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%.

Abstract Purpose It is essential for maritime safety, particularly in severe sea conditions, to investigate the nonlinear stability of a ship's pitch-roll motion. It reveals how nonlinear interactions can lead to unexpected instabilities, providing a more accurate design framework. Analysing two-degrees-of-freedom (2DOF) of roll-pitch motion of a vessel explicates complicated coupled dynamics, essential in comprehending nonlinear resonance events and parametric instabilities. Beyond linear approximations, nonlinear stability analysis of coupled pitch-roll ship motion helps capture genuine vessel behavior, particularly in strong sea conditions and large-amplitude waves. Consequently, the current study examines the 2DOF of an excited harmonically pendulum scheme, recognized in the works as an effective model of coupling between pitch and roll motions of a ship. We focus primarily on the dangerous condition of a vessel in which the excitation period is close to the pitch period, and the pitch frequency becomes twice as high as the roll frequency. Method The existing methodology is based mainly on a non-perturbative approach (NPA), which facilitates a unique analysis that is independent of Taylor expansion. He’s frequency formula (HFF) represents the principal tool employed in constructing NPA. The principal purpose of NPA is to transform weakly oscillating nonlinear ordinary differential equations (ODEs) into linear ones. The quick evaluation of frequency-amplitude correlation is essential in acquiring successive approximations of responses to parametric nonlinear variations. The inspiration for some criticisms on the stability of steady states is examined. A chaotic analysis of specified models is conducted using bifurcation diagrams, phase portraits, Poincaré maps, and Lyapunov spectrum. This strategy enables us to identify and distinguish distinct forms of motion exhibited by every system.

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 paper investigates a physics-informed surrogate modeling framework for multi-phase flow in porous media based on the Fourier Neural Operator. Traditional numerical simulators, though accurate, suffer from severe computational bottlenecks due to fine-grid discretizations and the iterative solution of highly nonlinear partial differential equations. By parameterizing the kernel integral directly in Fourier space, the operator provides a discretization-invariant mapping between function spaces, enabling efficient spectral convolutions. We introduce a Dual-Branch Adaptive Fourier Neural Operator with a shared Fourier encoder and two decoders: a saturation branch that uses an inverse Fourier transform followed by a multilayer perceptron and a pressure branch that uses a convolutional decoder. Temporal information is injected via Time2Vec embeddings and a causal temporal transformer, conditioning each forward pass on step index and time step to maintain consistent dynamics across horizons. Physics-informed losses couple data fidelity with residuals from mass conservation and Darcy pressure, enforcing the governing constraints in Fourier space; truncated spectral kernels promote generalization across meshes without retraining. On SPE10-style heterogeneities, the model shifts the infinity-norm error mass into the 10−2 to 10−1 band during early transients and sustains lower errors during pseudo-steady state. In zero-shot three-dimensional coarse-to-fine upscaling from 30×110×5 to 60×220×5, it attains R2=0.90, RMSE = 4.4×10−2, and MAE = 3.2×10−2, with more than 90% of voxels below five percent absolute error across five unseen layers, while the end-to-end pipeline runs about three times faster than a full-order fine-grid solve and preserves water-flood fronts and channel connectivity. Benchmarking against established baselines indicates a scalable, high-fidelity alternative for high-resolution multi-phase flow simulation in porous media.

Traditional transient stability-constrained optimal power flow (TSCOPF) methods rely on solving complex nonlinear differential equations, resulting in high computational demands and lengthy processing times. To address these issues, this paper proposes a TSCOPF model based on a cascaded CatBoost model (CatBoost-DF) and an improved seagull optimization algorithm (ISOA). First, a TSCOPF model is constructed. Second, the CatBoost-DF model is developed to establish a mapping relationship between the dynamic characteristics of the power system and the power angle of generators. The trained CatBoost-DF model is then employed as a surrogate model to handle transient stability constraints, thereby avoiding the computation of complex differential-algebraic equations traditionally required in transient stability constraint analysis. Then, the ISOA is employed to iteratively solve the TSCOPF model. This enables timely adjustment of generator output when transient instability risks arise, preventing accidents while maintaining system economic efficiency. Finally, simulations conducted on the IEEE 39 bus system demonstrate that this method effectively safeguards both system security and economic performance.

This research investigates the dynamic behavior of unsteady squeezing flow involving a non-Newtonian nanofluid permeating through a porous medium confined between two parallel plates, with particular emphasis on the influence of Hall currents and an internal heat source. The governing partial differential equations are reformulated into nonlinear ordinary differential equations using similarity transformations. Analytical solutions are derived employing advanced techniques such as the Homotopy Perturbation Method, Akbari-Ganji Method, and numerical solutions via the fourth-order Runge–Kutta method (RK4), all implemented in Python with the aid of its computational capabilities. Notably, the SymPy library is utilized to handle symbolic computations. The study provides a comprehensive analysis of axial velocity, radial velocity, temperature, and nanoparticle concentration distributions under varying parameter conditions. Key findings indicate that an increase in the Prandtl number or heat source intensity elevates the temperature, while a higher thermophoretic parameter results in reduced nanoparticle concentration. The minimum errors for velocity, temperature, and concentration are 2×10−8, while that for radial velocity is 1×10−8. The findings have potential applications in various fields such as industry, biology, and biomedical engineering (including drug delivery and photothermal therapy).

The nonlinear Helmholtz equation models nonparaxial electromagnetic wave propagation in Kerr-type media and presents numerical challenges in two-dimensional photonic crystals due to strong nonlinearities, high wave numbers, and material discontinuities. This work presents an efficient and robust numerical framework combining a fourth-order finite difference scheme with multiple iterative methods, including fixed-point, frozen nonlinearity, Newton’s, and modified Newton’s methods. A key novelty contribution is the high-order discretization of nonlinear interfaces, enabling accurate treatment of discontinuous coefficients. Enhanced boundary conditions further ensure stability at high frequencies. Numerical experiments validate the accuracy of the scheme using analytical solutions, and demonstrate that the modified Newton’s method provides superior convergence performance. The results offer an efficient and robust approach for simulating nonlinear wave propagation in complex periodic media and designing nonlinear photonic device.

Many studies on acoustic radiation forces, especially those applied to acoustic levitation, focus on characterizing the behaviour of acoustic fields. However, the dynamic response of the levitated objects, particularly those larger than the wavelength limit, remains relatively underexplored. Here, we look to bridge this gap by deriving nonlinear equations of motion for a spherical object trapped under acoustic radiation forces while subject to external excitation. For such a contemporary scenario, the otherwise elemental Gorkov formulation fails to provide accurate results. Using Sparse Identification of Nonlinear Dynamical Systems (SINDy), first, we derive the corresponding nonlinear equation of motion from analytical time series data obtained through the Gorkov formulation and external excitation for acoustically small objects. This approach recovers the governing equation with less than 0.05% error in coefficient values when compared to the analytical solution. Second, we conduct experiments with the TinyLev levitator with external excitation applied via an external actuator to generate the required time series for an acoustically large object. SINDy is applied to reconstruct governing equations from experimental data, allowing for the study of how excitation amplitude affects acoustically large objects. All obtained coefficients change with excitation amplitude, and the coefficients in the dynamic equation of motion should not be treated as constants. Strong velocity-dependent terms emerged, indicating a complex relationship between viscosity and object response, which classical models do not predict. The bifurcation diagram obtained using the SINDy-derived equation of motion shows closer agreement with that obtained experimentally. These results demonstrate that SINDy can recover equations consistent with Gorkov's formulation and extend beyond it, providing a pathway to derive analytical expressions directly from data for levitating and manipulating objects beyond the Rayleigh limit.

In this research work, we are interested in the discrete study of the blow-up time of the solution of certain nonlinear parabolic Partial Differential Equations (PDEs) subject to nonlinear boundary conditions. We have been able to establish the necessary and sufficient conditions under which the discrete solution of the problem blows up in a finite discrete time and, at the same time, we have given an estimate of this blow up time. Also, using a convergence study, we showed that the discrete time and solution converge respectively to the continuous time and solution when the discretization steps in space and time tend towards zero. Finally, we illustrated our analysis with graphical representations and some numerical results.

In this paper, we are concerned with the following initial-boundary value problem: (P) \left\{% \begin{array}{ll} \hbox{$u_t(x,t)- \Delta u(x,t)- G(x)|u|^{p-1}u  =0, \quad x\in  \Omega, t\in(0,T)$,} \hbox{$u(x,t)=0 \quad x \in \partial \Omega,  t\in(0,T)$,} \hbox{$u(x,0)=u_{0}(x), \quad x \in \Omega,$} \\ \end{array}% \right. where $ p \geq p_s :=\dfrac{ d + 2}{d-2} $, $ u_0 \in L^\infty(D_\mu) $, and $ G(r) \in C^1([0, \mu]), $  $0 <  \underline{C} \leq G(r) \leq \overline{C}  < \infty,$ $G^{'}(r) \leq 0 $. We study the initial value problem and boundary conditions for a nonlinear heat equation incorporating a potential term. Particularly, we focus on the asymptotic behavior of solutions during blow-ups. We extend existing results on this phenomenon, specifically in the case where the potential term is constant, based on the works of Matano-Merle (Matano  \& Merle, 2004). We  show that when $p_s \leq p < p^* $, the radial solutions of this problem always exhibit Type I blow-up. This result generalizes previous results for the case where $G \equiv 1 $, and its achievement is non-trivial due to the presence of the potential term $ G $. We use the contraction mapping principle to  show the existence of singular stationary solutions to an associated elliptic equation with a potential. Furthermore, our analysis of the properties of the zeros of the solutions lead to the nonexistence of type II singularity for the problem. We also delve into the study of critical solutions for a class of nonlinear  parabolic equations in a bounded domain, focusing on the construction of appropriate approximate solutions.

Abstract This article is concerned with the singularity formation of smooth solutions for a nonhomogeneous hyperbolic system arising in magnetohydrodynamics. The system owns four linearly degenerate characteristic fields that influence each other in the relations of second derivatives, making the problem difficult to handle. We develop the characteristic decomposition technique to apply the nonlinear hyperbolic equations with more than two wave characteristics. It is verified that the smooth solution can form a singularity in finite time and the density itself tends to infinity at the blowup point for a special kind of initial data.

In this work, a Susceptible-Infected-Recovered (SIR)-based epidemic model incorporating nonlinear incidence and recovery rates, with the consideration of limited medical resources (e.g., the availability of hospital beds) is examined. The model also emphasises the significance of factoring in distinct intervention strategies and considers some important epidemiological factors, in the light of COVID-19 endemicity. In particular, the study employs a Monod-type nonlinear incidence rate coupled with a nonlinear recovery equation, to uncover the intricate dynamics that emerge from the interplay of these epidemiological forces. The findings reveal the existence of disease-free and endemic equilibria, their stability conditions, and bifurcational changes in the dynamics of the system. Bifurcation analysis demonstrates the emergence of transcritical, saddle-node and Hopf bifurcations with the existence of distinct stable and unstable equilibria and limit cycles. Overall, this work highlights the importance of mathematical modeling and dynamical systems techniques in investigating the interplay among various epidemiological factors, thereby providing valuable insights to guide effective epidemic control strategies.

Abstract This study explores the modified Benjamin–Bona–Mahony equation using the new extended direct algebraic approach, a powerful analytical technique for solving nonlinear partial differential equations. The proposed methodology yields a diverse spectrum of exact solutions, categorized into 12 distinct classes, including rational, hyperbolic, and trigonometric functions, as well as mixed periodic, singular, shock-singular, complex solitary-shock, and plane-wave solutions. These solutions are systematically derived and validated using Mathematica , demonstrating the reliability and effectiveness of the method. A comparative analysis with existing techniques underscores the consistency and superiority of the proposed approach. Additionally, the Hamiltonian function is constructed to examine the system’s conservation properties, ensuring the physical relevance of the obtained solutions. A comprehensive sensitivity analysis is performed to assess the model response to variations in parameters and initial conditions. To further illustrate the dynamical characteristics of the solutions, three-dimensional, two-dimensional, and contour plots are presented, offering deeper insights into their physical behavior. The results contribute to the larger study of nonlinear wave phenomena in engineering and applied sciences, providing a robust analytical framework for future research in soliton theory and mathematical physics.

This paper introduces a hybrid numerical method for fractional differential equations combining the Caputo-Fabrizio and Atangana Baleanu derivatives through a blending parameter. The resulting hybrid operator captures a spectrum of memory effects from exponential to power-law types, offering flexible and realistic modeling of complex systems. We develop a predictor-corrector scheme that efficiently approximates the resulting nonlinear integral equation, achieving second-order accuracy and stable convergence. The method’s effectiveness is demonstrated via a detailed fractional logistic growth example, showcasing smooth interpolation between memory behaviors and fast convergence. This approach broadens the toolbox for fractional calculus applications where multiple memoryscales coexist.

In recent years data-driven machine learning techniques attract the attention of researchers in analyzing many complex systems. This study introduces a novel unsupervised deep neural network approach to predict the temperature and velocity behaviour of electro-magneto-hydrodynamics hybrid nanofluid flow for geothermal pipelines application.The exceptional flow and thermal characteristics of hybrid nanofluids making them ideal for use in geothermal energy extraction applications. The dynamics of hybrid nanofluid flow through a pipe are examined using a third-grade sodium alginate model, which has a lot of potential for geothermal applications. The copper oxide (CuO) and zinc oxide (ZnO) nanoparticles make up the nanofluid. It is also investigated how the flow dynamics are affected by electric and magnetic fields. The energy equation takes into account the effects of Joule heating and viscous dissipation as the fully developed incompressible fluid passes through the pipe. Consequently, an unsupervised deep neural network (DNN) method is used to predict the dynamics of nonlinear differential equations (DEs). The accuracy of the deep neural network ranges from 10^{-06} to 10^{-13} across different cases. The velocity profile exhibits a clear symmetrical pattern and is found to be significantly influenced by both the electric field and the thermal Grashof number. The overall thermal profile along the pipe’s length decreases as a result of the nanoparticles. Additionally, lowering the pressure has a similar effect on both velocities. This study makes important contributions to the comprehension of the intricate dynamics of electro-magneto-hydrodynamics hybrid nanofluid flow, thereby laying a foundational framework for optimizing thermodynamic systems in geothermal. The findings of this research hold significant practical implications for the design and engineering of systems aimed at energy conservation and improved heat transfer efficiency in geothermal pipelines.

In this paper, we prove the existence of extremals for the singular Trudinger–Moser inequality in the entire Euclidean space R n involved with the trapping potential: for any β ∈ ( 0 , n ) , sup u ∈ W 1 , n ( R n ) , ∫ R n ( | ∇u | n + V ( x ) | u | n ) d x ≤ 1 ⁡ ∫ R n Φ n ( α | u | n n − 1 ) | x | β d x < ∞ iff α ≤ α n , β := ( 1 − β n ) α n , where Φ n ( t ) = e t − ∑ k = 0 n − 2 t k k ! , α n = n ω n − 1 1 n − 1 and ω n − 1 is the surface area of unit sphere in R n , V ( x ) is the trapping type potential, that is, 0 < inf x ∈ R n ⁡ V ( x ) < sup x ∈ R n ⁡ V ( x ) = lim | x | → ∞ ⁡ V ( x ) . The proof is based on the method of blow-up analysis of the nonlinear Euler-Lagrange equations of the singular Trudinger–Moser functionals. Our result extends the recent work Chen et al. [Existence of extremals for Trudinger–Moser inequalities involved with a trapping potential. Calc Var Partial Differ Equations. 2023;62(5):150. doi: 10.1007/s00526-023-02477-8] to the singular case.

We consider the effect of nonreciprocity in a binary mixture of self-propelled particles with antialigning interactions, where a particle of type A reacts differently to a particle of type B than vice versa. Starting from a well-known microscopic Langevin model for the particles, setting up the corresponding exact N-particle Fokker-Planck equation, and making Boltzmann's assumptions of low density and one-sided molecular chaos, the nonlinear active Boltzmann equation with the exact collision operator is derived. In this derivation, the effect of phase-space compression and the buildup of pair correlations during binary interactions is explicitly taken into account, leading to a theoretical description beyond mean field. This extends previous results for reciprocal interactions, where it was found that orientational order can emerge in a system with purely antialigning interactions. Although the equations of motion are more complex than in the reciprocal system, the theory still leads to analytical expressions and predictions. Comparisons with agent-based simulations show excellent quantitative agreement of the dynamic and static behavior in the low-density and/or small coupling limit.

This paper introduces a new class of generalized metric structures, called interpolative b-metric spaces, which unify and extend both b-metric spaces and interpolative metric spaces in a non-trivial way. By incorporating a nonlinear correction term alongside a multiplicative scaling parameter into the triangle inequality, this framework enables broader contractive conditions and refined control of convergence behavior. We develop the foundational theory of interpolative b-metric spaces and establish a generalized Ćirić-type fixed point theorem, along with Banach, Kannan, and Bianchini-type results as corollaries. To highlight the originality and applicability of our approach, we apply the main theorem to a nonlinear Volterra-type integral equation, demonstrating that interpolative b-metrics effectively accommodate nonlinear solution structures beyond the scope of traditional metric models. This work offers a unified platform for fixed point analysis and opens new directions in nonlinear and functional analysis.

Abstract This study examines mass and heat transfer in a permeable ternary nanofluid flow over a stretching sheet, considering the combined effects of a chemical reaction, Joule heating, an exponentially space-dependent heat source, and an inclined magnetic field. Three types of water-based nanofluids are analysed: mono (Cu), hybrid (Cu + Al 2 O 3 ), and ternary (Cu + Al 2 O 3 + Ag). The governing nonlinear partial differential equations are reduced using similarity transformations and solved numerically via MATLAB’s BVP4c method. The results reveal that ternary nanofluids exhibit superior thermal performance, with significantly higher temperature profiles compared to mono and hybrid nanofluids. The influence of key parameters is also investigated. Increased suction and velocity slip reduce thermal and concentration boundary layers, while higher Biot numbers and heat source intensity enhance temperature profiles. Additionally, Joule heating and magnetic field inclination intensify the heat transfer rate. These findings provide valuable insights for optimizing thermal systems in applications such as solar energy collectors, thermoelectric devices, and chemical processing industries.