Methods For Ordinary Differential Equations Research Articles

On Newton and Lagrange Interpolation Method for first order ordinary differential equations

On Newton and Lagrange Interpolation Method for first order ordinary differential equations

Real, imaginary and complex branches of Lamb waves in p-type piezoelectric semiconductor GaAs plate: Numerical and experimental investigation

First, we experimentally measure the concentration of holes of the piezoelectric semiconductor (PSC) Gallium arsenide (GaAs) material based on the Hall effect measurement. We then numerically calculate the Lamb wave characteristics in a p-type PSC GaAs plate using this measured value of the concentration of holes (estimated at approximately p0=3.5431×1025m−3). To guarantee the coupling effect that makes the Lamb wave a piezo-active wave, the GaAs crystal is oriented in a specific orientation of (110) plane. Ordinary differential equation (ODE) method is employed to calculate the dispersion curves, 3D view of the mechanical displacements, 3D view of the electric potential and 3D view of the concentrations of holes. Results show that the appearance of the complex branches of the Lamb modes coincides with the vanishing of the imaginary part of the wavevector (Im(k1)) to zero, which is a very remarkable finding that has not been discussed in previous studies. Meanwhile, these complex branches start exactly at the point of Zero-group-velocity (ZGV). Moreover, the present work highlights the hole drift characteristics of PSC GaAs and provides a critical comparative discussion with those published by (Zhu et al., 2018) for the PSC ZnO plate. Accordingly, the “Screening effect” phenomenon that is based on the positive holes in the negative electric potential region is discussed. The present results provide a theoretical and fundamental guidance on the development of smart devices based on the PSC GaAs material.

Super implicit two-step collocation methods for ordinary differential equations

Super implicit two-step collocation methods for ordinary differential equations

Integro perturbation method for ordinary differential equations

Integro perturbation method for ordinary differential equations

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

On Approximate Matrix Factorization and TASE W-Methods for the Time Integration of Parabolic Partial Differential Equations

Linearly implicit methods for Ordinary Differential Equations combined with the application of Approximate Matrix Factorization (AMF) provide efficient numerical methods for the solution of large semi-discrete parabolic Partial Differential Equations in several spatial dimensions. Interesting particular subclasses of such linearly implicit methods are the so-called W-methods and the TASE W-methods recently introduced in González-Pinto et al. (Appl Numer Math, 188:129–145, 2023) with the aim of reducing the computational cost of the TASE Runge–Kutta methods in Bassenne et al. (J Comput Phys 424:109847, 2021) and Calvo et al. (J Comp Phys 436:110316, 2021). In this paper, we study the application of the AMF approach in combination with TASE W-methods. While for AMF W-methods the temporal order of consistency is immediately obtained from that of the underlying W-method, this property needs a more thorough analysis for the newly introduced AMF-TASE W-methods. For these latter methods it is described which are the additional order conditions to be fulfilled and it is shown that the parallel structure of the methods is crucial to retain the order of consistency of the underlying TASE W-method. Numerical experiments are presented in three spatial dimensions to assess the consistency result and to show that the proposed schemes are competitive with other well-known good performing AMF W-methods.

Forecasting dynamics by an incomplete equation of motion and an auto-encoder Koopman operator

The governing equation of mass–spring–damper dynamics usually consists of a known structure and undetermined nonlinear components. This work proposes a physics-informed method to learn the unknown component by modeling its evolution with an auto-encoder Koopman operator. The Koopman operator is well known for its global linearization of dynamics and sensitivity to spectral modes. The undetermined component in an oscillatory equation of motion (EOM) is first reconstructed by a time-delay neural network whose inputs stack past sequences of observables and the control. Then, the evolution of the undetermined component is globally linearized and propagated by the Koopman operator in a latent space created by an auto-encoder neural network. Ultimately, observables at the future timestep are forecasted by timestepping the original EOM. Uniting an incomplete EOM and the Koopman operator, the proposed physics-informed Koopman neural ordinary differential equation (PiKNODE) method shows competitive edges compared with several black- and grey-box benchmarks in four synthetic and experimental case studies. Additionally, an anomaly-detection technique based on the Jensen–Shannon distance is proposed to boost the practicality of the PiKNODE for structural health monitoring.

Continuous-discrete extended Kalman filtering based on the neural ordinary differential equations method

In bearings-only tracking (BOT), the accuracy of state estimation significantly diminishes when confronted with an unknown target motion model. Therefore, this study proposed a novel continuous-discrete filtering algorithm, the neural ordinary differential equations-based continuous-discrete extended Kalman filtering (NODE-CD-EKF). The approach utilizes the neural ordinary differential equation (NODE) to build the target's system model, which is then used in the time update to precisely predict the target state at the next time point within the continuous-discrete Kalman framework. The NODE concisely and accurately describes the motion model in the form of ordinary differential equations in a data-driven way. This approach addresses the limitations of traditional Kalman filters, which suffer from degraded performance due to a lack of a precise mathematical model of target states. Our method is evaluated and compared to existing multiple-model and other data-driven algorithms in simulations of BOT and another scenario using the Accumulated Root Mean Square Error (ARMSE) of position. The ARMSE of our method is only 16.30 % of the multiple-model methods at least. Overall, this approach outperforms other multiple-model and data-driven algorithms and possesses the potential for solving other nonlinear tasks in real-world applications.

Alleviated Shifted Gegenbauer Spectral Method for Ordinary and Fractional Differential Equations

This article introduces two numerical methods to address boundary value problems associated with secondorder and fractional differential equations. These methods employ two parameters related to shifted Gegenbauer polynomials as their basis functions. The process involves establishing a differentiation operational matrix for the shifted Gegenbauer polynomials. Subsequently, the initial/boundary value problems for ordinary and fractional differential equations are transformed into a system of equations through the Galerkin, collocation, and tau methods. The convergence analysis is ensured by leveraging theorems pertaining to the shifted Gegenbauer polynomials. To validate the accuracy of the approach, numerous numerical examples are presented.

Novel approach for solving higher-order differential equations with applications to the Van der Pol and Van der Pol–Duffing equations

BackgroundNumerical methods are used to solve differential equations, but few are effective for nonlinear ordinary differential equations (ODEs) of order higher than one. This paper proposes a new method for such ODEs, based on Taylor series expansion. The new method is a second-order method for second-order ODEs, and it is equivalent to the central difference method, a well-known method for solving differential equations. The new method is also simple to implement for higher-order differential equations. The proposed technique was applied to solve the Van der Pol and Van der Pol–Duffing equations. It is stable over a wide range of nonlinearity and produces accurate and reliable results. For the self-excitation Van der Pol equation, the proposed technique was applied with different values of nonlinear damping.ResultsThe results were compared with those obtained using the ODE15s solver in MATLAB. The two sets of results showed excellent agreement. For the forced Van der Pol–Duffing equation, the proposed technique was applied with different values of exciting force amplitude and frequency. It was found that for certain conditions, the solution obtained using the proposed technique differed from that obtained using ODE15s.ConclusionsThe solution obtained using the proposed technique showed good agreement with the solutions obtained using ODE45 and Runge–Kutta fourth order. The results show that the proposed approach is very simple to apply and produces acceptable error. It is a powerful and versatile tool for solving of high-order nonlinear differential equations accurately.

Impact of weak nonlocal nonlinearity on modulational instability and chirp-free soliton trains in parabolic law optical fiber

This study considers the generation and evolution of chirp-free soliton trains in a focusing cubic-quintic nonlinear optical fiber media (by utilizing experimental parameters), under the influence of weak nonlocal nonlinearity. Analysis of modulational instability (MI) reveals that the MI gain exist only in the anomalous dispersion regime, with the quintic nonlinear term increasing the maximum gain and sideband frequencies. The nonlocal nonlinearity parameter generally suppresses the impact of MI, because the maximum gain is greatly reduced with increase in nonlocality. By using the subsidiary ordinary differential equation method, stationary periodic soliton trains are obtained; with the intensity of the optical beam greatly reduced and frequency increased as nonlocality is gradually stepped up. However, an increase in nonlocality instead reduces the frequency of the traveling optical signals; derived via an efficient transformation method. Our results strongly suggest that nonlocal nonlinearity has potential applications in the control of laser beams; as nonlinear periodic optical signals can be easily transformed to very weak quasi-plane waves.

On the Solution of Blasius Boundary Layer Equations of Prandtl-Eyring Fluid Flow Past a Stretching Sheet

Objective: This paper investigates velocity profile for two-dimensional, incompressible, laminar forced convection flow of the fluid model for Prandtl-Eyring fluid past a stretching sheet in the presence of fluid parameters. Methods: The governing partial differential equation for the flow was transformed into non-linear ordinary differential equation by using the deductive one parameter group theoretic method and numerical solution of non-linear ordinary differential equation (ODE) is solved by MATLAB bvp4c solver. Findings: The solution of velocity profile obtained as a function of parameter and . The effect of the fluid parameter was discussed graphically. Novelty: The main goal of this article is to analyze boundary layer flow of Prandtl-Eyring fluid over a stretching surface. The conservation equations of mass, momentum are converted into non-linear ordinary differential equations along with boundary conditions using deductive one parameter group theoretic method and solved by MATLAB ODE solver. Comparisons with previously published works are made, and results show a high level of agreement. This type of research is applicable to extrusion, paper production, fiber glass production, hot rolling, condensation process, crystal growing, polymer sheets etc. Keywords: Boundary layer, laminar flow, Deductive one parameter Group theoretic method, Absolute invariant, Stretching Sheet, Prandtl-Eyring fluid

A new graph theoretic analytical method for nonlinear distributed order fractional ordinary differential equations by clique polynomial of cocktail party graph

In this paper, we presented a new analytical method for one of the rapidly emerging branches of fractional calculus, the distributed order fractional differential equations (DFDE). Due to its significant applications in modeling complex physical systems, researchers have shown profound interest in developing various analytical and numerical methods to study DFDEs. With this motivation, we proposed an easy computational technique with the help of graph theoretic polynomials from algebraic graph theory for nonlinear distributed order fractional ordinary differential equations (NDFODE). In the method, we used clique polynomials of the cocktail party graph as an approximation solution. With operational integration and fractional differentiation in the Caputo sense, the NDFODEs transformed into a system of algebraic equations and then solved by Newton–Raphson's method to determine the unknowns in the Clique polynomial approximation. The proficiency of the proposed Clique polynomial collocation method (CCM) is illustrated with four numerical examples. The convergence and error analysis are discussed in tabular and graphical depictions by comparing the CCM results with the results of existing numerical methods.

Efficient energy management in smart grids with finite horizon Q-learning

Efficient energy distribution in smart grids is an important problem driven by the need to manage increasing power consumption across the globe. This problem has been studied in the past using different frameworks including Markov Decision Processes (MDP) and Reinforcement Learning. However, existing algorithms in Reinforcement Learning Theory largely deal with infinite horizon decision making. On the other hand, smart grid problems are inherently finite horizon in nature and so can best be addressed using Finite Horizon algorithms. We therefore analyze the smart grid setup using a finite horizon MDP model and develop, for the first time, a finite horizon version of the popular Q-learning algorithm. We observe that our algorithm shows good empirical performance on the smart grid setting. We also theoretically analyze the full convergence of the algorithm by proving both the stability and the convergence of the same. Our analysis of stability and convergence of finite horizon Q-learning is based entirely on the ordinary differential equations (O.D.E) method. Apart from smart grids, we additionally demonstrate the performance of our algorithm on a setting of random MDPs indicating that the algorithm is more generally applicable and can be studied on other settings in the future.

Double diffusive effects on nanofluid flow toward a permeable stretching surface in presence of Thermophoresis and Brownian motion effects: A numerical study

AbstractThe present study explores the nanofluid boundary layer flow over a stretching sheet with the combined influence of the double diffusive effects of thermophoresis and Brownian motion effects. For the purpose of transforming nonlinear partial differential equations into the linear united ordinary differential equation method, the similarity transformation technique is used. The Runge–Kutta–Fehlberg method was used to solve the equations of flow, along with sufficient boundary conditions. The effect on hydrodynamic, thermal and solutes boundary layers of a number of related parameters is investigated and the effects are graphically displayed. In conclusion, a strong agreement between the current numerical findings and the previous literature results is sought.

Generalization to differential–algebraic equations of Lyapunov–Schmidt type reduction at Hopf bifurcations

The Lyapunov–Schmidt procedure, a well-known and powerful tool for the local reduction of nonlinear systems at bifurcation points or for ordinary differential equations (ODEs) at Hopf bifurcations, is extended to the context of strangeness-free differential–algebraic equations (DAEs), by generalizing the comprehensive presentation of the method for ODEs provided in the classical textbook by Golubitsky and Schaeffer (1985). The appropriate setting in the context of DAEs at Hopf bifurcations is first detailed, introducing suitable operators and addressing the question of appropriate numerical algorithms for their construction as well. The different steps of the reduction procedure are carefully reinterpreted in the light of the DAE context and detailed formulas are provided for systematic and rational construction of the bifurcating local periodic solution, whose stability is shown, likely to the ODE context, to be predicted by the reduced equations. As an illustrative example, a classical DAE model for an electric power system is considered, exhibiting both supercritical and subcritical Hopf bifurcations, demonstrating the prediction capability of the reduced system with regard to the global dynamics.

High-Gain-Observer-Based Predictive Output Feedback for Nonlinear Systems With Large Input-Delays

Abstract The predictor feedback has been demonstrated to be quite effective in large delay compensation. However, few researches in the field of predictor feedback for large delays focused on output feedback control (OFC). This paper develops the previous work to design high-gain-observer-based predictive output feedback for nonlinear systems with large delays. Two methods are employed for large delay compensation: the backstepping-based partial differential equation (PDE) method and the reduction-based ordinary differential equation (ODE) method. It appears that, for continuous-time control, the first method leads to simpler linear matrix inequality (LMI) conditions and deal with larger delays, whereas the second method is easily exploited for sampled-data implementation under continuous-time measurement. Lyapunov–Krasovskii method is presented to guarantee the exponential stability of the nonlinear systems under predictor-based controllers. Through a simulation example of pendulum, the proposed methods are demonstrated to be efficient when the input delays are too large for the system to be stabilized without a predictor.

常微分方程的多区域配置方法

常微分方程的多区域配置方法

Solutions of random ordinary differential equations with laplace transform - Adomian decomposition method

Combine the Laplace Adomian Decomposition method (LTADM) for non-linear random ordinary differential equations is utilised in a novel way to obtain accurate solutions.The iterative implementation of the Laplace transform decomposition is employed to approximate the solutions of ordinary differential equations with random components. We use a Normal distribution to analyse the parameters and initial conditions of ordinary differential equations with random components. Using the Mathematica 13.3 programme, the graphs of the approximate solutions and absolute error are presented. An investigation is conducted into the effects of the normal distribution on the results of random component differential equations. According to the results of the numerical investigations, this method is extremely effective. Two applications are presented as examples of how the proposed technique can be utilised to obtain analytical or numerical solutions for certain kinds of random differential equations in order to demonstrate its efficacy and potential.

Machine learning methods for autonomous ordinary differential equations

Machine learning methods for autonomous ordinary differential equations