Second-order Ordinary Differential Equations Research Articles

On the solutions of the Lane-Emden model via non-dyadic wavelet scheme to unravel astrophysical phenomena

The research focuses on exploring the characteristics of the Lane Emden equation, which is a second-order ordinary differential equation that has its roots in astrophysics. To approximate this equation, a technique that employs non-dyadic Haar wavelets in conjunction with quasi-linearization is utilized. By utilizing non-dyadic wavelets, the ordinary differential equation can be simplified into a system of algebraic equations. Error estimates are provided to assess the accuracy of the produced data. A comparison between the existing solutions and the numerical results obtained using the suggested approach is performed to showcase its efficiency and advantages. The non-dyadic Haar wavelet approach is found to be a rich structure for numerous solutions that span a wide range of physical parameter.

Leveraging feed-forward neural networks to enhance the hybrid block derivative methods for system of second-order ordinary differential equations

This study introduces an innovative method combining discrete hybrid block techniques and artificial intelligence to enhance the solution of second-order Ordinary Differential Equations (ODEs). By integrating feed-forward neural networks (FFNN) into the hybrid block derivative method (HBDM), the modified approach shows improved accuracy and efficiency compared to traditional methods. Through comprehensive comparisons with exact and existing solutions, the study demonstrates the effectiveness of the proposed approach. The evaluation, utilizing root mean square error (RMSE), confirms its superior performance, robustness, and applicability in diverse scenarios. This research sets a new standard for solving complex ODE systems, offering promising avenues for future research and practical implementations.

Equivalence problem for first-order and second-order ODEs with a quadratic restriction

We consider the equivalence problem for first-order ordinary differential equations (ODEs) and recover the well-known result that any two first-order ODEs are equivalent under a point transformation by Cartan’s equivalence method. Curvature properties of some first-order and second-order ODEs motivate us to define the equivalence problem for such equations as the metric equivalence problem which defines the [Formula: see text]-structure on the corresponding manifolds. Accordingly, we show that there are first-order ODEs and linear second-order ODEs which may not be equivalent under a class of fiber-preserving diffeomorphisms.

Periodic and subharmonic solutions in the motion of a bead on a rotating circular hoop

We establish the necessary conditions for the existence and multiplicity of periodic and subharmonic solutions to a second-order nonlinear ordinary differential equation (ODE). This ODE describes the motion of a bead on a rotating circular hoop subjected to a constant angular velocity ω and a T-periodic forcing. Our approach involves estimating bounds for the angular velocity and period using upper and lower solution methods.

Modeling monkeypox virus transmission: Stability analysis and comparison of analytical techniques

Abstract Monkeypox is a highly infectious disease and spreads very easily, hence posing several health concerns or risks as it may lead to outbreak. This article proposes a new mathematical model to simulate the transmission rate of the monkeypox virus-infected fractional-order differential equations using the Caputo–Fabrizio derivative. The existence, uniqueness, and stability under contraction mapping of the fixed point of the model are discussed using Krasnoselskii’s and Banach’s fixed point theorems. To verify the model proposed, we employ data that record the actual dynamics, and based on these data, the model can capture the observed transmission patterns in Ghana. Also, the analytic algorithm is used to find the result applying the Laplace Adomian decomposition method (LADM). Performance analysis of LADM is made regarding Runge-Kutta fourth order, which is the most commonly employed method for solving second-order ordinary differential equations. This comparison therefore offers information on the truth and reliability of the two techniques toward modeling the transmission pattern of the monkey pox virus. The information obtained through this study provides a better understanding of the antibodies linked to monkeypox virus spreading and provides effective strategies to doctors and politicians. This article helps shape better strategies about combating the impact of monkeypox virus in public health since it makes it easy to predict and prevent the occurrence of the disease.

Understanding Cyclotron Motion: A Visual Approach to Electron Dynamics in Electric and Magnetic Fields

Physics material is often material that is difficult for students to understand because of its abstract concepts, one of which is in the material of electric and magnetic fields that discuss the motion of cyclotrons in connecting theoretical concepts and practical applications. Cyclotron motion is accelerated by an electric field that is influenced by the Lorentz force on the combination of electric and magnetic fields. This paper is made with the aim to analyze and visualize the electric field and magnetic field around the cyclotron motion particles using Python animation. In this study, a helix-shaped trajectory is used to represent the movement of electron particles. The animation developed depicts the electric and magnetic field vectors moving around the helix trajectory. In running the particle trajectory animation, numerical equations are involved. The involved numerical equations are second-order ordinary differential equations. This research not only provides theoretical insights into the electrostatic and magnetostatic properties of electron particles but also provides effective visualizations for further education and research. The resulting animation facilitates to enhance the understanding of the concept of how electric and magnetic fields interact and move in the context of electron particles moving in a helix trajectory, opening up opportunities for further exploration in the development of knowledge related to cyclotron motion that is often difficult to understand only through theoretical approaches.

Going with the Flow

Given a sequence of poses of a body we study the motion resulting when the body is immersed in a (possibly) moving, incompressible medium. With the poses given, say, by an animator, the governing second-order ordinary differential equations are those of a rigid body with time-dependent inertia acted upon by various forces. Some of these forces, like lift and drag, depend on the motion of the body in the surrounding medium. Additionally, the inertia must encode the effect of the medium through its added mass. We derive the corresponding dynamics equations which generalize the standard rigid body dynamics equations. All forces are based on local computations using only physical parameters such as mass density. Notably, we approximate the effect of the medium on the body through local computations avoiding any global simulation of the medium. Consequently, the system of equations we must integrate in time is only 6 dimensional (rotation and translation). Our proposed algorithm displays linear complexity and captures intricate natural phenomena that depend on body-fluid interactions.

Stability and numerical solutions for second-order ordinary differential equations with application in mechanical systems

This study undertakes a comprehensive analysis of second-order Ordinary Differential Equations (ODEs) to examine animal avoidance behaviors, specifically emphasizing analytical and computational aspects. By using the Picard–Lindelöf and fixed-point theorems, we prove the existence of unique solutions and examine their stability according to the Ulam-Hyers criterion. We also investigate the effect of external forces and the system’s sensitivity to initial conditions. This investigation applies Euler and Runge–Kutta fourth-order (RK4) methods to a mass-spring-damper system for numerical approximation. A detailed analysis of the numerical approaches, including a rigorous evaluation of both absolute and relative errors, demonstrates the efficacy of these techniques compared to the exact solutions. This robust examination enhances the theoretical foundations and practical use of such ODEs in understanding complex behavioral patterns, showcasing the connection between theoretical understanding and real-world applications.

Solution of telegraph equation by double general integral transform

This paper presents a method for solving the Telegraph equation using the double general integral transform. The method involves transforming the equation into a second-order ordinary differential equation with variable coefficients, which is then solved using standard techniques. The results demonstrate the effectiveness of the method in solving the Telegraph equation.

A fifth order block methods for solving second-order stiff ordinary differential equations using trigonometric functions and polynomial function as the basis function

A fifth order block methods for solving second-order stiff ordinary differential equations using trigonometric functions and polynomial function as the basis function

A class of polynomial approximation methods to second-order delay differential equations

This paper studies special high-order methods to numerically solve special second-order ordinary differential equations and second-order differential equations with constant delay, which are based on vector field expansion along the shifted and scaled Legendre polynomial orthonormal basis. In the framework of method design, we obtain the second-order perturbation results by transforming the second-order differential equations into first-order ones, and the error accuracy estimation of numerical solutions is achieved by using perturbation results for the problem under consideration. In addition, some numerical tests on specific second-order delay Hamiltonian systems are shown, the purpose of which is to verify the effectiveness of the theoretical findings.

Solving Nonlinear Second-Order ODEs via the Eisenhart Lift and Linearization

The linearization of nonlinear differential equations represents a robust approach to solution derivation, typically achieved through Lie symmetry analysis. This study adopts a geometric methodology grounded in the Eisenhart lift, revealing transformative techniques that linearize a set of second-order ordinary differential equations. The research underscores the effectiveness of this geometric approach in the linearization of a class of Newtonian systems that cannot be linearized through symmetry analysis.

Comparative analysis of thermal behaviour in functionally graded material fin under convective-radiative heat transfer: Homotopy perturbation method

In this study, the thermal characteristics of a fully wet, moving longitudinal fin made from linear Functionally Graded Material (FGM) is explored. The fin response to convective-radiative heat transfer is examined through three distinct FGM scenarios namely, a Homogeneous material (HM), Functionally Graded Material I (FGM I) having greater thermal grading near the base, and Functionally Graded Material II (FGM II) having increased thermal grading towards the tip. The derived energy equation is a second-order nonlinear ordinary differential equation is solved using the Homotopy Perturbation Method (HPM) compared with results found in the existing literature. The fin thermal profile has been graphically analysed for the thermal conductivity grading parameter (β), Peclet number (Pe) and other pertinent parameters. It is found that the temperature distribution along the fin is highest for the FGM II fin and lowest for the HM fin, with the FGM I fin displaying intermediate values. Efficiency of the fin (η) is discovered to be lowest in the case of HM fin structures followed by FGM I and FGM II fin structures at the base and vice versa at the tip. The results show significant potential of HPM which benefits the manufacture and design of FGM fin structures.

Mathematical model of a lightweight three-axle off-road vehicle construction for Arctic zone of Russia

Introduction. The model and results of calculating the smoothness of a light three-axle off-road vehicle for the Arctic zone of Russia are considered. The model on standard approaches and uses a system of assumptions that limits the number of degrees of freedom for the vehicle body to three, as well as one degree of freedom for the unsprung masses is based. The mathematical model is a system of ordinary differential equations and is supplemented with the necessary algebraic equations, as well as initial conditions. The system is integrated using the 4th order RungeKutta method, for which a program was written in C++. The calculations presented in the article demonstrate the possibility of conducting research on the smoothness of a vehicle under conditions of arbitrary terrain, typical for off-road conditions in the winter conditions of the Arctic zone. The dimensions and other parameters of the vehicle were taken from a full-scale model that was used in real expeditions in 2003-2019. Based on the model, suspension characteristics will be developed for a new all-terrain vehicle.Theory. When operating a wheeled vehicle in a wide range of conditions, even in northern regions, transverseangular vibrations are very often insignificant, so only vertical linear and longitudinal-angular vibrations of the frame can be considered. This problem enables to construct a system of equations of vehicle motion using selected degrees of freedom. From a mathematical point of view, these equations are classified as second-order ordinary differential equations with a variable structure of the right-hand sides, which reflects the non-linear nature of the behavior of the suspension in terms of its geometric constraints.Methods. The work uses numerical methods to solve the equations of the constructed model, which enables to gradually weaken the accepted assumptions and build more general calculation algorithms. The main integration method for ensuring the stability of solutions is the multi-step Adams method, which, with the correct choice of step, ensures the necessary stability of the solution over sufficiently long model times. However, in this work, the 4th order Runge-Kutta method was adopted, which turned out to be quite sufficient.Results and conclusions. The paper presents the results of a numerical study of the oscillatory processes of an off-road vehicle during uniform translational motion of the vehicle on a horizontal surface with a given profile of irregularities. The graphs show a transition process of oscillations, which ends with reaching a steady state. The shape of oscillations in a steady state can be irregular and significantly depends on the given speed of the all-terrain vehicle. Analysis of the dependencies presented in the figures shows that the shape of the oscillations of the allterrain vehicle’s frame, as well as the amplitude and frequency, significantly depend on the speed of the vehicle (at a constant road profile). Changing the road profile leads to corresponding changes in the forms and characteristics of forced vibrations of a vehicle on a suspension, which makes it possible to build the necessary amplitude-frequency characteristics, optimize the elastic and dissipative parameters of suspensions, optimize their number and location, and also monitor the movements of arbitrary points at which various units are located.

On the Existence of Periodic Solutions of a System of Second-Order Ordinary Differential Equations with a Quasihomogeneous Nonlinearity

On the Existence of Periodic Solutions of a System of Second-Order Ordinary Differential Equations with a Quasihomogeneous Nonlinearity

A High-Resolution Exponential Spline Method and Its Convergence Analysis for Two-Point Mixed Boundary Value Problems

In this paper, we present a new numerical method for effectively solving general second-order ordinary differential equations with mixed boundary conditions. Our approach utilizes a quasi-variable mesh enabling us the flexibility to adapt the mesh density according to different boundary layer problems. By employing a discretization technique that incorporates the construction of exponential spline, we achieve third-order accuracy at internal grid points, while boundary points exhibit fourth-order accuracy. Since the method is based on off-step grid points, it eliminates the need to modify the method while employing it to singular problems. We provide computational results to various numerical problems which arise in different physical phenomena like Burger’s equation and Sturm–Liouville equation. A comparison with recent findings underscores the superior performance of our method.

Finding and implementing the numerical solution of an optimal control problem for oscillations in a coupled objects system

In the modern world, where efficiency, stability, and precision play a crucial role, the development and application of optimal control strategies in oscillatory systems hold significant importance. The issues related to the numerical solution of control problems associated with damping oscillatory systems consisting of two objects are considered. To numerically solve the discussed problem, the gradient projection method, based on the formula for the first variation of the functional, and the method of successive approximations, associated with the linearity of boundary problems describing oscillatory processes, are applied. The oscillations of one object are described by a wave equation with first-order boundary conditions, while a second-order ordinary differential equation models the oscillations of the other object. Furthermore, the original and the adjoint boundary value problems are solved using direct methods at each iteration step. An algorithm for the numerical solution of the problem is proposed, and based on this algorithm, a software code for implementation is developed. The numerical results obtained in the study demonstrate that there is convergence in terms of functionality, and the approximately optimal controls found in this process are minimizing sequences in the control space. The mechanism of controlling and regulating the operation of the system according to its input constraints is provided by observed feedback, allowing systems with limited excitation to maintain stability and optimal functioning in conditions of changing external or internal circumstances. The obtained results can also be used to forecast the system's behavior in the future, resource planning, prevention of emergencies, or optimization of production processes

On Periodic Solutions of a Second-Order Ordinary Differential Equation

On Periodic Solutions of a Second-Order Ordinary Differential Equation

One-step block scheme with optimal hybrid points for numerical integration of second-order ordinary differential equations

In this paper, a one-step block of optimized hybrid schemes for the numerical integration of second-order initial value problems (IVP) of ordinary differential equations (ODE) is constructed via collocation techniques. The developed scheme is obtained by considering two intra-step nodal points as hybrid points, which are chosen in order to achieve optimized errors of the main formulae approximating the solution such that 0 < v1 < v 2 < 1 where v1 and v2 are defined as hybrid points. The characteristics of the developed scheme are analyzed. Application of the new scheme on some second-order IVPs shows the accuracy and effectiveness of the scheme compared with some existing methods.

Double shooting method for FRCM reinforced systems in debonding problems

The debonding process in FRCM (Fiber Reinforced Cementitious Matrix) strengthened brittle substrates, such as concrete and masonry, is characterized by complex failure modes that include debonding at both the external and internal interfaces of the composite, as well as damage to the mortar matrix. This study addresses this problem classically as a mode-II fracture process by accounting for the one-dimensional non-linear interfacial shear stress-slip relationship combined with matrix axial non-linearity. The approach employed considers the longitudinal displacements of both the inner and outer layers of mortar, along with that of the fiber textile, as unknowns, and analytically solves the second-order ordinary differential equation (ODE) system that governs the debonding problem. Then, the boundary value problem (BVP) is transformed into an initial value problem (IVP) and a double shooting method is proposed to find displacements and stresses along the bonded length in the presence of non-linearity. The appropriate initial values of matrix layer displacements, satisfying the required boundary conditions, are efficiently determined through a two-dimensional bisection method working on rectangular and triangular patches. The softening behavior of both mortar layers and interfaces between mortar and fiber are taken into account using jagged constitutive relationships to enhance convergence. The numerical calculation speed, robustness, and key simulation parameters are thoroughly discussed. Moreover, the proposed numerical approach is compared with existing experimental data and models, on both concrete and masonry FRCM-reinforced specimens. The influence of mortar failure and interfacial strength is also explored. The model demonstrates its ability to effectively reproduce the global bond behavior observed in experimental studies, while capturing the local behavior and simulating various failure modes observed in the tests.