prior Newtonian or non‐Newtonian studies, and unveiling the effectiveness of intelligent methods for solving problems in computational fluid mechanics.

ABSTRACT The third fractional 3D Wazwaz–Benjamin–Bona–Mahony (WBBM) equation is examined in this paper, along with new waveforms and various analyses. This is important for understanding how waves move in plasma physics, shallow water, and nonlinear optics. We use a Galilean transformation to obtain the research output of this model. The planner dynamic system of the equation is also constructed, and all possible phase portrait analyses are described, including bifurcation and chaos. We observed chaotic, periodic, and quasi‐periodic behaviors by introducing a perturbed term for various parameter values. This study talks about multistability analysis, sensitivity analysis, and exact traveling wave solutions of the governing model. Fractal dimension, strange attractor, recurrence plot, power spectrum, return map, and Lyapunov exponent (LE) are some of the graphs that show how the model works. Additionally, this research work employs the unified solver technique to yield diverse solitary‐wave outcomes. We visually display the derived outcomes in 2D and 3D plots. We can conclude that these findings provide a solid foundation for further investigation and are valuable, useful, and reliable for dealing with future complex nonlinear problems. The approach employed in this work demonstrates a high level of reliability, robustness, and efficiency, making it suitable for addressing a vast area of nonlinear partial differential equations (NLPDEs) that have not been studied in any other research.