STUDENT MISCONCEPTIONS IN USING EULER'S METHOD IN ORDINARY DIFFERENTIAL EQUATIONS AND THE IMPORTANCE OF STEP SIZE

PRIMUS ◽  
2002 ◽  
Vol 12 (3) ◽  
pp. 262-276 ◽  
Author(s):  
William P. Fox
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Step Size ◽  
Euler’S Method ◽  
Student Misconceptions

scholarly journals A Quantitative Comparison of Numerical Method for Solving Stiff Ordinary Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
S. A. M. Yatim ◽  
Z. B. Ibrahim ◽  
K. I. Othman ◽  
M. B. Suleiman
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Computation Time ◽  
Quantitative Comparison ◽  
Initial Value ◽  
Step Size ◽  
Stiff Ordinary Differential Equations ◽  
Block Methods ◽  
Variable Step

We derive a variable step of the implicit block methods based on the backward differentiation formulae (BDF) for solving stiff initial value problems (IVPs). A simplified strategy in controlling the step size is proposed with the aim of optimizing the performance in terms of precision and computation time. The numerical results obtained support the enhancement of the method proposed as compared to MATLAB's suite of ordinary differential equations (ODEs) solvers, namely, ode15s and ode23s.

scholarly journals Step Change Strategies for Multistep Methods

1985 ◽  
Vol 14 (196) ◽  
Author(s):  
Ole Østerby
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multistep Methods ◽  
Step Change ◽  
Change Strategies ◽  
Step Size ◽  
Step Method ◽  
Adams Methods

When a system of ordinary differential equations is solved using a step-by-step method it is often desirable to change the step size during the course of the integration. We show that the commonly used formulas for calculating the new step sizes are not correct for multistep methods and we derive correct formulas for Adams methods.

One-leg Integration of Ordinary Differential Equations with Global Error Control

2005 ◽  
Vol 5 (1) ◽  
pp. 86-96 ◽  
Author(s):  
Gennady Yu. Kulikov ◽  
Sergey K. Shindin
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Error Control ◽  
Size Control ◽  
Adaptive Algorithms ◽  
Reasonable Accuracy ◽  
Step Size ◽  
Step Size Control ◽  
The Family

AbstractIn this paper we study the family of one-leg two-step second-order methods developed by Dahlquist et al., which possess the A-stability and G-stability properties on any grid. These methods are implemented with the local-global step size control derived by Kulikov and Shindin with the aim to obtain automatically the numerical solution with any reasonable accuracy set by the user. We show that the error control is more complicated in one-leg methods, especially when applied to stiffproblems. Thus, we adapt our local-global step size control for the methods indicated above and test these adaptive algorithms in practice.

scholarly journals Two-Point Block variable order step size Multistep Method for Solving Higher Order Ordinary Differential Equations Directly

2021 ◽  
pp. 101376
Author(s):  
Ahmad Fadly Nurullah Rasedee ◽  
Mohammad Hasan Abdul Sathar ◽  
Siti Raihana Hamzah ◽  
Norizarina Ishak ◽  
Wong Tze Jin ◽  
...  
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Higher Order ◽  
Multistep Method ◽  
Variable Order ◽  
Step Size

scholarly journals Numerical Solutions of Nonlinear Ordinary Differential Equations by Using Adaptive Runge-Kutta Method

2019 ◽  
Vol 17 ◽  
pp. 147-154
Author(s):  
Abhinandan Chowdhury ◽  
Sammie Clayton ◽  
Mulatu Lemma
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Solutions ◽  
Adaptive Methods ◽  
Integration Step ◽  
Nonlinear Ordinary Differential Equations ◽  
Kutta Method ◽  
Step Size ◽  
Runge Kutta Method ◽  
Runge Kutta

We present a study on numerical solutions of nonlinear ordinary differential equations by applying Runge-Kutta-Fehlberg (RKF) method, a well-known adaptive Runge-kutta method. The adaptive Runge-kutta methods use embedded integration formulas which appear in pairs. Typically adaptive methods monitor the truncation error at each integration step and automatically adjust the step size to keep the error within prescribed limit. Numerical solutions to different nonlinear initial value problems (IVPs) attained by RKF method are compared with corresponding classical Runge-Kutta (RK4) approximations in order to investigate the computational superiority of the former. The resulting gain in efficiency is compatible with the theoretical prediction. Moreover, with the aid of a suitable time-stepping scheme, we show that the RKF method invariably requires less number of steps to arrive at the right endpoint of the finite interval where the IVP is being considered.

scholarly journals A COMPARATIVE EXPLORATION ON DIFFERENT NUMERICAL METHODS FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS

2020 ◽  
Vol 15 (12) ◽  
Author(s):  
Mohammad Asif Arefin
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Solutions ◽  
Euler Method ◽  
Euler’S Method ◽  
Runge Kutta Method ◽  
Accuracy Level ◽  
Runge Kutta ◽  
Matlab Programming

In this paper, the initial value problem of Ordinary Differential Equations has been solved by using different Numerical Methods namely Euler’s method, Modified Euler method, and Runge-Kutta method. Here all of the three proposed methods have to be analyzed to determine the accuracy level of each method. By using MATLAB Programming language first we find out the approximate numerical solution of some ordinary differential equations and then to determine the accuracy level of the proposed methods we compare all these solutions with the exact solution. It is observed that numerical solutions are in good agreement with the exact solutions and numerical solutions become more accurate when taken step sizes are very much small. Lastly, the error of each proposed method is determined and represents them graphically which reveals the superiority among all the three methods. We fund that, among the proposed methods Runge-Kutta 4th order method gives the accurate result and minimum amount of error.

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