Search Results (1,702)

This article introduces two numerical methods based on the Theory of Functional Connections (TFC) for solving linear ordinary differential equations that involve step discontinuities in the forcing term. The novelty of the first proposed approach lies in the direct incorporation of discontinuities into the free function of the TFC framework, while the second proposed method resolves discontinuities through piecewise constrained expressions comprising particular weighted support functions systematically chosen to enforce continuity conditions. The accuracy of the proposed methods is validated for both a second-order initial value and boundary value problem. As a final demonstration, the methods are applied to a third-order differential equation with non-constant coefficients and multiple discontinuities, for which an analytical solution is known. The methods achieve error levels approaching machine precision, even in the case of equations involving functions whose Laplace transforms are not available. Full article

Viruses cause a large number of diseases. After penetrating into a host, the virus starts infecting healthy cells. Then it uses the RNA or DNA of the cell to replicate and afterward it explodes the infected cell, letting out many copies of the virus that can infect new cells. The innate and adaptive parts of the immune system defend the body by eliminating both the free viruses and the infected cells. Neutrophils, macrophages, natural killer cells, helper T cells (CD4+) and cytotoxic T lymphocytes (CD8+) are among the participating immune cells. The interactions are complex and not fully understood. In this paper, we present and study three mathematical models based on ordinary differential equations of virus and immune system interactions with different complexities, and also introduce possible treatments. We discuss the advantages and disadvantages of each model. We do global sensitivity analysis and numerical simulations. Finally, we present conclusions including comments about the complexity of mathematical models. Full article

Periodic solutions of high-order nonlinear differential equations are fundamental in dynamical systems, yet they remain challenging to establish with traditional methods. This paper addresses the existence of periodic solutions in general $2n$-order autonomous and nonautonomous ordinary differential equations. By extending Carathéodory’s variational technique from the calculus of variations, we reformulate the original periodic solution problem as an equivalent higher-order variational problem. The approach constructs a convex function and introduces an auxiliary transformation to enforce convexity in the highest-order term, enabling a tractable operator-theoretic analysis. Within this framework, we prove two main theorems that provide sufficient conditions for periodic solutions in both autonomous and nonautonomous cases. These results generalize the known theory for second-order equations to arbitrary higher-order systems and highlight a connection to the Hamilton–Jacobi theory, offering new insights into the underlying variational structure. Finally, numerical examples validate our theoretical results by confirming the periodic solutions predicted by the theory and demonstrating the approach’s practical applicability. Full article

Neural ordinary differential equations (neural ODEs) are a well-established tool for optimizing the parameters of dynamical systems, with applications in image classification, optimal control, and physics learning. Although dynamical systems of interest often evolve on Lie groups and more general differentiable manifolds, theoretical results for neural ODEs are frequently phrased on ${\mathbb{R}}^n$. We collect recent results for neural ODEs on manifolds and present a unifying derivation of various results that serves as a tutorial to extend existing methods to differentiable manifolds. We also extend the results to the recent class of neural ODEs on Lie groups, highlighting a non-trivial extension of manifold neural ODEs that exploits the Lie group structure. Full article

The main goal of this study is to create a computational solver that analyzes the effects of magnetohydrodynamics (MHD) on heat radiation in Cu–water-based Prandtl nanofluid flow using artificial neural networks. Copper nanoparticles are utilized to boost the water-based fluid’s thermal effect. This study primarily focuses on heat transfer over a horizontal sheet, exploring different scenarios by varying key factors such as the magnetic field and thermal radiation properties. The mathematical model is formulated using partial differential equations (PDEs), which are then transformed into a corresponding set of ordinary differential equations (ODEs) through appropriate similarity transformations. The bvp4c solver is then used to simulate the numerical behavior. The effects of relevant parameters on the temperature, velocity, skin friction, and local Nusselt number profiles are examined. It is discovered that the parameters of the Prandtl fluid have a considerable impact. The local skin friction and the local Nusselt number are improved by increasing these parameters. The dataset is split into 70% training, 15% validation, and 15% testing. The ANN model successfully predicts skin friction and Nusselt number profiles, showing good agreement with numerical simulations. This hybrid framework offers a robust predictive approach for heat management systems in industrial applications. This study provides important insights for researchers and engineers aiming to comprehend flow characteristics and their behavior and to develop accurate predictive models. Full article

Simulating nonlinear classical dynamics on a quantum computer is an inherently challenging task due to the linear operator formulation of quantum mechanics. In this work, we provide a systematic approach to alleviate this difficulty by developing an explicit quantum algorithm that implements the time evolution of a second-order time-discretized version of the Lorenz model. The Lorenz model is a celebrated system of nonlinear ordinary differential equations that has been extensively studied in the contexts of climate science, fluid dynamics, and chaos theory. Our algorithm possesses a recursive structure and requires only a linear number of copies of the initial state with respect to the number of integration time-steps. This provides a significant improvement over previous approaches, while preserving the characteristic quantum speed-up in terms of the dimensionality of the underlying differential equations system, which similar time-marching quantum algorithms have previously demonstrated. Notably, by classically implementing the proposed algorithm, we showcase that it accurately captures the structural characteristics of the Lorenz system, reproducing both regular attractors–limit cycles–and the chaotic attractor within the chosen parameter regime. Full article

The SARS-CoV-2 spike (S1) protein facilitates viral entry through binding to angiotensin-converting enzyme 2 (ACE2), but it also contains an Arg–Gly–Asp (RGD) motif that may enable interactions with RGD-binding integrins on ACE2-negative cells. Here, we provide quantitative evidence for this alternative binding pathway using a live-cell, label-free resonant waveguide grating (RWG) biosensor. RWG technology allowed us to monitor real-time adhesion kinetics of live cells to RGD-displaying substrates, as well as cell adhesion to S1-coated surfaces. To characterize the strength of the integrin–S1 interaction, we determined the dissociation constant using two complementary approaches. First, we performed a live-cell competitive binding assay on RGD-displaying surfaces, where varying concentrations of soluble S1 were added to cell suspensions. Second, we recorded the adhesion kinetics of cells on S1-coated surfaces and fitted the data using a kinetic model based on coupled ordinary differential equations. By comparing the results from both methods, we estimate that approximately 33% of the S1 molecules immobilized on the Nb₂O₅ biosensor surface are capable of initiating integrin-mediated adhesion. These findings support the existence of an alternative integrin-dependent entry route for SARS-CoV-2 and highlight the effectiveness of label-free RWG biosensing for quantitatively probing virus–host interactions under physiologically relevant conditions without the need of the isolation of the interaction partners from the cells. Full article

An asymptotic stability velocity tracking controller is designed to enable the autonomous maneuvering flight of unmanned helicopters. Firstly, taking the UH-60A without pilots as the research object, a high-efficient rotor aerodynamic modeling is developed, which incorporates a free-wake vortex method with the flap response of blades. The consummate flight dynamic model is complemented by wind tunnel-validated fuselage/tail rotor load regressions. Secondly, a linear state–space equation is derived via the small perturbation linearization method based on the flight dynamic model within the body coordinate system. A decoupled model is formulated based on the linear state–space equation by employing the implicit model approach. Subsequently, a system of ordinary differential equations is constructed, which is related to the deviation between actual velocity and its expected value, along with higher-order derivatives of this discrepancy. The I&I (immersion and invariance) theory is then employed to facilitate the design of a non-cascade control loop. Finally, the response of desired velocity in longitudinal channel is simulated with step signal to compare the control effect with a PID (proportional–integral–derivative) controller. By adjusting the coefficients, the response progress of the PID controller is similar to the effect of adaptive controller with I&I theory. However, there is no obvious overshoot in the process with I&I adaptive controller, and the average response amplitude accounts for 16.69% of the random white noise, which is 7.38% of the oscillation level under the PID controller. The parameter tuning complexity when employing I&I theory is significantly lower than that of the PID controller, which is evaluated by mathematical derivations and simulations. Meanwhile, the sidestep and pirouette maneuvers are simulated and analyzed to examine the controller in accordance with the performance criteria outlined in the ADS-33E-REF standards. The simulation results demonstrate that the speed expectation-oriented asymptotic stability control can achieve a fast response. Both sidestep and pirouette maneuvers can satisfy the desired performance requirements stipulated by ADS-33E-REF. Full article

The development of simple and yet accurate formulations of frequency–amplitude relationships for non-conservative nonlinear oscillators is an important issue. The present paper is concerned with integral-type frequency–amplitude formulas in the dimensionless time domain and time domain to accurately determine vibrational frequencies of nonlinear oscillators. The novel formulation is a balance of kinetic energy and the work during motion of the nonlinear oscillator within one period; its generalized formulation permits a weight function to appear in the integral formula. The exact values of frequencies can be obtained when exact solutions are inserted into the formulas. In general, the exact solution is not available; hence, low-order periodic functions as trial solutions are inserted into the formulas to obtain approximate values of true frequencies. For conservative nonlinear oscillators, a powerful technique is developed in terms of a weighted integral formula in the spatial domain, which is directly derived from the governing ordinary differential equation (ODE) multiplied by a weight function, and integrating the resulting equation after inserting a general trial ODE to acquire accurate frequency. The free parameter is involved in the frequency–amplitude formula, whose optimal value is achieved by minimizing the absolute error to fulfill the periodicity conditions. Several examples involving two typical non-conservative nonlinear oscillators are explored to display the effectiveness and accuracy of the proposed integral-type formulations. Full article

Fiber orientation is an important descriptor of the microstructure for short fiber polymer composite materials where accurate and efficient prediction of the orientation state is crucial when evaluating the bulk thermo-mechanical response of the material. Macroscopic fiber orientation models employ the moment-tensor form in representing the fiber orientation state, and they all require a closure approximation for the higher-order orientation tensors. In addition, various models have more recently been developed to account for rotary diffusion due to fiber-fiber and fiber-matrix interactions which can now more accurately simulate the experimentally observed slow fiber kinematics in polymer composite processing. It is common to use explicit numerical initial value problem-ordinary differential equation (IVP-ODE) solvers such as the 4th- and 5th-order Dormand Prince Runge–Kutta (RK45) method to predict the transient and steady-state fiber orientation response. Here, we propose a computationally efficient method based on the Newton-Raphson (NR) iterative technique for determining steady state orientation tensor values by evaluating exact derivatives of the moment-tensor evolution equation with respect to the independent components of the orientation tensor. We consider various existing macroscopic-fiber orientation models and several closure approximations to ensure the robustness and reliability of the method. The performance and stability of the approach for obtaining physical solutions in various homogeneous flow fields is demonstrated through several examples. Validation of our orientation tensor exact derivatives is performed by benchmarking with results of finite difference techniques. Overall, our results show that the proposed NR method accurately predicts the steady state orientation for all tensor models, closure approximations and flow types considered in this paper and was relatively faster compared to the RK45 method. The NR convergence and stability behavior was seen to be sensitive to the initial orientation tensor guess value, the fiber orientation tensor model type and complexity, the flow type and extension to shear rate ratio. Full article

This study explores the dynamic behavior of a vertical pipe conveying gas-liquid two-phase flow with rotationally restrained boundaries, employing the generalized integral transform technique (GITT). The rotationally restrained boundary conditions are more realistic for practical engineering applications in comparison to the classical simply-supported and clamped boundary conditions, which can be viewed as limiting scenarios of the rotationally restrained boundary conditions when rotational stiffness approaches zero and infinity, respectively. Utilizing the small-deflection Euler-Bernoulli beam theory, the governing equation of motion for the deflection of the pipe is transformed into an infinite set of coupled ordinary differential equations, which is then numerically solved following truncation at a finite order $NW$. The proposed integral transform solution was initially validated against extant literature results. Numerical findings demonstrate that as the gas volume fraction increases, there is a reduction in both the first-order critical flow velocity and the vibration frequency of the pipe conveying two-phase flow. Conversely, as the rotational stiffness factor enhances, both the first-order critical velocity and vibration frequency increase, resulting in improved stability of the pipe. The impact of the bottom-end rotational stiffness factor $r_2$ on the dynamic stability of the pipe is more pronounced compared to the top-end rotational factor $r_1$. The variation in two-phase flow parameters is closely associated with the damping and stiffness matrices. Modifying the gas volume fraction in the two-phase flow alters the distribution of centrifugal and Coriolis forces within the pipeline system, thereby affecting the pipeline’s natural frequency. The results illustrate that an increase in the gas volume fraction leads to a decrease in both the pipeline’s critical velocity and vibration frequency, culminating in reduced stability. The findings suggest that both the gas volume fraction and boundary rotational stiffness exert a significant influence on the dynamic behavior and stability of the pipe conveying gas-liquid two-phase flow. Full article

This paper is a short review on applications of non-linear dynamics in the concept of Heider balance, known also as structural balance. In all the papers listed here, the basic tools are ordinary differential equations. All papers pay attention to real social phenomena, which play the role of illustrations of the mathematical formalisms. Full article

As agricultural land continues to expand, the conversion of forests to farmland has intensified, significantly altering the structure and function of agroecosystems. However, the dynamic ecological responses and their interactions with economic outcomes remain insufficiently modeled. This study proposes an integrated framework that combines a dynamic food web model with the Eco-Economic Benefit and Sustainability (EEBS) model, utilizing empirical data from Brazil and Ghana. A system of ordinary differential equations solved using the fourth-order Runge–Kutta method was employed to simulate species interactions and energy flows under various land management strategies. Reintroducing key species (e.g., the seven-spot ladybird and ragweed) improved ecosystem stability to over 90%, with soil fertility recovery reaching 95%. In herbicide-free scenarios, introducing natural predators such as bats and birds mitigated disturbances and promoted ecological balance. Using XGBoost (Extreme Gradient Boosting) to analyze 200-day community dynamics, pest control, resource allocation, and chemical disturbance were identified as dominant drivers. EEBS-based multi-scenario optimization revealed that organic farming achieves the highest alignment between ecological restoration and economic benefits. The model demonstrated strong predictive power ($R^2$ = 0.9619, RMSE = 0.0330), offering a quantitative basis for green agricultural transitions and sustainable agroecosystem management. Full article

This work analyzes the aero-elastic oscillations of cantilever beams reinforced by carbon nanotubes (CNTs). Four different distributions of single-walled CNTs are assumed as the reinforcing phase, in the thickness direction of the polymeric matrix. A modified third-order piston theory is used as an accurate tool to model the supersonic air flow, rather than a first-order piston theory. The nonlinear dynamic equation governing the problem accounts for Von Kármán-type nonlinearities, and it is derived from Hamilton’s principle. Then, the Galerkin decomposition technique is adopted to discretize the nonlinear partial differential equation into a nonlinear ordinary differential equation. This is solved analytically according to a multiple time scale method. A comprehensive parametric analysis was conducted to assess the influence of CNT volume fraction, beam slenderness, Mach number, and thickness ratio on the fundamental frequency and lateral dynamic deflection. Results indicate that FG-X reinforcement yields the highest frequency response and lateral deflection, followed by UD and FG-A patterns, whereas FG-O consistently exhibits the lowest performance metrics. An increase in CNT volume fraction and a reduction in slenderness ratio enhance the system’s stiffness and frequency response up to a critical threshold, beyond which a damped beating phenomenon emerges. Moreover, higher Mach numbers and greater thickness ratios significantly amplify both frequency response and lateral deflections, although damping rates tend to decrease. These findings provide valuable insights into the optimization of CNTR composite structures for advanced aeroelastic applications under supersonic conditions, as useful for many engineering applications. Full article

This study investigates the dynamics of dust acoustic periodic waves in a three-component, unmagnetized dusty plasma system using generalized $(r^{\star },q^{\star })$ 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