Methods For Ordinary Differential Equations Research Articles

A Brief Discussion on the Research Progress of the Theory of Dynamic Vibration Reduction with Time Delay

In practical application, the phenomenon of time delay inevitably exists in all kinds of systems. Although the time delay is small, it has a certain influence on the stability and control performance of the control system. In this paper, the rise and development of the time-delay dynamic vibration reduction theory are summarized. Since the time-delay vibration reduction technology was proposed, scholars have conducted a lot of theoretical research, and the research field has expanded from the linear system when it was first proposed to the nonlinear system, and achieved good research results. At present, the main research methods of linear or nonlinear ordinary differential equations with time delay can be divided into time domain method and frequency domain method. The dynamics research of this kind of system mainly focuses on stability analysis, perturbation technology, Holp bifurcation, chaos and so on. The research shows that both linear and nonlinear systems have good application value in vehicle engineering. By selecting reasonable time-delay control parameters, the vehicle body vibration can be greatly suppressed.

Numerical Investigation on System of Ordinary Differential Equations Absolute Time Inference with Mathematica®

The purpose of this research is to perform a comparative numerical analysis of an efficient numerical methods for second-order ordinary differential equations, by reducing the second-order ODE to a system of first-order differential equations. Then we obtain approximate solutions to the system of ODE. To validate the accuracy of the algorithm, a comparison between Euler’s method and the Runge-Kutta method or order four was carried out and an exact solution was found to verify the efficiency, accuracy of the methods. Graphical representations of the parametric plots were also presented. Time inference analysis is taken to check the time taken to executes the algorithm in Mathematica®12.2.0. The obtained approximate solution using the algorithm shows that the Runge-Kutta method of order four is more efficient for solving system of linear ordinary differential equations.

Numerical integrators for Lagrangian oceanography

Abstract. A common task in Lagrangian oceanography is to calculate a large number of drifter trajectories from a velocity field precalculated with an ocean model. Mathematically, this is simply numerical integration of an ordinary differential equation (ODE), for which a wide range of different methods exist. However, the discrete nature of the modelled ocean currents requires interpolation of the velocity field in both space and time, and the choice of interpolation scheme has implications for the accuracy and efficiency of the different numerical ODE methods. We investigate trajectory calculation in modelled ocean currents with 800 m, 4 km, and 20 km horizontal resolution, in combination with linear, cubic and quintic spline interpolation. We use fixed-step Runge–Kutta integrators of orders 1–4, as well as three variable-step Runge–Kutta methods (Bogacki–Shampine 3(2), Dormand–Prince 5(4) and 8(7)). Additionally, we design and test modified special-purpose variants of the three variable-step integrators, which are better able to handle discontinuous derivatives in an interpolated velocity field. Our results show that the optimal choice of ODE integrator depends on the resolution of the ocean model, the degree of interpolation, and the desired accuracy. For cubic interpolation, the commonly used Dormand–Prince 5(4) is rarely the most efficient choice. We find that in many cases, our special-purpose integrators can improve accuracy by many orders of magnitude over their standard counterparts, with no increase in computational effort. Equivalently, the special-purpose integrators can provide the same accuracy as standard methods at a reduced computational cost. The best results are seen for coarser resolutions (4 and 20 km), thus the special-purpose integrators are particularly advantageous for research using regional to global ocean models to compute large numbers of trajectories. Our results are also applicable to trajectory computations on data from atmospheric models.

A discontinuous Galerkin method for systems of stochastic differential equations with applications to population biology, finance, and physics

In this paper, we propose a discontinuous Galerkin (DG) method for systems of stochastic differential equations (SDEs) driven by m-dimensional Brownian motion. We first construct a new approximate system of SDEs on each element using whose converges to the solution of the original system. The new system is then discretized using the standard DG method for deterministic ordinary differential equations (ODEs). For the case of additive noise, we prove that the proposed scheme is convergent in the mean-square sense. Our numerical experiments suggest that our results hold true for the case of multiplicative noise as well. Several linear and nonlinear test problems are presented to show the accuracy and effectiveness of the proposed method. In particular, the proposed scheme is illustrated by considering different examples arising in population biology, physics, and mathematical finance.

Traveling wave with beta derivative spatial-temporal evolution for describing the nonlinear directional couplers with metamaterials via two distinct methods

This work is reported the analytical solutions for describing the nonlinear directional couplers with metamaterials by including spatial–temporal fractional beta derivative evolution. The auxiliary ordinary differential equation method and the generalized Riccati method with the properties of beta derivative are implemented to secure such solutions. The solutions are obtained in the new forms by involving of some useful mathematical functions. The constraint conditions among the traveling wave parameters are evaluated. Some of the obtained solutions are presented graphically to illustrate the effectiveness of beta derivative parameter and mathematical techniques. It is investigated that the amplitudes of soliton are increased with the increase of fractional beta derivative parameter. The obtained results would be very useful to examine and understand the physical issues in nonlinear optics, especially in twin-core couplers with optical metamaterials.

Sparse Proximal Reinforcement Learning via Nested Optimization

We consider the tasks of feature selection and policy evaluation based on linear value function approximation in reinforcement learning problems. High-dimension feature vectors and limited number of samples can easily cause over-fitting and computation expensive. To prevent this problem, ${\ell _{1}}$ -regularized method obtains sparse solutions and thus improves generalization performance. We propose an efficient ${\ell _{1}}$ -regularized recursive least squares-based online algorithm with ${O}$ ( ${n} ^{{2}}$ ) complexity per time-step, termed ${\ell _{1}}$ -RC. With the help of nested optimization decomposition, ${\ell _{1}}$ -RC solves a series of standard optimization problems and avoids minimizing mean squares projected Bellman error with ${\ell _{1}}$ -regularization directly. In ${\ell _{1}}$ -RC, we propose RC with iterative refinement to minimize the operator error, and we propose an alternating direction method of multipliers with proximal operator to minimize the fixed-point error. The convergence of ${\ell _{1}}$ -RC is established based on ordinary differential equation method and some extensions are also given. In empirical computations, some state-of-the-art ${\ell _{1}}$ -regularized methods are chosen as the baselines, and ${\ell _{1}}$ -RC are tested in both policy evaluation and learning control benchmarks. The empirical results show the effectiveness and advantages of ${\ell _{1}}$ -RC.

Generalised Modified Taylor Series Approach of Developing k-step Block Methods for Solving Second Order Ordinary Differential Equations

Various algorithms have been proposed for developing block methods where the most adopted approach is the numerical integration and collocation approaches. However, there is another conventional approach known as the Taylor series approach, although it was utilised at inception for the development of linear multistep methods for first order differential equations. Thus, this article explores the adoption of this approach through the modification of the aforementioned conventional Taylor series approach. A new methodology is then presented for developing block methods, which is a more accurate method for solving second order ordinary differential equations, coined as the Modified Taylor Series (MTS) Approach. A further step is taken by presenting a generalised form of the MTS Approach that produces any k-step block method for solving second order ordinary differential equations. The computational complexity of this approach after being generalised to develop k-step block method for second order ordinary differential equations is calculated and the result shows that the generalised algorithm involves less computational burden, and hence is suitable for adoption when developing block methods for solving second order ordinary differential equations. Specifically, an alternate and easy-to-adopt approach to developing k-step block methods for solving second order ODEs with fewer computations has been introduced in this article with the developed block methods being suitable for solving second order differential equations directly.

Multistage Optimal Homotopy Asymptotic Method for the Nonlinear Riccati Ordinary Differential Equation in Nonlinear Physics

Multistage Optimal Homotopy Asymptotic Method for the Nonlinear Riccati Ordinary Differential Equation in Nonlinear Physics

On approximate implicit Taylor methods for ordinary differential equations

An efficient approximate version of implicit Taylor methods for initial-value problems of systems of ordinary differential equations (ODEs) is introduced. The approach, based on an approximate formulation of Taylor methods, produces a method that requires less evaluations of the function that defines the ODE and its derivatives than the usual version. On the other hand, an efficient numerical solution of the equation that arises from the discretization by means of Newton’s method is introduced for an implicit scheme of any order. Numerical experiments illustrate that the resulting algorithm is simpler to implement and has better performance than its exact counterpart.

An efficient numerical method for fractional ordinary differential equations based on exponentially decreasing random memory on uniform meshes

This paper proposes a new numerical method to solve non-linear fractional ordinary differential equations (FODEs) of the form Dγu(t) = f(t, u(t)), with initial conditions u(0) = u0. Here, f(t,u) is a continuous function, 0

Generalized Adams method for solving fractional delay differential equations

Based on fractional generalized Adams methods, a numerical method is constructed for solving fractional delay differential equations. The convergence of the method is analyzed in detail. The stability of the fractional generalized Adams methods for fractional ordinary differential equations is generalized to a general framework. Under such framework, the linear stability of the method is studied for fractional delay differential equations. Numerical experiments confirm the convergence and the stability of the method.

Interaction of complex short wave envelope and real long wave described by the coupled Schrödinger–Boussinesq equation with variable coefficients and beta space fractional evolution

This work deals with the interaction of complex short wave envelope and real long wave described by the coupled Schrödinger–Boussinesq equation with variable-coefficients and beta space fractional evolution. The above interacting wave phenomena are studied by evaluating new non-autonomous analytical solutions of the considered coupled equation. The solutions of the equations are obtained by modifying the auxiliary ordinary differential equation method and taking the conformable beta derivative properties into account. The behaviors of the resonance nonautonomous structures are discussed with physical insight for diverse consideration of the chance functions of time-dependent in the solutions.

Viral disease spreading in grouped population

Background and ObjectiveThe currently active COVID-19 pandemic has increased, among others, public interest in the computational techniques enabling the study of disease-spreading processes. Thus far, numerous approaches have been used to study the development of epidemics, with special attention paid to the identification of crucial elements that can strengthen or weaken the dynamics of the process. The main thread of this research is associated with the use of the ordinary differential equations method. There also exist several approaches based on the analysis of flows in the Cellular Automata (CA) approach. MethodsIn this paper, we propose a new approach to disease-spread modeling. We start by creating a network that reproduces contacts between individuals in a community. This assumption makes the presented model significantly different from the ones currently dominant in the field. It also changes the approach to the act of infection. Usually, some parameters that describe the rate of new infections by taking into account those infected in the previous time slot are considered. With our model, we can individualize this process, considering each contact individually. ResultsThe typical output from calculations of a similar type are epidemic curves. In our model, except of presenting the average curves, we show the deviations or ranges for particular results obtained in different simulation runs, which usually lead to significantly different results. This observation is the effect of the probabilistic character of the infection process, which can impact, in different runs, individuals with different significance to the community. We can also easily present the effects of different types of intervention. The effects are studied for different methods used to create the graph representing a community, which can correspond to different social bonds. ConclusionsWe see the potential usefulness of the proposition in the detailed study of epidemic development for specific environments and communities. The ease of entering new parameters enables the analysis of several specific scenarios for different contagious diseases.

A performance- and energy-oriented extended tuning process for time-step-based scientific applications

Scientific application codes are often long-running time- and energy-consuming parallel codes, and the tuning of these methods towards the characteristics of a specific hardware is essential for a good performance. However, since scientific software is often developed over many years, the application software usually survives several hardware generations, which might make a re-tuning of the existing codes necessary. To simplify the tuning process, it would be beneficial to have software with inherent tuning possibilities. In this article, we explore the possibilities of tuning methods for time-step-based applications. Two different time-step-based application classes are considered, which are solution methods for ordinary differential equations and particle simulation methods. The investigation comprises a broad range of tuning possibilities, starting from the choice of algorithms, the parallel programming model, static implementation variants, input characteristics as well as hardware parameters for parallel execution. An experimental investigation shows the different characteristics of the application classes on different multicore systems. The results show that a combination of offline and online tuning leads to good tuning results. However, due to the different input characteristics of the two application classes, regular versus irregular, different tuning aspects are most essential.

The evolution equation of non-linear waves and its exact solutions by subsidiary ordinary differential equation method

In this work, we investigate the dynamics of the equatorial Rossby waves by including the complete Coriolis force, external source and dissipation. The amplitude evolution of equatorial Rossby waves is described as an extended non-linear mKdV–Burgers equation from a potential vorticity equation and it is unlike the standard mKdV–Burgers equation. Built on the obtained model, the corresponding physical phenomena related to the non-linear Rossby waves are analyzed. Also, the subsidiary ordinary differential equation method is employed to solve the solitary solution of the mKdV equation. By analyzing the solution, we find that the horizontal component of Coriolis parameter works on the amplitude of the Rossby waves. Meanwhile, we use the Adomian decomposition method to obtain the approximate soliton solution of the model.

Mean-square convergence of numerical methods for random ordinary differential equations

The aim of this work is to analyze the mean-square convergence rates of numerical schemes for random ordinary differential equations (RODEs). First, a relation between the global and local mean-square convergence order of one-step explicit approximations is established. Then, the global mean-square convergence rates are investigated for RODE-Taylor schemes for general RODEs, Affine-RODE-Taylor schemes for RODEs with affine noise, and Ito-Taylor schemes for RODEs with Ito noise, respectively. The theoretical convergence results are demonstrated through numerical experiments.

Observer-Based Decentralized Predictor Control for Large-Scale Interconnected Systems With Large Delays

This article proposes a decentralized predictor-based output feedback to compensate for large delays in large-scale interconnected systems under the premise that the interconnections between subsystems are not strong. The full state of each subsystem is assumed to be unmeasurable and the observer-based output feedback is employed. Two methods are used to tackle the large delays: the backstepping-based partial differential equation (PDE) approach and the reduction-based ordinary differential equation (ODE) approach. The PDE method is used for continuous-time control and manages with larger delays, whereas the ODE method is applicable to sampled-data implementation under discrete-time measurement.

Bandwidth optimization of information application system under fine integral method of fuzzy fractional order ordinary differential equations

In order to solve the problems that the network bandwidth of previous information application systems can’t guarantee the quality of big data transmission, resulting in low transmission efficiency and slow data processing, etc., an improved information application system is proposed by optimizing the fine integration method of fuzzy fractional ordinary differential equations and combining with software-defined networking (SDN). Firstly, the fractional-order fuzzy differential equation fine integration method of Pade approximation is derived. Then, based on the error analysis theory, the optimization formula for the adaptive selection of weighted parameter N and expanded item number q is obtained. Combined with SDN, an improved information application system is designed, and the improved algorithm and system are detected by example simulation and performance test. The results show that the numerical accuracy and the computational efficiency of the improved algorithm are higher than that of the improved algorithm. The improved port data merge rate and task completion efficiency under different bandwidth are also significantly higher than the original system. It shows that the information application system proposed in this research can better solve the problems of insufficient bandwidth and low communication speed of traditional systems, and provide a new perspective for bandwidth optimization of big data.

Modified operational matrix method for second-order nonlinear ordinary differential equations with quadratic and cubic terms

In this study, by means of the matrix relations between the Laguerre polynomials, and their derivatives, a novel matrix method based on collocation points is modified and developed for solving a class of second-order nonlinear ordinary differential equations having quadratic and cubic terms, via mixed conditions. The method reduces the solution of the nonlinear equation to the solution of a matrix equation corresponding to system of nonlinear algebraic equations with the unknown Laguerre coefficients. Also, some illustrative examples along with an error analysis based on residual function are included to demonstrate the validity and applicability of the proposed method.

Numerical comparison by different methods (second order Runge Kutta methods, Heun method, fixed point method and Ralston method) to differential equations with initial condition

This manuscript contains a detailed comparison between numerical solution methods of ordinary differential equations, which start from the Taylor series method of order 2, stating that this series hinders calculations for higher order derivatives of functions of several variables, so that the Runge Kutta methods of order 2 are implemented, which achieve the required purpose avoiding the cumbersome calculations of higher order derivatives. In this document, different variants of the Runge-Kutta methods of order 2 will be exposed from an introduction and demonstration of the connection of these with the Taylor series of order 2, these methods are: the method of Heun, the method of midpoint and the Ralston method. It will be observed from the solution of test differential equations its respective error with respect to the analytical solution, obtaining an error index dictated by the mean square error EMC. Through this document we will know the best numerical approximation to the analytical solution of the different PVI (initial value problems) raised, also fixing a solution pattern for certain problems, that is, the appropriate method for each type of problem will be stipulated. It was observed that the Ralston method presented greater accuracy followed by the midpoint method and the Heun method, in the other PVI it is observed that the midpoint method yields the best numerical solution since it has a very low EMC and difficult to reach by the other methods.