scholarly journals A Zero-Dissipative Phase-Fitted Fourth Order Diagonally Implicit Runge-Kutta-Nyström Method for Solving Oscillatory Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
K. W. Moo ◽  
N. Senu ◽  
F. Ismail ◽  
M. Suleiman
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Variable Coefficient ◽  
Numerical Experiments ◽  
The Other ◽  
Phase Lag ◽  
Oscillatory Solutions ◽  
Second Order Differential Equations ◽  
Algebraic Order ◽  
Runge Kutta

A new diagonally implicit Runge-Kutta-Nyström (DIRKN) method is constructed for solving second order differential equations with oscillatory solutions. The method is originally based on existing DIRKN method derived by Senu et al. which is three-stage and fourth algebraic order. The new derived method has a variable coefficient with phase-lag of order infinity. The numerical experiments are carried out and the results show the efficiency and accuracy of the new method in comparison with the other DIRKN methods in the literature.

scholarly journals Phase-Fitted and Amplification-Fitted Higher Order Two-Derivative Runge-Kutta Method for the Numerical Solution of Orbital and Related Periodical IVPs

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
N. A. Ahmad ◽  
N. Senu ◽  
F. Ismail
Keyword(s):  
Numerical Solution ◽  
Numerical Experiments ◽  
Initial Value ◽  
Oscillatory Solutions ◽  
First Order ◽  
Runge Kutta Method ◽  
Algebraic Order ◽  
Runge Kutta ◽  
The Stability ◽  
Order Four

A phase-fitted and amplification-fitted two-derivative Runge-Kutta (PFAFTDRK) method of high algebraic order for the numerical solution of first-order Initial Value Problems (IVPs) which possesses oscillatory solutions is derived. We present a sixth-order four-stage two-derivative Runge-Kutta (TDRK) method designed using the phase-fitted and amplification-fitted property. The stability of the new method is analyzed. The numerical experiments are carried out to show the efficiency of the derived methods in comparison with other existing Runge-Kutta (RK) methods.

scholarly journals A Zero-Dissipative Runge-Kutta-Nyström Method with Minimal Phase-Lag

2010 ◽  
Vol 2010 ◽  
pp. 1-15 ◽  
Author(s):  
Norazak Senu ◽  
Mohamed Suleiman ◽  
Fudziah Ismail ◽  
Mohamed Othman
Keyword(s):  
Differential Equations ◽  
Second Order ◽  
Phase Lag ◽  
Nyström Method ◽  
Numerical Comparisons ◽  
Nystrom Method ◽  
Second Order Differential Equations ◽  
Runge Kutta ◽  
Minimal Phase

An explicit Runge-Kutta-Nyström method is developed for solving second-order differential equations of the formq′′=f(t,q)where the solutions are oscillatory. The method has zero-dissipation with minimal phase-lag at a cost of three-function evaluations per step of integration. Numerical comparisons with RKN3HS, RKN3V, RKN4G, and RKN4C methods show the preciseness and effectiveness of the method developed.

scholarly journals New 4(3) Pairs Diagonally Implicit Runge-Kutta-Nyström Method for Periodic IVPs

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Norazak Senu ◽  
Mohamed Suleiman ◽  
Fudziah Ismail ◽  
Norihan Md Arifin
Keyword(s):  
Differential Equations ◽  
Truncation Error ◽  
Second Order ◽  
Phase Lag ◽  
Initial Value ◽  
Third Order ◽  
Second Order Differential Equations ◽  
New Methods ◽  
Oscillating Solutions ◽  
Runge Kutta

New 4(3) pairs Diagonally Implicit Runge-Kutta-Nyström (DIRKN) methods with reduced phase-lag are developed for the integration of initial value problems for second-order ordinary differential equations possessing oscillating solutions. Two DIRKN pairs which are three- and four-stage with high order of dispersion embedded with the third-order formula for the estimation of the local truncation error. These new methods are more efficient when compared with current methods of similar type and with the L-stable Runge-Kutta pair derived by Butcher and Chen (2000) for the numerical integration of second-order differential equations with periodic solutions.

scholarly journals New Phase-Fitted and Amplification-Fitted Fourth-Order and Fifth-Order Runge-Kutta-Nyström Methods for Oscillatory Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
K. W. Moo ◽  
N. Senu ◽  
F. Ismail ◽  
M. Suleiman
Keyword(s):  
Fourth Order ◽  
Variable Coefficients ◽  
Phase Lag ◽  
Oscillatory Solutions ◽  
Numerical Tests ◽  
New Methods ◽  
Nyström Methods ◽  
Runge Kutta ◽  
Fifth Order ◽  
New Phase

Two new Runge-Kutta-Nyström (RKN) methods are constructed for solving second-order differential equations with oscillatory solutions. These two new methods are constructed based on two existing RKN methods. Firstly, a three-stage fourth-order Garcia’s RKN method. Another method is Hairer’s RKN method of four-stage fifth-order. Both new derived methods have two variable coefficients with phase-lag of order infinity and zero amplification error (zero dissipative). Numerical tests are performed and the results show that the new methods are more accurate than the other methods in the literature.

scholarly journals A New Family of Phase-Fitted and Amplification-Fitted Runge-Kutta Type Methods for Oscillators

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Zhaoxia Chen ◽  
Xiong You ◽  
Xin Shu ◽  
Mei Zhang
Keyword(s):  
Initial Value Problems ◽  
Numerical Experiments ◽  
Order Conditions ◽  
Initial Value ◽  
Oscillatory Solutions ◽  
New Methods ◽  
New Family ◽  
Algebraic Order ◽  
Runge Kutta ◽  
Phase Fitting

In order to solve initial value problems of differential equations with oscillatory solutions, this paper improves traditional Runge-Kutta (RK) methods by introducing frequency-depending weights in the update. New practical RK integrators are obtained with the phase-fitting and amplification-fitting conditions and algebraic order conditions. Two of the new methods have updates that are also phase-fitted and amplification-fitted. The linear stability and phase properties of the new methods are examined. The results of numerical experiments on physical and biological problems show the robustness and competence of the new methods compared to some highly efficient integrators in the literature.

EXPONENTIALLY FITTED RUNGE–KUTTA FOURTH ALGEBRAIC ORDER METHODS FOR THE NUMERICAL SOLUTION OF THE SCHRÖDINGER EQUATION AND RELATED PROBLEMS

2000 ◽  
Vol 11 (04) ◽  
pp. 785-807 ◽  
Author(s):  
P. S. WILLIAMS ◽  
T. E. SIMOS
Keyword(s):  
Schrödinger Equation ◽  
Numerical Solution ◽  
Fourth Order ◽  
Numerical Experiments ◽  
Schrodinger Equation ◽  
Variable Step ◽  
Algebraic Order ◽  
Exponentially Fitted ◽  
Runge Kutta ◽  
Step Algorithm

Fourth order exponential and trigonometric fitted Runge–Kutta methods are developed in this paper. They are applied to problems involving the Schrödinger equation and to other related problems. Numerical results show the superiority of these methods over conventional fourth order Runge–Kutta methods. Based on the methods developed in this paper, a variable-step algorithm is proposed. Numerical experiments show the efficiency of the new algorithm.

scholarly journals Sixth Order Numerov-Type Methods with Coefficients Trained to Perform Best on Problems with Oscillating Solutions

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2756
Author(s):  
Vladislav N. Kovalnogov ◽  
Ruslan V. Fedorov ◽  
Tamara V. Karpukhina ◽  
Theodore E. Simos ◽  
Charalampos Tsitouras
Keyword(s):  
Differential Evolution ◽  
Test Problem ◽  
Interval Of Periodicity ◽  
The Other ◽  
Sixth Order ◽  
Phase Lag ◽  
Oscillatory Solutions ◽  
Oscillating Solutions ◽  
Algebraic Order ◽  
Free Parameters

Numerov-type methods using four stages per step and sharing sixth algebraic order are considered. The coefficients of such methods are depended on two free parameters. For addressing problems with oscillatory solutions, we traditionally try to satisfy some specific properties such as reduce the phase-lag error, extend the interval of periodicity or even nullify the amplification. All of these latter properties come from a test problem that poses as a solution to an ideal trigonometric orbit. Here, we propose the training of the coefficients of the selected family of methods in a wide set of relevant problems. After performing this training using the differential evolution technique, we arrive at a certain method that outperforms the other ones from this family in an even wider set of oscillatory problems.

FOURTH ORDER SYMPLECTIC INTEGRATION WITH REDUCED PHASE ERROR

2008 ◽  
Vol 19 (08) ◽  
pp. 1257-1268 ◽  
Author(s):  
HANS VAN DE VYVER
Keyword(s):  
Hamiltonian Systems ◽  
Fourth Order ◽  
Numerical Experiments ◽  
Phase Error ◽  
Phase Lag ◽  
Order Method ◽  
Symplectic Integration ◽  
Algebraic Order ◽  
Fourth Order Method ◽  
Order Four

In this paper we introduce a symplectic explicit RKN method for Hamiltonian systems with periodical solutions. The method has algebraic order four and phase-lag order six at a cost of four function evaluations per step. Numerical experiments show the relevance of the developed algorithm. It is found that the new method is much more efficient than the standard symplectic fourth-order method.

scholarly journals New oscillation criteria for second-order neutral differential equations with distributed deviating arguments

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Osama Moaaz ◽  
Choonkil Park ◽  
Elmetwally M. Elabbasy ◽  
Waed Muhsin
Keyword(s):  
Differential Equations ◽  
Second Order ◽  
The Other ◽  
Riccati Transformation ◽  
Transformation Technique ◽  
Deviating Arguments ◽  
Neutral Differential Equations ◽  
Second Order Differential Equations ◽  
Continuous Delay ◽  
New Criteria

AbstractIn this work, we create new oscillation conditions for solutions of second-order differential equations with continuous delay. The new criteria were created based on Riccati transformation technique and comparison principles. Furthermore, we obtain iterative criteria that can be applied even when the other criteria fail. The results obtained in this paper improve and extend the relevant previous results as illustrated by examples.

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