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

2014 ◽

Vol 2014 ◽

pp. 1-10 ◽

Cited By ~ 6

Author(s):

Ahmad Fadly Nurullah Rasedee ◽

Mohamed bin Suleiman ◽

Zarina Bibi Ibrahim

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Higher Order ◽

Variable Order ◽

Numerical Techniques ◽

Step Size ◽

The Relationship

The current numerical techniques for solving a system of higher order ordinary differential equations (ODEs) directly calculate the integration coefficients at every step. Here, we propose a method to solve higher order ODEs directly by calculating the integration coefficients only once at the beginning of the integration and if required once more at the end. The formulae will be derived in terms of backward difference in a constant step size formulation. The method developed will be validated by solving some higher order ODEs directly using variable order step size. To simplify the evaluations of the integration coefficients, we find the relationship between various orders. The results presented confirmed our hypothesis.

Download Full-text

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

International Journal of Modeling Simulation and Scientific Computing ◽

10.1142/s1793962320500014 ◽

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.

Download Full-text

A Numerical Algorithm for Solving Stiff Ordinary Differential Equations

Mathematical Problems in Engineering ◽

10.1155/2013/989381 ◽

2013 ◽

Vol 2013 ◽

pp. 1-11 ◽

Cited By ~ 4

Author(s):

S. A. M. Yatim ◽

Z. B. Ibrahim ◽

K. I. Othman ◽

M. B. Suleiman

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Computation Time ◽

Test Problems ◽

Variable Order ◽

Step Size ◽

Differentiation Formula ◽

Stiff Ordinary Differential Equations ◽

Backward Differentiation Formula ◽

The Stability

An advanced method using block backward differentiation formula (BBDF) is introduced with efficient strategy in choosing the step size and order of the method. Variable step and variable order block backward differentiation formula (VSVO-BBDF) approach is applied throughout the numerical computation. The stability regions of the VSVO-BBDF method are investigated and presented in distinct graphs. The improved performances in terms of accuracy and computation time are presented in the numerical results with different sets of test problems. Comparisons are made between the proposed method and MATLAB’s suite of ordinary differential equations (ODEs) solvers, namely, ode15s and ode23s.

Download Full-text

Solving Directly Higher Order Ordinary Differential Equations by Using Variable Order Block Backward Differentiation Formulae

Symmetry ◽

10.3390/sym11101289 ◽

2019 ◽

Vol 11 (10) ◽

pp. 1289

Author(s):

Asnor ◽

Mohd Yatim ◽

Ibrahim

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Higher Order ◽

Computational Effort ◽

Variable Order ◽

Numerical Efficiency ◽

Stiff Odes ◽

Backward Differentiation Formulae ◽

Bdf Method ◽

Selection Of

Variable order block backward differentiation formulae (VOHOBBDF) method is employedfor treating numerically higher order Ordinary Differential Equations (ODEs). In this respect, the purpose of this research is to treat initial value problem (IVP) of higher order stiff ODEs directly. BBDF method is symmetrical to BDF method but it has the advantage of producing more than one solutions simultaneously. Order three, four, and five of VOHOBBDF are developed and implemented as a single code by applying adaptive order approach to enhance the computational efficiency. This approach enables the selection of the least computed LTE among the three orders of VOHOBBDF and switch the code to the method that produces the least LTE for the next step. A few numerical experiments on the focused problem were performed to investigate the numerical efficiency of implementing VOHOBBDF methods in a single code. The analysis of the experimental results reveals the numerical efficiency of this approach as it yielded better performances with less computational effort when compared with built-in stiff Matlab codes. The superior performances demonstrated by the application of adaptive orders selection in a single code thus indicate its reliability as a direct solver for higher order stiff ODEs.

Download Full-text

Direct Two-Point Block One-Step Method for Solving General Second-Order Ordinary Differential Equations

Mathematical Problems in Engineering ◽

10.1155/2012/184253 ◽

2012 ◽

Vol 2012 ◽

pp. 1-16 ◽

Cited By ~ 8

Author(s):

Zanariah Abdul Majid ◽

Nur Zahidah Mokhtar ◽

Mohamed Suleiman

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Multistep Method ◽

Second Order ◽

Variable Step Size ◽

Step Size ◽

Step Method ◽

One Step ◽

The Stability ◽

The One

A direct two-point block one-step method for solving general second-order ordinary differential equations (ODEs) directly is presented in this paper. The one-step block method will solve the second-order ODEs without reducing to first-order equations. The direct solutions of the general second-order ODEs will be calculated at two points simultaneously using variable step size. The method is formulated using the linear multistep method, but the new method possesses the desirable feature of the one-step method. The implementation is based on the predictor and corrector formulas in thePE(CE)mmode. The stability and precision of this method will also be analyzed and deliberated. Numerical results are given to show the efficiency of the proposed method and will be compared with the existing method.

Download Full-text

Solution for nonlinear Duffing oscillator using variable order variable stepsize block method

MATEMATIKA ◽

10.11113/matematika.v33.n2.1015 ◽

2017 ◽

Vol 33 (2) ◽

pp. 165 ◽

Cited By ~ 2

Author(s):

Ahmad Fadly Nurullah Rasedee ◽

Mohamad Hasan Abdul Sathar ◽

Norizarina Ishak ◽

Nur Shuhada Kamarudin ◽

Muhamad Azrin Nazri ◽

...

Keyword(s):

Differential Equations ◽

Numerical Solutions ◽

Duffing Oscillator ◽

Computational Cost ◽

Real Life ◽

Multistep Method ◽

Variable Order ◽

Block Method ◽

Step Size ◽

Variable Stepsize

Real life phenomena found in various fields such as engineering, physics, biology and communication theory can be modeled as nonlinear higher order ordinary differential equations, particularly the Duffing oscillator. Analytical solutions for these differential equations can be time consuming whereas, conventional numerical solutions may lack accuracy. This research propose a block multistep method integrated with a variable order step size (VOS) algorithm for solving these Duffing oscillators directly. The proposed VOS Block method provides an alternative numerical solution by reducing computational cost (time) but without loss of accuracy. Numerical simulations are compared with known exact solutions for proof of accuracy and against current numerical methods for proof of efficiency (steps taken).

Download Full-text