scholarly journals Numerical Algorithm of Block Method for General Second Order ODEs using Variable Step Size

2017 ◽  
Vol 46 (5) ◽  
pp. 817-824 ◽  
Author(s):  
Nazreen Waeleh ◽  
Zanariah Abdul Majid
Keyword(s):  
Numerical Algorithm ◽  
Second Order ◽  
Variable Step Size ◽  
Block Method ◽  
Step Size ◽  
Variable Step

An efficient variable step-size rational Falkner-type method for solving the special second-order IVP

2016 ◽  
Vol 291 ◽  
pp. 39-51 ◽  
Author(s):  
Higinio Ramos ◽  
Gurjinder Singh ◽  
V. Kanwar ◽  
Saurabh Bhatia
Keyword(s):  
Second Order ◽  
Variable Step Size ◽  
Step Size ◽  
Type Method ◽  
Variable Step

Variable-step-size second-order-derivative multistep method for solving first-order ordinary differential equations in system simulation

2020 ◽  
Vol 11 (01) ◽  
pp. 2050001
Author(s):  
Lei Zhang ◽  
Chaofeng Zhang ◽  
Mengya Liu
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Multistep Method ◽  
Second Order ◽  
Order Derivative ◽  
Variable Step Size ◽  
Variable Order ◽  
Step Size ◽  
Variable Step ◽  
The Stability

According to the relationship between truncation error and step size of two implicit second-order-derivative multistep formulas based on Hermite interpolation polynomial, a variable-order and variable-step-size numerical method for solving differential equations is designed. The stability properties of the formulas are discussed and the stability regions are analyzed. The deduced methods are applied to a simulation problem. The results show that the numerical method can satisfy calculation accuracy, reduce the number of calculation steps and accelerate calculation speed.

A Novel Variable Step-Size Algorithm for Implicit Integration

2011 ◽  
Vol 255-260 ◽  
pp. 2159-2163
Author(s):  
Jian Xiao Zou ◽  
Cui Yun Zhou ◽  
Gang Zheng
Keyword(s):  
Numerical Algorithm ◽  
Variable Coefficient ◽  
Variable Step Size ◽  
Implicit Integration ◽  
Output Variable ◽  
Step Size ◽  
Variable Step ◽  
The Common ◽  
The Difference ◽  
Coefficient Method

A new variable step-size numerical algorithm for implicit integration is discussed in this paper. The scheme for increase and decrease of step size is discussed according to the difference of output variable value. The next step size is calculated through the variable coefficient method and the limitation rules. The convergence and accuracy are testified by the simulation result. Compared with the common used ode45 algorithm, the algorithm has more computing efficiency with a certain calculation precision.

scholarly journals A 4-Point Block Method for Solving Higher Order Ordinary Differential Equations Directly

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Nazreen Waeleh ◽  
Zanariah Abdul Majid
Keyword(s):  
Initial Value Problems ◽  
Adaptive Strategy ◽  
Variable Step Size ◽  
Block Method ◽  
Initial Value ◽  
Step Size ◽  
First Order ◽  
Variable Step ◽  
Improved Performance ◽  
Fifth Order

An alternative block method for solving fifth-order initial value problems (IVPs) is proposed with an adaptive strategy of implementing variable step size. The derived method is designed to compute four solutions simultaneously without reducing the problem to a system of first-order IVPs. To validate the proposed method, the consistency and zero stability are also discussed. The improved performance of the developed method is demonstrated by comparing it with the existing methods and the results showed that the 4-point block method is suitable for solving fifth-order IVPs.

scholarly journals Three-Step Block Method for Solving Nonlinear Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Phang Pei See ◽  
Zanariah Abdul Majid ◽  
Mohamed Suleiman
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Second Order ◽  
Multiple Shooting ◽  
Variable Step Size ◽  
Nonlinear Boundary Value Problems ◽  
Block Method ◽  
Step Size ◽  
Multiple Shooting Techniques

We propose a three-step block method of Adam’s type to solve nonlinear second-order two-point boundary value problems of Dirichlet type and Neumann type directly. We also extend this method to solve the system of second-order boundary value problems which have the same or different two boundary conditions. The method will be implemented in predictor corrector mode and obtain the approximate solutions at three points simultaneously using variable step size strategy. The proposed block method will be adapted with multiple shooting techniques via the three-step iterative method. The boundary value problem will be solved without reducing to first-order equations. The numerical results are presented to demonstrate the effectiveness of the proposed method.

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