DYNAMICS OF NUMERICAL METHODS FOR COSYMMETRIC ORDINARY DIFFERENTIAL EQUATIONS

2001 ◽  
Vol 11 (09) ◽  
pp. 2339-2357 ◽  
Author(s):  
V. N. GOVORUKHIN ◽  
V. G. TSYBULIN ◽  
B. KARASÖZEN
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Equilibrium Points ◽  
Two Dimensions ◽  
Model Systems ◽  
Continuous Family ◽  
First Order ◽  
Spurious Solutions ◽  
Overall Performance ◽  
The Stability

The dynamics of numerical approximation of cosymmetric ordinary differential equations with a continuous family of equilibria is investigated. Nonconservative and Hamiltonian model systems in two dimensions are considered and these systems are integrated with several first-order Runge–Kutta methods. The preservation of symmetry and cosymmetry, the stability of equilibrium points, spurious solutions and transition to chaos are investigated by presenting analytical and numerical results. The overall performance of the methods for different parameters is discussed.

scholarly journals BBDF-Alpha for solving stiff ordinary differential equations with oscillating solutions

2020 ◽  
Vol 51 (2) ◽  
pp. 123-136
Author(s):  
Iskandar Shah Mohd Zawawi
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Experiments ◽  
Stability Region ◽  
Newton Iteration ◽  
Computational Time ◽  
First Order ◽  
Stiff Ordinary Differential Equations ◽  
Oscillating Solutions ◽  
The Stability

In this paper, the block backward differentiation α formulas (BBDF-α) is derived for solving first order stiff ordinary differential equations with oscillating solutions. The consistency and zero stability conditions are investigated to prove the convergence of the method. The stability region in the entire negative half plane shows that the derived method is A-stable for certain values of α. The implementation of the method using Newton iteration is also discussed. Several numerical experiments are conducted to demonstrate the performance of the method in terms of accuracy and computational time.

scholarly journals A New Two Derivative FSAL Runge-Kutta Method of Order Five in Four Stages

2020 ◽  
Vol 17 (1) ◽  
pp. 0166
Author(s):  
Hussain Et al.
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Numerical Results ◽  
New Method ◽  
Kutta Method ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability

A new efficient Two Derivative Runge-Kutta method (TDRK) of order five is developed for the numerical solution of the special first order ordinary differential equations (ODEs). The new method is derived using the property of First Same As Last (FSAL). We analyzed the stability of our method. The numerical results are presented to illustrate the efficiency of the new method in comparison with some well-known RK methods.

STABILITY ANALYSIS OF A PROPOSED SCHEME OF ORDER FIVE FOR FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 6 (2) ◽  
pp. 898
Author(s):  
Sunday Emmanuel Fadugba ◽  
Roseline Bosede Ogunrinde ◽  
Rowland Rotimi Ogunrinde
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Step Length ◽  
Step Method ◽  
First Order ◽  
Absolute Relative Error ◽  
One Step ◽  
Test Equation ◽  
The Stability

This paper presents the stability analysis of a proposed scheme of order five (FCM) for first order Ordinary Differential Equations (ODEs). The proposed FCM is derived by means of an interpolating function of polynomial and exponential forms. The properties of FCM were discussed extensively. The linear stability of FCM in the context of the Third Order One-Step Method (TCM) and Second Order One-Step Method (SCM) for the solution of initial value problems of first order differential equations is presented. The stability region of FCM, TCM and SCM is investigated using the Dahlquist’s test equation. The numerical results obtained via FCM are compared with TCM and SCM. Moreover, by varying the step length, the accuracy and convergence of the methods in terms of the final absolute relative error are measured. The results show that FCM converges faster and more stable than its counterparts.

scholarly journals A New Method for Numerical Integration of Higher-Order Ordinary Differential Equations Without Losing the Periodic Responses

2021 ◽  
Vol 7 ◽  
Author(s):  
John T. Katsikadelis
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Variable Coefficients ◽  
Space Representation ◽  
Parabolic Differential Equations ◽  
Linear Ordinary Differential Equations ◽  
First Order ◽  
Symmetrical Matrices ◽  
The Stability

A new numerical method is presented for the solution of initial value problems described by systems of N linear ordinary differential equations (ODEs). Using the state-space representation, a differential equation of order n > 1 is transformed into a system of L = n×N first-order equations, thus the numerical method developed recently by Katsikadelis for first-order parabolic differential equations can be applied. The stability condition of the numerical scheme is derived and is investigated using several well-corroborated examples, which demonstrate also its convergence and accuracy. The method is simply implemented. It is accurate and has no numerical damping. The stability does not require symmetrical and positive definite coefficient matrices. This advantage is important because the scheme can find the solution of differential equations resulting from methods in which the space discretization does not result in symmetrical matrices, for example, the boundary element method. It captures the periodic behavior of the solution, where many of the standard numerical methods may fail or are highly inaccurate. The present method also solves equations having variable coefficients as well as non-linear ones. It performs well when motions of long duration are considered, and it can be employed for the integration of stiff differential equations as well as equations exhibiting softening where widely used methods may not be effective. The presented examples demonstrate the efficiency and accuracy of the method.

scholarly journals L-Stable Block Hybrid Numerical Algorithm for First-Order Ordinary Differential Equations

2020 ◽  
pp. 160-165
Author(s):  
B. I. Akinnukawe ◽  
K. O. Muka
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Hybrid Methods ◽  
Initial Conditions ◽  
Multistep Method ◽  
Stability And Convergence ◽  
First Order ◽  
One Step ◽  
Grid Points ◽  
The Stability

In this work, a one-step L-stable Block Hybrid Multistep Method (BHMM) of order five was developed. The method is constructed for solving first order Ordinary Differential Equations with given initial conditions. Interpolation and collocation techniques, with power series as a basis function, are employed for the derivation of the continuous form of the hybrid methods. The discrete scheme and its second derivative are derived by evaluating at the specific grid and off-grid points to form the main and additional methods respectively. Both hybrid methods generated are composed in matrix form and implemented as a block method. The stability and convergence properties of BHMM are discussed and presented. The numerical results of BHMM have proven its efficiency when compared to some existing methods.

scholarly journals PHASE BEHAVIOUR OF FIRST - ORDER SYSTEMS

2018 ◽  
Vol 16 (2) ◽  
pp. 131-137 ◽  
Author(s):  
Kaloyan Yankov
Keyword(s):  
Differential Equations ◽  
Response Curve ◽  
Phase Plane ◽  
Nonlinear Differential Equations ◽  
Equilibrium Points ◽  
Phase Behaviour ◽  
First Order ◽  
Phase Plane Method ◽  
The Stability ◽  
First Order Differential Equations

The phase-plane method gives possibility to study the stability of systems described by linear and nonlinear differential equations. The article is devoted to the capabilities of MathCad for analysis of first order differential equations. An algorithm is proposed and Mathcad's specific operators for the construction and analysis of phase trajectories are described. Approaches for calculation of equilibrium points and determination the type of bifurcation in function of parameter are described. The proposed algorithm is applied to the dose-response curve of the antibiotic tubazid.

scholarly journals Diagonally Implicit Block Backward Differentiation Formulas for Solving Ordinary Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
I. S. M. Zawawi ◽  
Z. B. Ibrahim ◽  
F. Ismail ◽  
Z. A. Majid
Keyword(s):  
Differential Equations ◽  
Fixed Points ◽  
Ordinary Differential Equations ◽  
Initial Value Problem ◽  
Initial Value ◽  
First Order ◽  
Backward Differentiation Formulas ◽  
The Stability ◽  
Differentiation Formulas ◽  
A Performance

This paper focuses on the derivation of diagonally implicit two-point block backward differentiation formulas (DI2BBDF) for solving first-order initial value problem (IVP) with two fixed points. The method approximates the solution at two points simultaneously. The implementation and the stability of the proposed method are also discussed. A performance of the DI2BBDF is compared with the existing methods.

Sign in / Sign up

Export Citation Format

Share Document