Improved Euler Method Research Articles

An hybrid initial value method for singularly perturbed delay differential equations with interior layers and weak boundary layer

Abstract In this paper, an hybrid initial value method on Shishkin mesh is suggested to solve singularly perturbed boundary value problem for second order ordinary delay differential equation with discontinuous convection coefficient and source term. In this method, the original problem of solving the second order differential equation is reduced to solving four first order differential equations. Among the four first order differential equations, three of them are singularly perturbed differential equations without delay and other one is a regular differential equation with a delay term. The singularly perturbed differential equations are solved by the second order hybrid finite difference schemes, whereas the delay differential equation is solved by the improved Euler method. An error estimate is derived by using the supremum norm and it is of almost second order convergence. Numerical results are provided to illustrate the theoretical results.

Numerical Investigations on Hybrid Fuzzy Fractional Differential Equations by Improved Fractional Euler Method

In this paper, the improved Euler method is used for solving hybrid fuzzy fractional differential equations (HFFDE) of order $q \in (0, 1) $ under Caputo-type fuzzy fractional derivatives. This method is based on the fractional Euler method and generalized Taylor's formula. The accuracy and efficiency of the proposed method is demonstrated by solving numerical examples.

Numerical Simulations for Solving Fuzzy Fractional Differential Equations by Max-Min Improved Euler Methods

Abstract In this paper, an extension is introduced into Max-Min Improved Euler methods for solving initial value problems of fuzzy fractional differential equations (FFDEs). Two modified fractional Euler type methods have been proposed and investigated to obtain numerical solutions of linear and nonlinear FFDEs. The proposed algorithms are tested on various illustrative examples. Exact values are also simulated to compare and discuss the closeness and accuracy of approximations so obtained. Comparatively, tables and graphs results reveal the complete reliability, efficiency and accuracy of the proposed methods.

Curvature weight method of solving the point reactor neutron kinetic equations

The point kinetic equations are the system of a couple stiff ordinary differential equations. Many studies have focused on the development of more advanced and efficient methods of solving the equations, such as the high order Taylor polynomials method, the Haar wavelet operational method, the fractional point-neutron kinetic model method, the basis function method, the homotopy analysis method, and other methods. Most of these methods are successful in some specific problems, but still have, more or less, disadvantages. For example, the accuracy of the Haar wavelet operational method is limited by the collocation points, and it needs more computing time for a high precision. Aiming at the requirements that some numerical calculation results must have the higher precision and only the positive error in the nuclear reactor safety engineering and ship reactor for the maneuverability, in this paper we try to look for a new numerical method to satisfy that the calculation value is slightly higher than the real value when the actual curve is upward convex or downward concave, and the error is not greater than that by the Euler and improved Euler method. The new method is so-called the curvature weight (CW) method, which is based on the curvature circle method and considers the contributions of two curvatures at the interval step point to the average curvature inside the interval step. Using the decoupling method to remove the stiffness of equations and the instantaneous jump approximation to derive the neutron differential equations, the first and second derivative of neutron density are obtained. Then the CW method is used to solve the point reactor neutron kinetic equations, and thus obtaining the numerical solution. Compared with the results by the Euler and improved Euler method, the numerical calculation results by the CW method are always higher than the real value, and the calculation accuracy and speed are improved significantly. When this new method is used to solve the point reactor neutron differential equations with the step and linear reactivity inserted into the subcritical reactor, the numerical results which satisfy the requirements of positive calculation error and high precision can be obtained quickly. After improving the calculation step length, the precision reduction by the CW method is significantly lower than that by the Euler and improved Euler method. So the CW method can greatly shorten the total computing time, and it is also effective for most of differential equation systems.

Improved Euler method for the interpretation of potential data based on the ratio of the vertical first derivative to analytic signal

We propose a new automatic method for the interpretation of potential field data, called the RDAS-Euler method, which is based on Euler’s deconvolution and analytic signal methods. The proposed method can estimate the horizontal and vertical extent of geophysical anomalies without prior information of the nature of the anomalies (structural index). It also avoids inversion errors because of the erroneous choice of the structural index N in the conventional Euler deconvolution method. The method was tested using model gravity anomalies. In all cases, the misfit between theoretical values and inversion results is less than 10%. Relative to the conventional Euler deconvolution method, the RDAS-Euler method produces inversion results that are more stable and accurate. Finally, we demonstrate the practicability of the method by applying it to Hulin Basin in Heilongjiang province, where the proposed method produced more accurate data regarding the distribution of faults.

Time-Domain Simulation of Power System Frequency Fluctuations Caused by Intermittent Wind Power

Wind power is uncertain. When the proportion of wind power capacity is large, the impact of the randomness of the output power on the grid frequency is more significant. Aiming at system stability influence after integrating a high capacity of wind turbines, this paper introduces the dynamic simulation model for electric power system with wind power, and makes an equivalent transformation processing for the wind turbine and generator. By using improved Euler method, this paper analyzes the impact of system frequency through time-domain simulation when disturbance occurs at wind farm side. Simulation results show that when wind speed changes, the established model and algorithm can effectively simulate the dynamic response of the frequency, and also these results can provide a reference for the frequency stability when wind turbines integrate into power system.

On those ordinary differential equations that are solved exactly by the improved Euler method

As a numerical method for solving ordinary differential equations $y^{\prime }=f(x,y)$, the improved Euler method is not assumed to give exact solutions. In this paper we classify all cases where this method gives the exact solution for all initial conditions. We reduce an infinite system of partial differential equations for $f(x,y)$ to a finite system that is sufficient and necessary for the improved Euler method to give the exact solution. The improved Euler method is the simplest explicit second order Runge-Kutta method.

A New Approach to Fuzzy Initial Value Problem by Improved Euler Method

This paper targets to investigate the solution of linear and nonlinear ordinary differential equations with fuzzy initial condition. Here, two improved Euler type methods have been proposed in order to obtain numerical solution of the problem. Along with this, an exact methodology is also discussed. The obtained results are depicted in term of plots to show the efficiency of the proposed methods. The solutions are compared with the known results and are found that those obtained by the proposed methods are tighter than the results from the existing method.

Numerical solution of random differential models

This paper deals with the construction of a numerical solution of random initial value problems by means of a random improved Euler method. Conditions for the mean square convergence of the proposed method are established. Finally, an illustrative example is included in which the main statistics properties such as the mean and the variance of the stochastic approximation solution process are given.

컴퓨터 게임을 위한 물리 엔진의 성능 향상 및 이를 적용한 지능적인 게임 캐릭터에 관한 연구

이 논문은 컴퓨터 게임을 위한 물리 엔진의 성능 향상 및 이를 적용한 지능적인 게임 캐릭터에 관한 연구를 서술한다. 물리적 상황을 자동으로 인식하는 알고리즘으로는 Momentum back-propagation을 적용하였다. 또한 우리는 각 상황에 따른 적분 방식의 실험 결과를 제시한다. 실험을 위하여 Euler Method, Improved Euler Method, 및 Runge-kutta Method의 세 가지의 적분 방식을 적용하였다. 각 적분 방식의 실험 결과에서 충돌이 없는 상황에서는 Euler Method가 최적의 성능을 보여주었다. 또한 충돌 상황에서는 세 가지 방식이 모두 비슷한 성능을 보여주었지만, Runge-kutta Method가 최적의 정확도를 보여주었다. 물리 상황인식에 대한 실험결과에서는 입력 층과 출력 층이 고정된 상태에서 은닉 층이 3일 때 가장 좋은 성능을 보여주었고, 또한 학습횟수가 30000일 때 최적의 성능을 보여주었다. 앞으로 우리는 다른 장르의 게임에 이러한 물리적 컨텍스트(context)를 인식하는 연구를 진행할 것이며 또한 전체 게임의 성능을 증가할 수 있도록 M-BP이외의 인식 알고리즘을 적용할 것이다. This paper describes research on intelligent game character through performance enhancements of physics engine in computer games. The algorithm that recognizes the physics situation uses momentum back-propagation neural networks. Also, we present an experiment and its results, integration methods that display optimum performance based on the physics situation. In this experiment on integration methods, the Euler method was shown to produce the best results in terms of fps in a simulation environment with collision detection. Simulation with collision detection was shown similar fps for all three methods and the Runge-kutta method was shown the greatest accuracy. In the experiment on physics situation recognition, a physics situation recognition algorithm where the number of input layers (number of physical parameters) and output layers (destruction value for the master car) is fixed has shown the best performance when the number of hidden layers is 3 and the learning count number is 30,000. Since we tested with rigid bodies only, we are currently studying efficient physics situation recognition for soft body objects.

Windows version of the ion trap simulation program ITSIM: a powerful heuristic and predictive tool in ion trap mass spectrometry

The multi-particle simulation program ITSIM version 4.0 takes advantage of the enhanced performance of the Windows 95 and NT operating systems in areas such as memory management, user friendliness, flexibility of graphics and speed, to investigate the motion of ions in the quadrupole ion trap. New features and capabilities significantly broaden its applicability. The simulation program can provide help in understanding fundamental aspects of ion trap mass spectrometry and both precede experiments and assist in directing their course. It also has didactic value in elucidating and allowing visualization of ion behavior under a variety of experimental conditions. ITSIM 4.0 provides easy access to ion simulations for all users through a dramatically improved user interface. The program uses the improved Euler method to calculate ion trajectories as a numerical solution to the Mathieu differential equation. The Windows version can simultaneously simulate the trajectories of ions with a virtually unlimited number of different mass-to-charge ratios, up to a maximum of 600000 ions, and hence allow realistic mass spectra, ion kinetic energy distributions, phase-of-ejection distributions and other experimentally measurable properties to be simulated. The simulated data are used to obtain mass spectra from mass-selective instability scans and by Fourier transformation of image currents induced by coherently moving ion clouds. Field inhomogeneities arising from exit holes, electrode misalignment, imperfect electrode surfaces or alternative trap geometries can be simulated with the program. Non-zero angle scattering in the hard-sphere collision model allows simulations involving collisional cooling to be performed. Complete instruments, from an ion source through the ion trap mass analyzer to a detector, can be simulated. Some typical applications of the simulation program are presented and discussed. Such features as the mass-selective instability scan mode, mass-range extension via resonant ion ejection, r.f. and d.c. ion isolation and non-destructive detection are shown. Comparisons are made between the simulated and experimental results, for example in mass-selective photodissociation. Fourier transform experiments and a novel six-electrode ion trap mass spectrometer illustrate cases in which simulations precede reduction to practice. © 1998 John Wiley & Sons, Ltd.

Dynamics of constant and variable stepsize methods for a nonlinear population model with delay

Hutchinson's equation is a reaction-diffusion model where the quadratic reaction term involves a delay. It is a natural extension of the logistic equation (no diffusion, no delay) and Fisher's equation (no delay), both of which have been used to illustrate the potential for spurious long-term dynamics in numerical methods. For the case where initial conditions and periodic boundary conditions are supplied, we look at the use of central differences in space and either Euler's method or the Improved Euler method in time. Our aim is to investigate the impact of the delay on the long-term behaviour of the scheme. After studying the fixed points of the methods in constant stepsize mode, we consider an adaptive time-stepping approach, using a standard local error control strategy. Applying ideas of Hall (1985) we are able to explain the fine detail of the time-step selection process.

On the Application of Runge-Kutta Methods to Transport Calculations

Under a definition suitable to the transport equation, it is shown that the (two-stage explicit) Runge-Kutta (RK) methods having order of at least 2, and requiring essentially only one source evaluation per cell, consist of a one-parameter family, plus two additional methods. Two of these, the midpoint corrector and improved Euler methods, are selected for detailed computational comparison with the classical diamond-difference and step characteristic methods. Extensive monodirectional calculations reveal that the RK methods display absolute instability for cell path lengths exceeding 2 mfp, but that they are nearly competitive with the classical methods for small cell widths. It is shown how the two subject RK methods can be augmented by closure approximations, so as to permit their use in source iteration for multiple-direction calculations. The results of such calculations show that for small cell widths, the RK methods again are nearly competitive in accuracy, although the absolute stability requirement can impose a stringent upper bound on the acceptable cell widths; the RK methods interact well with source iteration, even though they do not conserve particles; and the particular closure approximations selected retain the second-order accuracy of the basic underlying methods.

New explicit and implicit “improved euler” methods for the integration of ordinary differential equations

The new improved Euler methods given here offer several advantages for the solution of ordinary differential equations. The explicit form is of second-order global accuracy but requires only one derivative evaluation per step. It is therefore more efficient than either simple Euler or Runge—Kutta two. This new method is conditionally stable, provides an estimate of local accuracy for error control, is well-suited to global extrapolation and is applicable to real-time problems. Potential important applications also include optimization, continuation problems and parameter estimation. The implicit form is unconditionally stable, offering second-order global accuracy with a stability which is in between backward Euler and the trapezoidal rule. This method should be valuable for stiff problems, and in particular it should serve as an improvement to the well-known Crank—Nicolson method for partial differential equations.

Inelastic post-buckling of tapered flexible bars

Large deformations of buckled elasto-plastic flexible cantilever bars of variable stiffness with rectangular and circular cross-sections are considered. An improved Euler method is used to solve the resulting first-order, nonlinear differential equations. Deflections at the free end, and hence the values of critical loads are presented for the tapered bars.

Spatial integration of strains using finite elements

The finite element method for spatial integration of strains was conceived and then evolved as an intuitively reasonable means of calculating the pretectonic shape of a region. Here we show that the technique is basically an improved Euler method for integrating differential equations, and that unstrained elements (parallelepipeds) should therefore be fitted together so that face centres of adjoining elements coincide; or where this is not possible, the interfacial distances should be minimized. Minimization yields the translation and rotation an element must undergo in order to fit well into the hole enclosed by neighbouring elements. Techniques are described for fitting of an array of elements by computer. A fastpacking routine gives a good but approximate result; whereas a slower but more rigorous procedure enables the result to be improved. Finally, a short discussion of errors is given.

Numerical procedure for the deflections of buckled elasto-plastic flexible bars

In this paper, a numerical procedure to determine the deflections at the free end and along the central line of buckled elasto-plastic flexible bars is introduced. Because of the large deflection of the bar, the nonlinear theory of bending is used to solve the problem at hand. The exact expression of the curvature is used in the moment-curvature relationship for both the elastic (elastica) and the elasto-plastic states of stress. For the elastica problem, a closed-form solution may be found, if any. However, for elasto-plastic flexible bars, where a region of the bar yields, and yielding progresses along the bar, a closed-form solution is not possible because of the complexity of the problem. Therefore, the resulting first-order, nonlinear differential equations, for the problem at hand, are numerically integrated, using the improved Euler method, from the base of the bar where boundary values are known, to the free end where boundary value must initially be guessed. The integration process is repeated until the computed value of deflection at the free end of the bar is close enough to the trial value within acceptable tolerance. Deflections at the free end of the bar in the x and y directions are presented in graphical form. A computer program is developed, and the results are compared to those obtained from the experimental data.

Post Lateral Buckling Behavior of Beams

The large deflections which occur when a uniform rectangular cantilever beam has an end load greater than the lateral buckling load are found by solving a set of three simultaneous, first-order, nonlinear, differential equations. These equations, derived through considerations of moment equilibrium, are numerically integrated using the improved Euler method from the fixed end where boundary values are known, to the free end where boundary values must initially be guessed. The integration process is repeated until the computed boundary values at the free end are sufficiently close to the guessed values. Convergence problems arise for beam cross sections which are almost square. Deflections in three directions are presented graphically as a function of the length, width, height, and lateral buckling load of the beam.