Purpose This paper introduces a novel algorithm, the Differential Evolution-based Backtracking Search Algorithm (DEBSA), designed to address the computational limitations of the Backtracking Search Algorithm (BSA) for complex constraint satisfaction problems. Design/methodology/approach DEBSA merges the core structure of BSA with three innovative mutation strategies derived from Differential Evolution (DE). These strategies focus on directing a random individual toward a historical individual and utilizing a random individual in conjunction with a perturbation vector as well as leveraging a historical best position with two perturbation vectors. Furthermore, DEBSA incorporates a unique crossover mechanism for combining solutions and a strategy selection approach to dynamically choose the most suitable mutation strategy during the search process. Findings DEBSA’s performance is evaluated on constrained optimization problems and systems of nonlinear equations. The results demonstrate exceptional performance, particularly in terms of convergence speed, surpassing traditional benchmark evolutionary algorithms. DEBSA exhibits a high success rate in achieving globally optimal solutions. Originality/value The proposed DEBSA offers a potentially efficient solution for tackling general optimization challenges in engineering design and solving nonlinear equations in applied mathematics due to its enhanced performance and ability to find global optima.

Two novel cold-start multistage neural solvers for constrained nonlinear equations with extended time horizons.

ABSTRACT The study examines the applicability of the Dzhumabaev parametrization method to multipoint boundary value problems for a Duffing‐type integro‐differential operator equation. The original problem is reformulated as an equivalent parametric multipoint problem, which was then decomposed into two subproblems: A nonlinear special Cauchy problem and a system of nonlinear algebraic equations. The special Cauchy problem is addressed via linearization at fixed parameter values and solved through a sequence of stepwise linear approximations. The solutions obtained are subsequently employed to construct the right‐hand side of the algebraic system and its associated Jacobi matrix. On this basis, a new approach for solving the original problem is developed. The efficiency and convergence of the proposed approach were confirmed through a numerical example.

Using Bielecki's idea, we begin by introducing generalized locally convex structures on $n$-Cartesian product of the set of continuous functions defined on the half-axis. Within this frame, we prove new boundedness results for generalized proportional fractional (GPF) integral operators of vector order involving maxima and deviating arguments. As a consequence, one of the well-known boundedness results for scalar Riemann-Liouville fractional integral operators is generalized and improved. As an application, a vector approach for coupled systems of nonlinear (GPF) differential equations with maxima is adopted. Based on our findings related to boundedness and using Perov's type fixed point theorem, we establish global existence-uniqueness results under less restrictive conditions compared to those commonly imposed in the literature.

Purpose The shape factor of nanoparticles is a parameter of interest in the variation of the thermophysical properties of nanofluids, and it affects their fluid flow and temperature distribution. Hence, this study aims to focus on analysing the influence of the shape factor on the convective heat and mass transfer over a nonlinear stretching sheet under the influence of magnetohydronamics. Design/methodology/approach By using similarity transformations, the governing system of partial differential equations was simplified to a nonlinear ordinary differential equation system, which was solved numerically using an explicit finite difference method (Keller box method). The behaviour of the fluid velocity and thermal profile at the boundary, as a result of slip conditions, is studied through a comprehensive parameter exploration. Findings The behaviour of the fluid velocity and thermal profile at the boundary, as a result of slip conditions, is studied by a very extensive parameter exploration. The major findings reveal that increasing values of magnetic parameters promote Lorentz’s force, which slows down the velocity profile. Increasing the stretching rate parameter causes deformation reducing both the velocity and temperature profile. It is found that the temperature rise for elevated values of a thermophoretic parameter is greater for brick-shaped nanoparticles. It is also observed that the heat transfer (Nusselt number) for augmented values of Eckert number is lower in brick-shaped nanoparticles compared to platelet-shaped nanoparticles. Research limitations/implications The main limitation of this work has been the two-dimensional nature of the modelling, the analysis of a certain range of parameters that may not suit all the reader interests and the assumption of specific functions for the stretched sheet velocity and temperature, as well as the magnetic field. Also, regarding the nanofluid characteristics, it has been considered as single-phase, with Fe3O4 particles only and Newtonian. Originality/value This manuscript analyzes mathematically important aspects of the behaviour of a nanofluid with nanoparticles in magnetohydrodynamics of a non-linearly stretched sheet. This work is a detailed parametric exploration (magnetic parameter, stretching parameter, slip parameters, Brownian motion parameter, thermophoretic parameter, Ecker number, Lewis number and solid volume fraction) of the behaviour of two different nanoparticle shapes (brick and platelet), which sheds light on relevant aspects such as, skin friction, heat transfer and mass transfer. These are valuable results for the scientific community for either perform their numerical analysis upon these results and methodology, or perform experimental prototyping according to the behaviour described in this manuscript.

The current manuscript introduces an efficient modification scheme, based on the Adomian decomposition method, for the numerical solution of systems of two-point boundary-value problems. Indeed, this introduced method is characterized by high computational flexibility in accurately solving both the systems of linear and nonlinear differential equations with two-point boundary conditions. Moreover, to assess the robustness of these schemes, several test numerical examples, including a well-known model in fluid dynamics, have been sought and treated. Comparatively, the results from the proposed method have been compared with those of the available exact solutions and those of deployed existing numerical methods in the literature. Lastly, in the presence of the methods used for comparison, the assessment of this method turned out to be positive, with rapid convergence and high precision speedily attained.

Abstract Superlinear convergence occurs frequently as a desirable second stage in a Krylov iteration, and its understanding has undergone a deep development. This paper aims to contribute to this long history of analysis for a class of nonlinear systems of partial differential equations (PDEs) covering reaction–diffusion–convection processes, discretized with finite elements (FE), linearized with a Newton–Krylov method and applying suitable operator preconditioning. We give practical estimates of the rate of superlinear convergence of the arising Krylov iterations in terms of the accessible data of the PDEs. We are in particular interested in the effect of integrability properties of possibly unbounded sources. We obtain robust superlinear rates both in the sense of mesh independence and of uniform behaviour w.r.t. the outer Newton iterations. The numerical examples reinforce the theoretical results.

Comprehending the interaction of transport phenomena across inclined cylinder is essential for enhancing engineering systems such as heat exchangers, pollution dispersion mechanisms, and bioreactors, where fluid flow, heat transfer, and mass transport are interconnected. This study examines flow, heat, and mass transfer with entropy generation in bioconvection nanofluid flow over an inclined cylinder, considering the influences of thermal radiation, mass suction, magnetohydrodynamics (MHD), Joule heating, viscous dissipation, heat absorption, Brownian motion, thermophoresis, discharge concentration, and stratification phenomena. The system of partial differential equations is rehabilitated into system of non-linear ordinary differential equations by sufficient transformations. Keller box technique is therefore an implicit finite difference strategy used numerically to solve similarity equations. This research carefully examines the impact of several dimensionless factors on velocity, temperature, concentration, entropy production, skin friction, Nusselt number, Sherwood number, and microbiological density profiles. A detailed parametric study demonstrates that inclined geometry intensifies axial gravity forces and alters boundary layer dynamics, improving velocity profiles at steeper angles (gamma) and facilitating magnetic forces (M). Thermal profiles demonstrate dual dependencies: viscous dissipation (Ec) and radiation (Rd) increase temperatures, whilst stratification (δ) and heat absorption (H) decrease them. The interactions of nanoparticles underscore a trade-off: Brownian motion (Nb) disperses particles, reducing concentration, while enhancing thermal conductivity and increasing temperature; conversely, thermophoresis (Nt) concentrates particles and heat at the surface. The number of microbes increases by about 327.3% when the Péclet number goes from 0.5 to 2.0. The Sherwood number goes up by around 436.3% and the Nusselt number goes up by about 51.7% as the Brownian motion parameter goes from 0.2 to 0.8. When the thermophoresis parameter goes from 0.1 to 0.4, the Sherwood number goes down by 57.4% and the Nusselt number goes down by 2.1%.

Triply super stable kneading sequences (TSSKS) are very important kernel concept in the study of symbolic dynamics of 1D trimodal maps. For a given period n, there are six types of TSSKS in which two of them decide the six cyclic star products, others supplemented the ‘joints’ in the symbolic space. For the former, start products provide the method to research metric universalities in the period-n-tupling process, the devil’s staircase of topological entropy and self-similar bifurcation structure in classical dynamical systems, the later will be calculated and obtained the corresponding parameters which can occupy so called the admissibe region. In this paper, firstly, for a period n takes 3-12, we produce all the permutations of six types , , , , and , by the famous admissibilty conditions, the admissible sets are obtained respectively, here m stands for the mode of the TSSKS and takes integer 0-5; second, for , detemines a system of nonlinear equations uniquely by passing three critical points , m ensures six different modes of equations, for an proper initial value, the newton-iteration method is applied to get the three parameters of . For m takes 2-5, these parameters of TSSKS in are calculated firstly in the paper, it would describe the parameter space and boundaries and enhence the knowledge of symbolic dynamics of 1D trimodal maps.

ABSTRACT In this paper, the stochastic Davey–Stewartson mathematical model of hydrodynamics, nonlinear optics, and plasma physics is considered. This model is impressive in that it can describe complex multidimensional wave processes under the action of random factors, which is typical for natural physical systems. The primary aim of this study is to acquire and examine exact stochastic solutions of the Davey–Stewartson equation via an analytical method. The problem is initially decomposed into real and imaginary parts, yielding a system of nonlinear partial differential equations (NLPDEs). The system is then reduced to a set of linear equations and associated polynomial versions. The resulting linear system gives some sets of solutions with both the model parameters and the form of the proposed solution. An appropriate set of solutions is determined, and a wave transformation is performed to allow the solutions to be obtained. The Jacobi elliptic function expansion method, a powerful analytical method, is used to get exact solutions of the Davey–Stewartson equation. This method offers a wide range of solution forms, such as singular, periodic, and trigonometric waveforms. In addition, numerical solutions are established for the study of the influence of noise on the reached solutions, and the results are presented in terms of 3D, 2D, and contour plots based on parameters obtained by an analytical procedure. The results provide new exact solutions in a stochastic environment, highlighting the importance of the process used. These findings represent novel results never previously presented in the literature.

This study presents a numerical investigation of entropy generation in a magnetohydrodynamic (MHD) flow of a Maxwell dusty nanofluid over an inclined stretching sheet, with a focused analysis on the previously overlooked parameters of nanoparticle radius and inter-particle spacing. The model incorporates the effects of viscous dissipation and thermal buoyancy on the flow dynamics. The governing partial differential equations are transformed into a system of nonlinear ordinary differential equations via similarity transformations and solved computationally using MATLAB’s bvp4c solver, with validation against published results confirming high accuracy. The findings quantitatively show that nanoscale particle geometry is a key factor influencing thermal performance and irreversibility. A reduction in the nanoparticle radius from 3.6 nm to 1.6 nm under standard conditions (h_p=0.5, beta =0.5, M=3.0) suppresses total entropy generation by approximately 20%. Conversely, increasing the nanoparticle radius beyond 2.5 nm enhances both the fluid and dust phase velocities by nearly 18%, which is beneficial for flow applications, but concurrently reduces the effective thermal conductivity by almost 12% due to a diminished surface-area-to-volume ratio. Furthermore, the analysis shows that increasing inter-particle spacing decreases entropy generation by reducing particle clustering. This study bridges a crucial research gap in the literature by quantifying the role of nanoparticle microstructure. It provides an operational framework for developing high-efficiency, low-irreversible thermal control systems in industries such as advanced manufacturing and energy production.

This article is devoted to the presentation of two numerical methods which give the solution of a two-dimensional nonlinear Volterra integral equation of the second kind. The first method, Bernoulli collocation, depend on approximating the unknown function using Bernoulli polynomials, while applying the collocation technique at shifted Chebyshev points over the interval[0,1]. The second method, Hermite-Galerkin method, relies on constructing an operational matrices and applying the Galerkin projection, which we have a system of nonlinear algebraic equations from Volterra integral equation. Discussion on the existence and uniqueness of the solution is provided. Finally, the effect of that two numerical methods is described. To illustrate the previously described methods, several numerical examples are provided. Numerical results show that the Bernoulli collocation method consistently provides more accurate and efficient results than the Hermite–Galerkin method for the same number of collocation points. Comparisons with previously published approaches further demonstrate the superiority of the proposed methods in terms of convergence and stability.

Research examining heavy metal concentrations in cacao products has raised concerns about potential toxicity for frequent chocolate consumers. In this study, conducted as part of SCUDEM IX Challenge, a nonlinear system of differential equations was developed to predict lead dynamics and bioaccumulation in the body. The model considers time-dependent lead concentrations in the GI tract M, bloodstream A, soft tissue S, and hard tissue H. Lead diffusion between systems follows logistic-based osmosis transport equations. Diffusion parameters were loosely selected from literature and iteratively refined to preserve model stability and ensure outputs align with surveyed values. Two primary consumption patterns, or inhomogeneities, were selected for test cases; daily “stress” and weekend “binge” consumption patterns were defined by a sinusoid and periodic step function, respectively. The model surveyed a year period, additionally considering increased consumption for American holidays and birthdays. Runge-Kutta fourth-order approximations were implemented to plot the solution curves developed via numerical methods. According to model predictions, maintaining daily chocolate consumption below 50g is unlikely to induce lead toxicity, as compared with the CDC Blood Lead Reference Value (BLRV) of 3.5 μg/dL.

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.

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.

This paper investigates a robust optimal reinsurance and investment problem for an insurance company operating in a Markov-modulated financial market. The insurer’s surplus process is modeled by a diffusion process with jumps, which is correlated with financial risky assets through a common shock structure. The economic regime switches according to a continuous-time Markov chain. To address model uncertainty concerning both diffusion and jump components, we formulate the problem within a robust optimal control framework. By applying the Girsanov theorem for semimartingales, we derive the dynamics of the wealth process under an equivalent martingale measure. We then establish the associated Hamilton–Jacobi–Bellman (HJB) equation, which constitutes a coupled system of nonlinear second-order integro-differential equations. An explicit form of the relative entropy penalty function is provided to quantify the cost of deviating from the reference model. The theoretical results furnish a foundation for numerical solutions using actor–critic reinforcement learning algorithms.

Solving nonlinear systems of equations is a central challenge in scientific computing, impacting a wide range of fields such as engineering, physics, and applied mathematics.Although the Newton-Raphson method is popular for its quadratic convergence near solutions, it faces notable difficulties, including reliance on the initial guess, potential failure with ill-conditioned Jacobians, and complications when multiple or closely situated roots are present. In this study, we investigate the creation of new iterative algorithms aimed at overcoming these obstacles by promoting better global convergence and improving numerical stability. The proposed approaches utilize adaptive step-size management, quasi-Newton techniques, and hybrid strategies that integrate trust-region and homotopy concepts. Results from numerical tests on standard benchmark problems show that these algorithms provide enhanced robustness for a wide array of nonlinear systems. Compared to Newton-Raphson, the new methods expand the convergence domain and frequently deliver equal or better accuracy and computational speed. This work paves the way for developing more trustworthy solvers for complex nonlinear systems in contemporary computational practice.

Analysis of binding kinetics and mass transport in SPR-based biosensor using the Generalized Integral Transform Technique and the Markov Chain Monte Carlo Method.

Generalized model of closed microecosystem «alga - micro-consumers».

Global bifurcation in symmetric systems of nonlinear wave equations