Nonlinear Equations Research Articles

In this article, we examine the stability of self-rewetting films (SRF) flowing along a soft vertical cylinder where the flow is driven by the combined action of gravity and thermocapillarity. A long-wave model is formulated to capture the evolution of the liquid layer thickness and the substrate deformation where the film flow interacts with the soft structure through gravity, thermocapillary forces, and the elasticity of the soft fibre with the Winkler-based framework. Using the model, we explore the impact of quadratic thermocapillarity of SRF films including elasticity of the soft fibre and film thickness on the temporal stability. The conventional thermocapillarity (when Ti < Tm) along with elasticity of soft fibre augments the instability whereas the anomalous thermocapillarity (when Ti > Tm) suppresses the instability with Ti and Tm denoting the interface temperature and minimum surface tension temperature respectively. The time-dependent computations of the coupled nonlinear partial differential equation (PDE) of the interface reveal soft-layer deformation which may also lead to localized bulging in the interface due to conventional thermocapillarity and elasticity of the soft fibre. The results of numerical simulations are consistent with our linear theory.

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 A thermomechanical model has been developed and implemented to couple temperature, thermal expansion, contact, and wear using the finite element method. This three-dimensional, transient, and nonlinear model is unconditionally stable because of the use of an implicit solver. The weak form of the nonlinear heat transfer and elasticity equations has been derived, incorporating penalty contact, conduction, convection, radiation, and deformation. Several boundary conditions, including Dirichlet, Neumann, and Robin conditions, have been applied. We focus on mechanical train brakes due to their strong thermomechanical coupling effect. The simulation results have been validated against full-scale experimental data. Additionally, mesh sensitivity and time step sensitivity analyses are conducted to further assess the model’s accuracy. Friction heat is calculated at each time step through contact conditions, enabling the identification of hot spots and thermal cracks. The results demonstrate that this thermomechanical model is both efficient and robust. This model has been open-sourced, providing a powerful tool for advanced research in thermomechanical multiphysics analysis.

This study presents a fundamental theoretical investigation of gradient elution chromatography employing core-shell particles and variable mobile phase composition. An extended form of the general rate model (GRM) is developed to examine the influence of column overloading on elution performance. The linear solvent strength (LSS) model is incorporated to describe variations in Henry's constant and the nonlinearity coefficient with solvent composition, while accounting for intraparticle diffusion, film mass transfer resistance, and axial dispersion. Core-shell particles enhance separation efficiency by reducing the accessible pore volume and diffusion path lengths, thereby allowing higher flow rates. To approximate the resulting nonlinear model equations, a semidiscrete high-resolution finite volume scheme is adapted and applied. The numerical framework enables a detailed analysis of the effects of key model parameters on the behavior and shape of the elution profiles, providing valuable insights into chromatographic dynamics. Validation of the proposed model and evaluation of the numerical scheme are conducted through benchmark test problems. Specific performance metrics are employed to identify the most influential parameters. The study utilizes binary mixtures as a model system to establish a fundamental understanding of elution behavior, refine numerical strategies, and provide insights that support the optimization of experimental conditions. The findings offer a foundational framework for enhancing separation performance with broader implications for more complex systems.

We present a new solution to the nonlinear shallow water equations (NSWEs) and show that it accurately predicts the swash flow due to obliquely approaching bores in large-scale wave basin experiments. The solution is based on an application of Snell’s law of refraction in settings where the bore approach angle $\theta$ is small. We therefore use the weakly two-dimensional NSWEs (Ryrie 1983 J. Fluid Mech. 129 , 193), where the cross-shore dynamics are independent of, and act as a forcing to, the alongshore dynamics. Using a known solution to the cross-shore dynamics (Antuono 2010 J. Fluid Mech. 658 , 166), we solve for the alongshore flow using the method of characteristics and show that it differs from previous solutions. Since the cross-shore solution assumes a constant forward-moving characteristic variable, $\alpha$ , we call our solution the ‘small- $\theta$ , constant- $\alpha$ ’ solution. We test our solution in large-scale experiments with data from 16 wave cases, including both normally and obliquely incident waves generated using the wall reflection method. We measure water depths and fluid velocities using in situ sensors within the surf and swash zones, and track shoreline motion using quantitative imaging. The data show that the basic assumptions of the theory (Snell’s law of refraction and constant- $\alpha$ ) are satisfied and that our solution accurately predicts the swash flow. In particular, the data agrees well with our expression for the time-averaged alongshore velocity, which is expected to improve predictions of alongshore transport at coastlines.

Estimating thermoelectric (TE) performance is vital for guiding thermoelectric research. A thermoelectric generator (TEG) is made of p-type and n-type semiconductor materials and the material properties are typically temperature dependent. Estimating the performance of a TEG involves solving the thermoelectric heat balance equation, and since the TE material properties are temperature dependent, this equation is a second order non-linear partial differential equation. Such equations can only be solved numerically using methods such as finite element methods. As these numerical methods are time consuming and costly, some approximate analytical methods exist, the constant properties model (CPM) being the backbone of all of them. Different papers suggest different modifications and comments on the CPM and it is not clear which assumptions and modifications work for all scenarios. In addition, the physics behind these models is not always completely analyzed. This work presents a critical view on the physics of thermoelectric generation and provides perspective on the commonly claimed errors and misconceptions with CPM, reviewing the popular approaches and suggestions for modifications on the use of CPM. Initially a brief overview of the relevant concepts for theoretical performance estimation of TEGs is presented in order to understand the physics of TEG. Finally, recommendations for the use of these models is provided.

Abstract This perspective deals with real scalar fields in two-dimensional spacetime. We focus on models described by one and two real scalar fields, paying closer attention to kinks and lumps, which are localized structures of current interest in high energy physics and in other areas of nonlinear science. We briefly review some of the main results presented in the literature and then focus on some new issues concerning the compact and long-range behavior of solutions and the presence of geometric constraints, suggesting how they can be used in applications in several areas of nonlinear science.

Optical fiber radiation sensing probes made using inorganic scintillator materials have notable advantages in achieving high spatial resolution and building sensing arrays due to their small size and excellent linearity, serving as a key tool for dose measurement in precision radiotherapy. This study establishes a theoretical model for scintillator luminescence coupling into optical fibers, and derives a fluorescence intensity calculation formula based on the fiber’s numerical aperture and fluorescence self-absorption. The light intensity response to scintillator length for different absorption coefficients is established based on numerical simulation, providing a nonlinear fitting equation, resulting in a novel “effective length of scintillator” concept. Five probes with scintillator lengths of 0.2 mm, 0.5 mm, 1.0 mm, 1.5 mm, and 2.0 mm were prepared in the laboratory using a 3:1 mass ratio mixture of UV-setting epoxy and Gd2O2S:Tb powder. Tests in a clinical radiation delivery setting showed good agreement between experimental data and theory, confirming optimum effective length of the scintillator as 0.62 mm. This study indicates that inorganic scintillators for end-constructed probes do need not need to be excessively long. Analyzing the effective length can reduce scintillator usage, simplify fabrication and processing, and enhance the probe’s spatial resolution without decreasing the signal-to-noise ratio, thus offering new insights for optimizing optical fiber radiation probes.

ABSTRACT This study presents a coupled numerical–statistical investigation of steady, laminar, incompressible magnetohydrodynamic (MHD) free convective heat and mass transfer in an electrically conducting micropolar fluid over a semi‐infinite vertical plate, incorporating double stratification, wall suction/injection, and combined thermal–solutal buoyancy effects. Micropolar behavior is modeled using Eringen's theory, which accounts for both translational and microrotational dynamics. The governing equations are reduced through Lie group similarity transformations into coupled nonlinear ordinary differential equations. These are solved using the Keller–box scheme to accurately resolve near‐wall gradients and asymptotic far‐field behavior. Parametric analysis reveals that increasing the micropolar coupling parameter K enhances the peak velocity (10%) and microrotation (100%) while marginally reducing the thermal and solutal fields via stronger convective removal. The magnetic parameter M suppresses velocity (up to 18%) and microrotation (40%), thickens boundary layers, and lowers Nusselt and Sherwood numbers. Thermal stratification ε 1 and solutal stratification ε 2 diminish buoyancy, lowering velocity by over 40% and 15%–20%, respectively. Suction ( f 0 &gt; 0) improves transport, increasing Nusselt and Sherwood numbers by 15%–30%, while injection ( f 0 &lt; 0) produces the opposite effect. Response surface methodology (RSM) is applied for multi‐objective optimization of Nu x , Sh x , C f *, and g peak . Quadratic models ( R 2 &gt; 0.99, Adeq Precision &gt; 79) capture significant linear, interaction, and quadratic effects of K , M , and f 0 . The optimal solution, with overall desirability 0.751, occurs at K ≈ 1.60, M ≈ 0.96, and f 0 ≈ −0.27, yielding Nu x = 1.3047, Sh x = 1.1433, C f * = 0.5001, and g peak = 0.7087, all within the 95% prediction intervals. The integrated findings demonstrate that strategic tuning of micropolar coupling, magnetic field strength, and wall mass flux can enhance thermal and mass transport while controlling frictional and microrotational effects, offering valuable design guidance for MHD micropolar systems in energy, materials processing, and thermal management applications.

The article presents a study of the computational complexity and efficiency of various parallel algorithms that implement the numerical solution of the equation in the hereditary α(t)-model of radon volumetric activity (RVA) in a storage chamber. As a test example, a problem based on such a model is solved, which is a Cauchy problem for a nonlinear fractional differential equation with a Gerasimov–Caputo derivative of a variable order and variable coefficients. Such equations arise in problems of modeling anomalous RVA variations. Anomalous RVA can be considered one of the short-term precursors to earthquakes as an indicator of geological processes. However, the mechanisms of such anomalies are still poorly understood, and direct observations are impossible. This determines the importance of such mathematical modeling tasks and, therefore, of effective algorithms for their solution. This subsequently allows us to move on to inverse problems based on RVA data, where it is important to choose the most suitable algorithm for solving the direct problem in terms of computational resource costs. An analysis and an evaluation of various algorithms are based on data on the average time taken to solve a test problem in a series of computational experiments. To analyze effectiveness, the acceleration, efficiency, and cost of algorithms are determined, and the efficiency of CPU thread loading is evaluated. The results show that parallel algorithms demonstrate a significant increase in calculation speed compared to sequential analogs; hybrid parallel CPU–GPU algorithms provide a significant performance advantage when solving computationally complex problems, and it is possible to determine the optimal number of CPU threads for calculations. For sequential and parallel algorithms implementing numerical solutions, asymptotic complexity estimates are given, showing that, for most of the proposed algorithm implementations, the complexity tends to be n2 in terms of both computation time and memory consumption.

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.

The comparative analysis of the results of low-temperature mechanical tests of the samples of amorphous and amorphous-crystalline polyimide of the Kapton H type was carried out. In the experiments [East Eur. J. Phys., No. 4, 144 (2020)], tensile diagrams σ(ɛ;T,ε˙) of such samples were recorded, namely, the dependences of the deforming stress on the strain ɛ = ε˙t at constant values of the strain rate ε˙ = 7⋅10–5, 7⋅10–4, 6⋅10–3 s–1, and temperature T = 293, 77, and 4.2 K. The initial aim of these experiments was to study the effect of moderate (77 K) and deep (4.2 K) cooling on the structure and some mechanical characteristics of polyimide, important for its use in cryogenic and aerospace engineering. Later (Low Temp. Phys. 49, 521 (2023) [Fiz. Nyzk. Temp. 49, 569 (2023)]), there was a need and opportunity to supplement the experimental results with additional analysis in order to use them to test new aspects of the theory of low-temperature elastic-viscous deformation of polymers, in particular, the description of the effects of forced elasticity and their competition with brittle fracture processes. A detailed comparison of the tensile diagrams of the polyimide samples with amorphous and amorphous-crystalline mole­cular structures performed in this study showed that at T = 293 K both structures have clearly pronounced pro­perties of the elastomers, namely, the rubber-like materials with high elasticity and the ability to reversible deformation. It has been established that amorphous samples retain these properties up to deep cooling at T = 4.2 K, and amorphous-crystalline ones only to a state of moderate cooling: at T &amp;lt; 77 K they acquire the properties of glassy materials with brittle fracture at the initial stage of elastic deformation. It is also shown that the kinetics of highly elastic deformation of polyimide with molecular structures of both types is due to the thermomechanical activation of soliton-like elaston excitations on molecular chains in the amorphous component of the material and is described by a nonlinear rheological equation derived earlier for the molecular model of an amorphous polymer: Low Temp. Phys. 48, 253 (2022) [Fiz. Nyzk. Temp. 48, 281 (2022)], Low Temp. Phys. 49, 228 (2023) [Fiz. Nyzk. Temp. 49, 246 (2023)]. By comparing the results of experiments and theory, an analytical description of the tension diagrams σ(ɛ;T,ε˙) of polyimide samples with molecular structures of both types was obtained, as well as empirical estimates of their rheological characteristics and microscopic parameters of elaston excitations. During low-temperature deformation of a polymer with a mixed structure, rigid crystalline fibrils immersed in the softer amorphous medium undergo only minor elastic deformations, but significantly increase the intensity of elaston activation and fracture processes in the amorphous component. Upon cooling, this leads to the convergence of critical stresses of highly elastic relaxation and fracture and to the transformation of an elastomer with such a structure into a glassy brittle material.

Corrigendum to “Sharkovskii theorem for infinite dimensional dynamical systems” [Communications in Nonlinear Science and Numerical Simulation 146 (2025) 108770

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

Li-Yau estimates and Harnack inequalities for nonlinear slow diffusion equations on a smooth metric measure space

Existence of very weak solutions to nonlinear elliptic equation with nonstandard growth and global weighted gradient estimates

Piecewise logarithmic Chebyshev cardinal functions: Application for nonlinear integral equations with a logarithmic singular kernel

Bound-preserving and entropy stable enriched Galerkin methods for nonlinear hyperbolic equations

A reliable strategy for a category of third-kind nonlinear fractional integro-differential equations