First-order Ordinary Differential Equations Research Articles

A Six-Point y-Function Hybrid Block Method for Direct Solution of Third Order Ordinary Differential Equations

In this article, a new continuous numerical method was developed for the numerical integration of general third-order initial value problems (IVPs) of ordinary differential equations (ODEs). The method was derived by adopting interpolation and collocation approach using a combination of power series and exponential functions as the basis function for the approximate solution to the ODEs. The approximate solution was interpolated at both nodal and off-nodal points while the differential systems from the approximate solution was collocated at all nodal points to obtain the set of methods with given step-numbers. This derived approach ensured that the hybrid points are obtained at the y-values of the methods. This class of hybrid linear multi-step method satisfied the conditions for usability and accuracy of any given method. These conditions were confirmed by testing the consistency, stability and convergence of the developed method. The method was applied to solve directly linear and non-linear third order ODEs without reduction to system of first-order ordinary differential equations. Results from the computation compared favorably with some existing numerical methods in literature with comparable properties to confirm the superiority of the newly developed method.

Symmetry Reductions of the (1 + 1)-Dimensional Broer–Kaup System Using the Generalized Double Reduction Method

The generalized theory of the double reduction of systems of partial differential equations (PDEs) based on the association of conservation laws with Lie–Bäcklund symmetries is one of the most effective algorithms for performing symmetry reductions of PDEs. In this article, we apply the theory to a (1 + 1)-dimensional Broer–Kaup (BK) system, which is a pair of nonlinear PDEs that arise in the modeling of the propagation of long waves in shallow water. We find symmetries and construct six local conservation laws of the BK system arising from low-order multipliers. We establish associations between the Lie point symmetries and conservation laws and exploit the association to perform double reductions of the system, reducing it to first-order ordinary differential equations or algebraic equations. Our paper contributes to the broader understanding and application of the generalized double reduction method in the analysis of nonlinear PDEs.

A new approach to determine occupational accident dynamics by using ordinary differential equations based on SIR model

The motivation of this study is to develop and establish an occupational accident dynamical model (OA model) based on Susceptible-Infected-Recovered framework. In order to investigate the dynamics of the OA model, monthly occupational accident data from Turkey between 2013 and 2020 has been selected as dataset. The OA model is defined by a coupled first-order ordinary nonlinear differential equation with four variables. In addition, the relationships between these variables are described with ten parameters. The OA model’s characterization of the equilibrium points is analyzed by investigating the behaviors of these points according to the eigenvalues derived from the Jacobian matrix. Also, the stability of these points is obtained according to the eigenvalues. These results show the behavior of the system near equilibrium points. After that, the reproduction number is computed by using the next-generation matrix method. The calculated reproduction number for given parameters reveals that the OA model is unstable. The OA model is numerically solved in 96 steps, with a time interval of 1 month, using the ODE45 Matlab routine based on the explicit Runge–Kutta algorithm. In addition, a modified OA model is developed by adding the Occupational Health and Safety (OHS) re-training parameter to the OA model to observe a reducing effect on occupational accident numbers. The main results of this study provide a new approach about the future estimation of the number of occupational accidents. Furthermore, through the comparison of numerical results from both models, the study demonstrates that national safety policies, particularly those enhancing the efficacy of OHS training, can effectively mitigate accidents.

A fast search method of buckling load for telescopic boom structure

The telescopic boom structures are extensively utilized in engineering applications involving large mobile cranes, aerial work platforms of significant height, and similar mechanical devices. These are predominantly fabricated using solid-webbed box types, latticed truss designs, or a combination thereof. Under load, they exhibit pronounced geometric nonlinear effects, with the relationship between load and displacement demonstrating marked nonlinear characteristics. This paper focuses on the geometric nonlinear modeling of slender multi flexible beam structures. The overall structural system is divided into several substructures, and the concept of multi flexible system modeling is borrowed to establish a follow-up connected body foundation on each substructure. By decomposing the node displacement and rotation in the substructure, the large displacement and large rotation of the node are decomposed into rigid body motion of the connected base and small displacement and small rotation relative to the connected base, effectively representing the large displacement and large rotation of the substructure as rigid body motion of the connected base. A modeling method for the geometric nonlinear analysis of slender and flexible multibody beam structures is proposed. By decomposing the nodal displacement and rotation within the substructures, the large displacement and rotation of the nodes are broken down into the rigid body motion of the body-fixed coordinate and small displacement and rotation relative to the body-fixed coordinate. This approach establishes the application conditions for calculating the structure’s virtual power of deformation using traditional beam elements with linear strain. A method for solving the instability load of slender and flexible multibody structures is proposed. This method integrates the characteristics of the system’s nonlinear equilibrium equations with respect to load, by deriving the system equations with respect to a load increment control parameter. This paper converts nonlinear equilibrium equations into first-order ordinary differential equation (ODE). The method proposed in this paper provides a more efficient solution for the buckling load of the telescopic boom. It can be used to systematically and quickly calculate the structure of the telescopic boom.

A qualitative analysis of the artificial neural network model and numerical solution for the nanofluid flow through an exponentially stretched surface

This article aims to analyze the two-dimensional (2D) nanofluid (Ag/C2H6O2) flow past an exponentially stretched sheet. The magnetic field impact, heat source/sink, and convection in the thermal profile are taken into account. The complexity of the problem is reduced by introducing a dimensionless group of functions. The reduced model is transformed into a system of first-order ordinary differential equations (ODEs). This system is further analyzed with the artificial neural network (ANN), which is trained using the Levenberg–Marquardt algorithm. The whole dataset is sub divided into three parts: training (70%), validation (15%), and testing (15%). The impact of nonlinear heat source/sink parameter, magnetic parameter, volume fraction of nanoparticles, and Prandtl number is displayed through graphs. The heat source, volume fraction, and the Prandtl number cause an increase in the thermal profile with its larger values. The magnetic parameter causes a decline in both the thermal and momentum boundary layers with its higher values. The analysis shows that the thermal energy profile is enhanced with the larger values of the volume fraction of silver nanoparticles and heat source. For each case study, the residual error (RE), regression line, and validation of the results are presented. The performance of the proposed methodology is numerically tabulated for the nanoparticle volume fraction shown in Table 3, where the minimum absolute error (AE) is 5.3373e−11 at ϕ=0.05. Based on this, we recommend ϕ=0.05 for better performance. The AEs for the ANN and bvp4c are computed for the state variables in Tables for the magnetic parameter M=5,10, and 15. These tables show the overall performance of the ANN and further validate the present study. We have also validated the results of the ANN through the mean squared error graphically, where the accuracy of the proposed methodology is proven.

Improved State-Space Approach Based on Lumped Mass Matrix for Transient Analysis of Large-Scale Locally Nonlinear Structures

Due to the assumption of acceleration variation in traditional step-by-step integration methods such as Newmark, sufficiently small time steps are required to ensure numerical stability and accuracy in dynamic systems. In contrast, the state-space approach, based on piecewise interpolation of discrete load functions, does not rely on predetermined acceleration assumptions and has demonstrated high efficiency in terms of stability and accuracy. The original state-space method requires the calculation of the inverse of the structural mass in the transition matrix. However, when a lumped mass matrix is used, this computation renders the entire mass matrix singular, resulting in an invalid solution expression. To address this issue, this study proposes an improved state-space approach for the transient analysis of large-scale structural systems with local nonlinearities. In this approach, a nonlinear force corrector is introduced as an external force term applied to the linear elastic system to account for the nonlinear behavior of locally yielding components. Consequently, the original nonlinear dynamic system can be transformed into an equivalent linear elastic transient system. Furthermore, based on the lumped mass matrix, a first-order ordinary differential state-space equation for such an equivalent linear elastic transient system is derived. Simulation results from three transient system examples show that the state-space approach outperforms the Newmark method in terms of accuracy and stability for dynamic systems characterized by high frequency and low damping. The prediction results show that the state-space approach appears to be insignificantly affected by the choice of the consistent or lumped mass matrix. The numerical results show that the root-mean-square errors between the consistent and lumped matrices in the top displacement time histories of a 15-storey plane frame under various seismic intensities are all less than 1%, and in the base reaction time histories responses the discrepancies are only about 0.5%, indicating that the use of lumped mass matrices is quite reliable. When many nodes or degrees of freedom have no assigned mass, the dimensionality of the state-space equation can be significantly reduced using the lumped mass approach. Therefore, the simulation of large-scale systems can be simplified by employing the improved state-space approach with lumped mass matrices, yielding results nearly identical to those obtained using traditional methods. In conclusion, the improved state-space approach has great potential for the simulation of transient behavior in large-scale systems with local nonlinearities.

On a priori estimate of periodic solutions of a system of nonlinear ordinary differential equations of the second order

The question of a priori estimation of periodic solutions for a system of non-linear ordinary differential equations of the second order with a distinguished main positively homogeneous part is investigated. In this question, the well-known methods for deriving an a priori estimation of periodic solutions for similar systems of first-order ordinary differential equations are not directly applicable.Combining these methods with the idea of qualitative research of singularly perturbed ordinary differential equations, conditions are found that provide an a priori estimation of periodic solutions for the system of second-order equations.The conditions for the a priori estimation are formulated in terms of the properties of the main positively homogeneous part of the system of equations. The existence of periodic solutions is proved to be invariant under a continuous change of the main positively homogeneous part and the conditions of a priori estimation are preserved. Based on the results obtained, in the future, it is possible to investigate the existence of periodic solutions.

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.

Retarded and advanced Green functions for first-order differential equations

Retarded and advanced Green functions are examined through the lens of linear first-order ordinary differential equations. They are constructed for a finite interval with emphasis on the imposition of homogeneous Dirichlet boundary conditions at the interval ends. It is shown that both types of Green functions exhibit the property of facilitating solutions.

Частица Штюкельберга в электрическом поле, решения с цилиндрической симметрией

In the present paper, the system of 11 equations for massive Stueckelberg particle is studied in presence of the external uniform electric field. We apply covariant formalism according to the general tetrad approach by Tetrode-Weyl-Fock-Ivanenko specified for cylindrical coordinates. After separating the variables, we derive the system of the first-order differential equations in partial derivatives with respect to coordinates (r, z). To resolve this system, we apply the Fedorov- Gronskiy method, thereby we consider the 11-dimensional spin operator and find on this base three projective operators, which permit us to expand the complete wave function in the sum of three parts. Besides, according to the general method, dependence of each projective constituent on the variable r should be determined by only one function. Also, in accordance with the general method we impose the first-order constraints which permit us to transform all differential equations in partial derivatives with respect to coordinates (r, z) into the system of 11 first-order ordinary differential equations in the variable z. The last system is solved in terms of confluent hypergeometric functions. In total, four independent types of solutions have been constructed, in contrast to the case of the ordinary spin 1 particle described by Daffin- Kemmer equation when only three types of solutions are possible.

Investigation of thermal radiation and joule heating effects on variable viscosity Casson nanofluid flow over a stretching sheet in Darcy–Forchheimer porous medium

Nanofluids possess unique thermal and rheological properties that offer significant advantages in enhancing heat transfer efficiency in various applications, such as heat exchangers, cooling systems, and microdevices. In this study, we investigate the combined effects of thermal radiation, Joule heating, variable viscosity, and a two-phase model on the flow behavior and heat transfer characteristics of a magnetohydrodynamic Casson nanofluid flowing over a stretching sheet in Darcy-Forchheimer porous media. The governing partial differential equations are converted into a system of first-order ordinary differential equations numerically solved using the bvp5c package in MATLAB. Consequently, the result indicates that the velocity profile increases with thermal radiation, variable viscosity, and viscous dissipation, decreasing with the magnetic field, permeability index, and Forchheimer coefficient. Also, the temperature profile shows an increase with the permeability index, viscous dissipation, and thermal radiation, but a decrease with the Forchheimer Coefficient, variable viscosity, and magnetic field. Additionally, the concentration profile exhibits an increase with the Forchheimer coefficient and permeability index, and variable viscosity while it decreases with viscous dissipation, magnetic field, and thermal radiation. Analyzing the skin friction coefficient reveals an increase with magnetic field and thermal radiation and a decrease with permeability index and Forchheimer coefficient as well as variable viscosity. Moreover, the local heat transfer rate demonstrates an increase with the variable viscosity, and magnetic field as well as the Forchheimer coefficient and permeability index. At the same time, it decreases with the viscous dissipation. Furthermore, the local mass transfer rate displays an increase with the magnetic field and thermal radiation and viscous dissipation, while it decreases with the variable viscosity as well as permeability index, viscous dissipation, and Forchheimer coefficient. Finally, the current findings are compared with that of the existing literature and a sound agreement was established.

Macroscopic fundamental diagram with volume–delay relationship: Model derivation, empirical validation and invariance property

This paper presents a macroscopic fundamental diagram model with volume–delay relationship (MFD-VD) for road traffic networks, by exploring two new data sources: license plate cameras (LPCs) and road congestion indices (RCIs). We derive a first-order, nonlinear and implicit ordinary differential equation involving the network accumulation (the volume) and average congestion index (the delay), and use empirical data from a 266 km2 urban network to fit an accumulation-based MFD with R2>0.9. The issue of incomplete traffic volume observed by the LPCs is addressed with a theoretical derivation of the observability-invariant property: The ratio of traffic volume to the critical value (corresponding to the peak of the MFD) is independent of the (unknown) proportion of those detected vehicles. Conditions for such a property to hold are discussed in theory and verified empirically. This offers a practical way to estimate the ratio-to-critical-value, which is an important indicator of network saturation and efficiency, by simply working with a finite set of LPCs. The significance of our work is the introduction of two new data sources widely available to study empirical MFDs, as well as the removal of the assumptions of full observability, known detection rates, and spatially uniform sensors, which are typically required in conventional approaches based on loop detector and floating car data.

Truncated Taylor Solvers for Simple First-Order Ordinary Differential Equations

Certain exact solutions of first-order ordinary differential equations naturally become corresponding solvers for simple firstorder ordinary differential equations whose form is the equality of a twice continuously differentiable function of the dependent variable to the derivative with respect the independent variable of the dependent variable. When the twice continuously differentiable function of the dependent variable is replaced by its truncated Taylor expansions through second order about its initial value, the resulting first-order ordinary differential equations have exact solutions that naturally become corresponding solvers for those simple first-order ordinary differential equations

A Study of Movable Singularities in Non-Algebraic First-Order Autonomous Ordinary Differential Equations

We studied the movable singularities of solutions of autonomous non-algebraic first-order ordinary differential equations in the form of y′=I(y(t)) and y′=I1(y(t))+I2(y(t))+⋯+In(y(t)), aiming to prove that all movable singularities of all complex solutions of these equations are at most algebraic branch points. This study explores the use of the constructing triangle method to analyze complex solutions of autonomous non-algebraic first-order ordinary differential equations. For complex solutions in the form of y=w+iv, we treat the constructing triangle method as a way to construct a right-angled triangle in the complex plane, with the lengths of the adjacent sides being w and v. We use the definitions of the trigonometric functions sin and cos (the ratio of the adjacent side to the hypotenuse) to represent the trigonometric functions of complex solutions y=w+iv. Since the movable singularities of the inverse functions of trigonometric functions are easy to analyze, the properties of the movable singularities of the complex solutions are then easy to deal with.

A Comparative Analysis of Two Semi Analytic Approaches in Solving Systems of First-Order Differential Equations

The resolution of systems of first-order ordinary differential equations (ODEs) stands as a pivotal pursuit with extensive implications across scientific and engineering domains. In tackling this fundamental task, this study undertakes a rigorous comparative assessment of two semi-analytic methodologies, the Variational Iterative Method (VIM) and the New Iterative Method (NIM). Motivated by the need to address a critical research gap, our investigation delves into these approaches' relative merits and demerits. Firstly, it conducts a comprehensive examination of VIM, a well-established method, juxtaposed with NIM, a relatively unexplored approach, to uncover their comparative strengths and limitations. Secondly, the study contributes to the existing knowledge in numerical methods for ODEs by shedding light on essential performance characteristics, including convergence properties, computational efficiency, and accuracy, across a diverse array of ODE systems. Through meticulous numerical experimentation, we not only reveal practical insights into the efficacy of VIM and NIM but also bridge a significant knowledge gap in the field of numerical ODE solvers. Our findings highlight VIM as the more effective method, thus advancing our understanding of semi-analytic approaches for solving ODE systems and furnishing valuable guidance for practitioners and researchers in selecting the most suitable method for their specific applications

Pump System Model Parameter Identification Based on Experimental and Simulation Data

In this paper, the entire downhole fluid-sucker rod-pump system is replaced with a viscoelastic vibration model, namely a third-order differential equation with an inhomogeneous forcing term. Both Kelvin’s and Maxwell’s viscoelastic models can be implemented along with the dynamic behaviors of a mass point attached to the viscoelastic model. By employing the time-dependent polished rod force measured with a dynamometer as the input to the viscoelastic dynamic model, we have obtained the displacement responses, which match closely with the experimental measurements in actual operations, through an iterative process. The key discovery of this work is the feasibility of the so-called inverse optimization procedure, which can be utilized to identify the equivalent scaling factor and viscoelastic system parameters. The proposed Newton–Raphson iterative method, with some terms in the Jacobian matrix expressed with averaged rates of changes based on perturbations of up to two independent parameters, provides a feasible tool for optimization issues related to complex engineering problems with mere information of input and output data from either experiments or comprehensive simulations. The same inverse optimization procedure is also implemented to model the entire fluid delivery system of a very viscous non-Newtonian polymer modeled as a first-order ordinary differential equation (ODE) system similar to the transient entrance developing flow. The convergent parameter reproduces transient solutions that match very well with those from fully fledged computational fluid dynamics models with the required inlet volume flow rate and outlet pressure conditions.

Arbitrary High Order ADER-DG Method with Local DG Predictor for Solutions of Initial Value Problems for Systems of First-Order Ordinary Differential Equations

Arbitrary High Order ADER-DG Method with Local DG Predictor for Solutions of Initial Value Problems for Systems of First-Order Ordinary Differential Equations

Inclined magnetized infinite shear rate viscosity of non-Newtonian tetra hybrid nanofluid in stenosed artery with non-uniform heat sink/source

Abstract Many scholars performed the analysis by using the non-Newtonian fluids based on the nano and hybrid nano particles in blood arteries to investigate the heat transport for cure in several diseases. These performances are presented to investigate the blood flow behaviour with extended form of the novel tetra hybrid Das and Tiwari nanofluid system attached by the Carreau fluid. The assessment of energy transport has been achieved based on the thermal radiation, heat source/sink, Joule heating, and viscous dissipation. The obtained partial differential equation from physical problem is transformed into ordinary differential equations (ODEs) by using the similarity variables. Furthermore, system of nonlinear ODEs attached with boundary conditions are transported into the system of first-order ODEs with initial conditions. For the numerical solution of obtained ODEs, the numerical solutions have been performed based on the RK method. The numerical results are plotted through figures, tables, and statistical graphs. Magnetic forces and inclined magnetic effects are caused to reduce velocity of blood. Temperature of blood within the arteries is increased by increasing the parameter of thermal radiation.

A dynamical systems formulation for inhomogeneous LRS-II spacetimes

We present a dynamical system formulation for inhomogeneous LRS-II spacetimes using the covariant 1+1+2 decomposition approach. Our approach describes the LRS-II dynamics from the point of view of a comoving observer. Promoting the covariant radial derivatives of the covariant dynamical quantities to new dynamical variables and utilizing the commutation relation between the covariant temporal and radial derivatives, we were able to construct an autonomous system of first-order ordinary differential equations along with some purely algebraic constraints. Using our dynamical system formulation we found several interesting features in the LRS-II phase space with dust, one of them being that the homogeneous solutions constitute an invariant submanifold. For the particular case of LTB, we were also able to recover the previously known result that an expanding LTB tends to Milne in the absence of a cosmological constant, providing a potential validation of our formalism.

Hamiltonian formalism for Bose excitations in a plasma with a non-Abelian interaction I: Plasmon – hard particle scattering

Hamiltonian theory for collective longitudinally polarized gluon excitations (plasmons) interacting with classical high-energy test color-charged particle propagating through a high-temperature gluon plasma is developed. A generalization of the Lie-Poisson bracket to the case of a continuous medium involving bosonic normal field variable ak⁎a and a non-Abelian color charge Qa is performed and the corresponding Hamilton equations are presented. The canonical transformations including simultaneously both bosonic degrees of freedom of the soft collective excitations and degree of freedom of hard test particle connecting with its color charge in the hot gluon plasma are written out. A complete system of the canonicity conditions for these transformations is derived. The notion of the plasmon number density Nkaa1′, which is a nontrivial matrix in the color space, is introduced. An explicit form of the effective fourth-order Hamiltonian describing the elastic scattering of a plasmon off a hard color particle is found and the self-consistent system of Boltzmann-type kinetic equations taking into account the time evolution of the mean value of the color charge of the hard particle is obtained. On the basis of these equations, a model problem of the interaction of two infinitely narrow wave packets is considered. A system of nonlinear first-order ordinary differential equations defining the dynamics of the interaction of the colorless Nkl and color Wkl components of the plasmon number density is derived. The problem of determining the third- and fourth-order coefficient functions entering into the canonical transformations of the original bosonic variable ak⁎a and color charge Qa is discussed. With the help of the coefficient functions obtained, a complete effective amplitude of the elastic scattering of plasmon off hard test particle is written out.