Search Results (1,688)

This study investigates the dynamics of dust acoustic periodic waves in a three-component, unmagnetized dusty plasma system using generalized $(r^{★},q^{★})$ distributions. First, boundary conditions are applied to reduce the model to a second-order nonlinear ordinary differential equation. The Galilean transformation is subsequently applied to reformulate the second-order ordinary differential equation into an unperturbed dynamical system. Next, phase portraits of the system are examined under all possible conditions of the discriminant of the associated cubic polynomial, identifying regions of stability and instability. The Runge–Kutta method is employed to construct the phase portraits of the system. The Hamiltonian function of the unperturbed system is subsequently derived and used to analyze energy levels and verify the phase portraits. Under the influence of an external periodic perturbation, the quasi-periodic and chaotic dynamics of dust ion acoustic waves are explored. Chaos detection tools confirm the presence of quasi-periodic and chaotic patterns using Basin of attraction, Lyapunov exponents, Fractal Dimension, Bifurcation diagram, Poincaré map, Time analysis, Multi-stability analysis, Chaotic attractor, Return map, Power spectrum, and 3D and 2D phase portraits. In addition, the model’s response to different initial conditions was examined through sensitivity analysis. Full article

In this paper, we investigate how adaptive time-integration strategies can be effectively combined with parallel-in-time numerical methods for solving systems of ordinary differential equations. Our focus is particularly on their influence on tolerance proportionality. We examine various grid-refinement strategies within the multigrid reduction-in-time (MGRIT) framework. Our results show that a simple adjustment to the original refinement factor can substantially improve computational stability and reliability. Through numerical experiments on standard test problems using the XBraid library, we demonstrate that parallel-in-time solutions closely match their sequential counterparts. Moreover, with the use of multiple processors, computing time can be significantly reduced. Full article

This research explores the application of NiTiNOL-steel (NiTi–ST) wire ropes as nonlinear damping devices for mitigating vibrations in composite stiffened panels. A dynamic model is formulated by coupling the composite panel with a modified Bouc–Wen hysteresis representation and employing the first-order shear deformation theory (FSDT), based on Hamilton’s principle. Using the Galerkin truncation method (GTM), the model is converted into a system of nonlinear ordinary differential equations. The dynamic response to axial harmonic excitations is analyzed, emphasizing the vibration reduction provided by the embedded NiTi–ST ropes. Finite element analysis (FEA) validates the model by comparing natural frequencies and force responses with and without ropes. A newly developed experimental apparatus demonstrates that NiTi–ST cables provide outstanding vibration damping while barely affecting the system’s inherent frequency. The N3a configuration of NiTi–ST ropes demonstrates optimal vibration reduction, influenced by excitation frequency, amplitude, length-to-width ratio, and composite layering. Full article

In the present paper, a mathematical analysis of the Gardner equation with varying coefficients has been performed to give a more realistic model of physical phenomena, especially in regards to plasma physics. First, a Lie symmetry analysis was carried out, as a result of which a symmetry classification following the different representations of the variable coefficients was systematically derived. The reduced ordinary differential equation obtained is solved using the power-series method and solutions to the equation are represented graphically to give an idea of their dynamical behavior. Moreover, a fully connected neural network has been included as an efficient computation method to deal with the complexity of the reduced equation, by using traveling-wave transformation. The validity and correctness of the solutions provided by the neural networks have been rigorously tested and the solutions provided by the neural networks have been thoroughly compared with those generated by the Runge–Kutta method, which is a conventional and well-recognized numerical method. The impact of a variation in the loss function of different coefficients has also been discussed, and it has also been found that the dispersive coefficient affects the convergence rate of the loss contribution considerably compared to the other coefficients. The results of the current work can be used to improve knowledge on the nonlinear dynamics of waves in plasma physics. They also show how efficient it is to combine the approaches, which consists in the use of analytical and semi-analytical methods and methods based on neural networks, to solve nonlinear differential equations with variable coefficients of a complex nature. Full article

The Korteweg–de Vries (KdV) equation is a nonlinear evolution equation that reflects a wide variety of dispersive wave occurrences with limited amplitude. It has also been used to describe a range of major physical phenomena, such as shallow water waves that interact weakly and nonlinearly, acoustic waves on a crystal lattice, lengthy internal waves in density-graded oceans, and ion acoustic waves in plasma. The KdV equation is one of the most well-known soliton models, and it provides a good platform for further research into other equations. The KdV equation has several forms. The aim of this study is to introduce and investigate a (2+1)-dimensional generalized fifth-order KdV equation with power law nonlinearity (gFKdVp). The research methodology employed is the Lie group analysis. Using the point symmetries of the gFKdVp equation, we transform this equation into several nonlinear ordinary differential equations (ODEs), which we solve by employing different strategies that include Kudryashov’s method, the $(G^{\prime }/G)$ expansion method, and the power series expansion method. To demonstrate the physical behavior of the equation, 3D, density, and 2D graphs of the obtained solutions are presented. Finally, utilizing the multiplier technique and Ibragimov’s method, we derive conserved vectors of the gFKdVp equation. These include the conservation of energy and momentum. Thus, the major conclusion of the study is that analytic solutions and conservation laws of the gFKdVp equation are determined. Full article

Topical drug administration is a common method of delivering medications to the eye to treat various ocular conditions, including glaucoma, dry eye, and inflammation. Drug efficacy following topical administration, including the drug’s distribution within the eye, absorption and elimination rates, and physiological responses can be predicted using physiologically based pharmacokinetic (PBPK) modeling. High-resolution computational models of the eye are desirable to improve simulations of drug delivery; however, these approaches can have long run times. In this study, a fast-running computational quasi-3D (Q3D) model of the human tear film was developed to account for absorption, blinking, drainage, and evaporation. Visualization of blinking mechanics and flow distributions throughout the tear film were enabled using this Q3D approach. Average drug absorption throughout the tear film subregions was quantified using a high-resolution compartment model based on a system of ordinary differential equations (ODEs). Simulations were validated by comparing them with experimental data from topical administration of 0.1% dexamethasone suspension in the tear film (R² = 0.76, RMSE = 8.7, AARD = 28.8%). Overall, the Q3D tear film model accounts for critical mechanistic factors (e.g., blinking and drainage) not previously included in fast-running models. Further, this work demonstrated methods toward improved computational efficiency, where central processing unit (CPU) time was decreased while maintaining accuracy. Building upon this work, this Q3D approach applied to the tear film will allow for more seamless integration into full-body models, which will be an extremely valuable tool in the development of treatments for ocular conditions. Full article

A model for the flow and thermal explosion of a micropolar gas is investigated, assuming the equation of state for a real gas. This model describes the dynamics of a gas mixture (fuel and oxidant) undergoing a one-step irreversible chemical reaction. The real gas model is particularly suitable in this context because it more accurately reflects reality under extreme conditions, such as high temperatures and high pressures. Micropolarity introduces local rotational dynamic effects of particles dispersed within the gas mixture. In this paper, we first derive the initial-boundary value system of partial differential equations (PDEs) under the assumption of spherical symmetry and homogeneous boundary conditions. We explain the underlying physical relationships and then construct a corresponding approximate system of ordinary differential equations (ODEs) using the Faedo–Galerkin projection. The existence of solutions for the full PDE model is established by analyzing the limit of the solutions of the ODE system using a priori estimates and compactness theory. Additionally, we propose a numerical scheme for the problem based on the same approximate system. Finally, numerical simulations are performed and discussed in both physical and mathematical contexts. Full article

This study addresses the standardization of Singular Distributed Parameter Systems (SDPSs). It focuses on classifying and simplifying first- and second-order linear SDPSs using characteristic matrix theory. First, the study classifies first-order linear SDPSs into three canonical forms based on characteristic curve theory, with an example illustrating the standardization process for parabolic SDPSs. Second, under regular conditions, first-order SDPSs can be decomposed into fast and slow subsystems, where the fast subsystem reduces to an Ordinary Differential Equation (ODE) system, while the slow subsystem retains the spatiotemporal characteristics of the original system. Third, the standardization and classification of second-order SDPSs is proposed using three reversible transformations that achieve structural equivalence. Finally, an illustrative example of a building temperature control is built with SDPSs. The simulation results show the importance of system standardization in real-world applications. This research provides a theoretical foundation for SDPS standardization and offers insights into the practical implementation of distributed temperature systems. Full article

In ecosystems, spatial structure plays a fundamental role in shaping the observed dynamics. In particular, the availability and distribution of unoccupied sites—potential habitats—can strongly affect species persistence. However, mathematical models of ecosystems based on ordinary differential equations (ODEs) often neglect the explicit representation of these unoccupied sites. Here, probabilistic cellular automata (PCA) are used to reproduce two basic ecological scenarios: competition between two species and a predator–prey relationship. In these PCA-based models, unoccupied sites are taken into account. Subsequently, a mean field approximation of the PCA behavior is formulated in terms of ODEs. The variables of these ODEs are the numbers of individuals of both species and the number of empty cells in the PCA lattice. Including the empty cells in the ODEs leads to a modified version of the Lotka–Volterra system. The long-term behavior of the solutions of the ODE-based models is examined analytically. In addition, numerical simulations are carried out to compare the time evolutions generated by these two modeling approaches. The impact of explicitly considering unoccupied sites is discussed from a modeling perspective. Full article

To investigate the oscillation modes of iced transmission lines, this study introduces a forcing term into the galloping equation and applies two discretization approaches: Discrete Method I (DMI), which directly transforms the partial differential equation into an ordinary differential form, and Discrete Method II (DMII), which first averages dynamic tension along the span. The finite element method is employed to validate the analytical solutions. Using a multiscale approach, amplitude-frequency responses under primary, harmonic, and internal resonance are derived. Results show that DMII yields larger galloping amplitudes and trajectories than DMI, with lower resonant frequencies and weaker geometric nonlinearities. In harmonic resonance, superharmonic and subharmonic modes (notably 1/2) are more easily excited. Under 2:1:2 internal resonance, amplitude differences in the vertical (z) direction are more sensitive to the discretization method, whereas the 1:1:1 case shows minimal variation across directions. These findings suggest that the choice of discretization significantly influences galloping behavior, with DMII offering a more conservative prediction. Full article

In this article, we investigate the fractional soliton solutions as well as the chaotic analysis of the fourth-order nonlinear Ablowitz–Kaup–Newell–Segur wave equation. This model is considered an intriguing high-order nonlinear partial differential equation that integrates additional spatial and dispersive effects to extend the concepts to more intricate wave dynamics, relevant in engineering and science for understanding complex phenomena. To examine the solitary wave solutions of the proposed model, we employ sophisticated analytical techniques, including the generalized projective Riccati equation method, the new improved generalized exponential rational function method, and the modified F-expansion method, along with mathematical simulations, to obtain a deeper insight into wave propagation. To explore desirable soliton solutions, the nonlinear partial differential equation is converted into its respective ordinary differential equations by wave transforms utilizing $\beta$-fractional derivatives. Further, the solutions in the forms of bright, dark, singular, combined, and complex solitons are secured. Various physical parameter values and arrangements are employed to investigate the soliton solutions of the system. Variations in parameter values result in specific behaviors of the solutions, which we illustrate via various types of visualizations. Additionally, a key aspect of this research involves analyzing the chaotic behavior of the governing model. A perturbed version of the system is derived and then analyzed using chaos detection techniques such as power spectrum analysis, Poincaré return maps, and basin attractor visualization. The study of nonlinear dynamics reveals the system’s sensitivity to initial conditions and its dependence on time-decay effects. This indicates that the system exhibits chaotic behavior under perturbations, where even minor variations in the starting conditions can lead to drastically different outcomes as time progresses. Such behavior underscores the complexity and unpredictability inherent in the system, highlighting the importance of understanding its chaotic dynamics. This study evaluates the effectiveness of currently employed methodologies and elucidates the specific behaviors of the system’s nonlinear dynamics, thus providing new insights into the field of high-dimensional nonlinear scientific wave phenomena. The results demonstrate the effectiveness and versatility of the approach used to address complex nonlinear partial differential equations. Full article

To characterize a phenomenological model of a mechanical oscillator, it is important to know the properties of the envelope of the three main physical motion variables: deviation from equilibrium, velocity, and acceleration. Experimental data show that friction forces restrict the shape of these functions. A linear, exponential, or more abrupt decay can be observed depending on the different physical systems and conditions. This paper aimed to contribute to clarifying the role that some types of friction forces play in these shapes. Three types of friction—constant sliding friction, pressure drag proportional to the square of velocity, and friction drag proportional to velocity—were considered to characterize the line connecting the maxima and minima of displacement for a generic mechanical harmonic oscillator. The ordinary differential equation (ODE), describing the harmonic oscillator simultaneously containing the three types of dissipative forces (constant, viscous, and quadratic), was numerically solved to obtain energy dissipation, and the extrema of both displacement and velocity. The differential equation ruling the behavior of the amplitude, as a function of the friction force coefficients, was obtained from energy considerations. Solving this equation, we obtained analytical functions, parametrized by the force coefficients that describe the oscillator tail. A comparison between these functions and the predicted oscillator ODE extrema was made, and the results were in agreement for all the situations tested. Information from the velocity extrema and nulls was enough to obtain a second function that rules completely the ODE solution. The correlations obtained allow for the reverse operation: from the identified extremum data, it was possible to identify univocally the three friction coefficients fitting used in the model. Motion equations were solved, and some physical properties, namely energy conservation and work of friction forces, were revisited. Full article

In this paper, we study a geometric framework for second-order differential systems arising in classical and relativistic mechanics. For this class of systems, we derive necessary and sufficient conditions for their Lagrangian description. The main objectives of this work are to construct efficient structure-preserving variational integrators in a variational framework. To achieve this, we develop new variational integrators through Lagrangian splitting and prove their equivalence to composition methods. We display the superiority of the newly derived numerical methods for the Kepler problem and provide rigorous error estimates by analysing the Laplace–Runge–Lenz vector. The framework provides tools applicable to geometric numerical integration of both ordinary and partial differential equations. Full article

This paper presents a uniformly convergent cubic spline numerical method for a singularly perturbed two-perturbation parameter ordinary differential equation. The considered differential equation is discretized using the cubic spline numerical method on a Bakhvalov-type mesh. The uniform convergence via the error analysis is established very well. The numerical findings indicate that the proposed method achieves second-order uniform convergence. Four test examples have been considered to perform numerical experimentations. Full article

This paper explores the complex dynamics of mixed convective flow in a Casson fluid saturated in a non-Darcy porous medium, focusing on the influence of gyrotactic microorganisms, internal heat generation, and multiple convective mechanisms. Casson fluids, known for their non-Newtonian behavior, are relevant in various industrial and biological contexts where traditional fluid models are insufficient. This study addresses the limitations of the standard Darcy’s law by examining non-Darcy flow, which accounts for nonlinear inertial effects in porous media. The governing equations, derived from conservation laws, are transformed into a system of no linear ordinary differential equations (ODEs) using similarity transformations. These ODEs are solved numerically using a finite differencing method that incorporates central differencing, tridiagonal matrix manipulation, and iterative procedures to ensure accuracy across various convective regimes. The reliability of this method is confirmed through validation with the MATLAB (R2024b) bvp4c scheme. The investigation analyzes the impact of key parameters (such as the Casson fluid parameter, Darcy number, Biot numbers, and heat generation) on velocity, temperature, and microorganism concentration profiles. This study reveals that the Casson fluid parameter significantly improves the velocity, concentration, and motile microorganism profiles while decreasing the temperature profile. Additionally, the Biot number is shown to considerably increase the concentration and dispersion of motile microorganisms, as well as the heat transfer rate. The findings provide valuable insights into non-Newtonian fluid behavior in porous environments, with applications in bioengineering, environmental remediation, and energy systems, such as bioreactor design and geothermal energy extraction. Full article