Methods For Ordinary Differential Equations Research Articles

Super implicit two-step collocation methods for ordinary differential equations

Super implicit two-step collocation methods for ordinary differential equations

Super implicit two-step collocation methods for ordinary differential equations

Super implicit two-step collocation methods for ordinary differential equations

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

AbstractIn 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.

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

The Newtonian heating effect on MHD free convective boundary layer flow of magnetic nanofluids past a moving inclined plate

The effect of magnetic strength on the MHD free convection flow of nanofluids over a moving inclined plate with Newtonian heating is analyzed. The governing partial differential equations with Newtonian heating boundary conditions are transformed into a system of nonlinear coupled ordinary differential equations (ODEs) by using similarity transformations. The Keller Box method was used as a solvation method for ODEs. The skin friction and Nusselt number are evaluated analytically as well as numerically in a tabular form. Numerical results for velocity and temperature are shown graphically for various parameters of interest, and the physics of the problem is well explored. The significant findings of this study are promoting an angle of an aligned magnetic field, magnetic strength parameter, the angle of inclination parameter, local Grashof number, the volume fraction of nanoparticles, and Newtonian heating parameter. The result shows that the moving inclined plate in the same direction increases the skin friction coefficient and reduces the Nusselt number. It is also observed that the velocity of moving an inclined plate with the flow is higher compared to the velocity of moving an inclined plate against the flow. The temperature of a moving inclined plate with the flow is decreased much quicker than the temperature of a moving inclined plate against the flow. The other noteworthy observation of this study demonstrates that the Nusselt number in the Newtonian heating parameter shows that Fe3O4-kerosene is better than Fe3O4-water.

NEW IMPLICIT MULTISTEP METHOD FOR ODE’S

A new class of multistep methods for stiff ordinary differential equations is presented. The method is based on the transformation of the arguments of the original system into purely algebraic combinations of the solutions of previous steps. The scheme differs from the classical multistep methods in that the state variables, instead of functions of them, are approximated by means of linear combination of previous steps. A family of coefficients is deduced from a combined analysis of convergence order and stability. Numerical results are presented for three test problems.

Mathematical Modeling of Dynamic Supply Chains Subject to Demand Fluctuations

This research work aims to develop the mathematical modeling for a class of dynamic supply chains. Demand fluctuation corresponds to product demand volatility, which increases or decreases over a given time frame. Industrial engineering practitioners should consider the function that applied mathematical modeling plays in providing approximations of solutions that may be used in simulations and technical implementations at the strategic, tactical, and operational levels of an organization. In order to achieve proper results, two mathematical models are presented in this paper: In addition to a finite-dimensional system of Ordinary Differential Equations (ODEs) for coupled dynamic pricing, production rate, and inventory level, which properly integrates Lyapunov stability analysis of the dynamical system and simulations, there is an infinite-dimensional Partial Differential Equation (PDE) production level modeling system available. Infinite and finite-dimensional systems incorporate a dynamic pricing approach in the mathematical modeling. The main research goal of this work is to explore the dynamic nature of supply chains applying PDE and ODE methods, with proper analytical analysis and simulations for both systems.

Frobenius methods for analytic second order linear partial differential equations

The main subject of this text is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods inspired by the Frobenius method for second order linear ordinary differential equations. We introduce a notion of Euler type partial differential equation. To such a PDE we associate an indicial conic, which is an affine plane curve of degree two. Then comes the concept of regular singularity and finally convergence theorems, which must necessarily take into account the type of PDE (parabolic, elliptical or hyperbolic) and a nonresonance condition. This condition gives a new geometric interpretation of the original condition between the roots of the original Frobenius theorem for second order ODEs. The interpretation is something like, a certain reticulate has or not vertices on the indexical conic. Finally, we retrieve the solution of all the classical PDEs by this method (heat diffusion, wave propagation and Laplace equation), and also increase the class of those that have explicit algorithmic solution to far beyond those admitting separable variables. The last part of the text is dedicated to the construction of PDE models for the classical ODEs like Airy, Legendre, Laguerre, Hermite and Chebyshev by two different means. One model is based on the requirement that the restriction of the PDE to lines through the origin must be the classical ODE model. The second is based on the idea of having symmetries on the PDE model and imitating the ODE model. We study these PDEs and obtain their solutions, obtaining for the framework of PDEs some of the classical results, like existence of polynomial solutions (Laguerre, Hermite and Chebyshev polynomials).

Phase Function Methods for Second Order Inhomogeneous Linear Ordinary Differential Equations

Phase Function Methods for Second Order Inhomogeneous Linear Ordinary Differential Equations

Variable substitution methods in the solution of stochastic ordinary differential equations and their applications

Abstract In this paper, we first propose numerical solution methods for stochastic ordinary differential equations by using the two-step Maruyama method and Euler-Maruyama method in variable substitution, and analyze the mean-square compatibility, mean-square convergence and mean-square linear stability of the corresponding numerical methods, respectively. Finally, 10,000 times value experiments are conducted to verify the convergence accuracy and stability of the variable substitution methods. The results show that the Euler method simulates this equation when taking steps h = 0.05, β = 5 and σ = 2 for numerical experiments, and the numerical results obtained are convergent but unstable. On the other hand, the Euler-Maruyama method with variable substitution is consistent with the real solution. It shows that variable substitution is an important method for solving stochastic ordinary differential equations with high convergence and stability, which is of great significance for the solution of stochastic ordinary differential equations.