scholarly journals A Phase-Fitted and Amplification-Fitted Explicit Runge–Kutta–Nyström Pair for Oscillating Systems

2021 ◽  
Vol 26 (3) ◽  
pp. 59
Author(s):  
Musa Ahmed Demba ◽  
Higinio Ramos ◽  
Poom Kumam ◽  
Wiboonsak Watthayu
Keyword(s):  
Stability Analysis ◽  
Numerical Experiments ◽  
Order Of Convergence ◽  
Truncation Error ◽  
Local Truncation Error ◽  
Nyström Methods ◽  
Oscillating Systems ◽  
The Common ◽  
Runge Kutta ◽  
The Stability

An optimized embedded 5(3) pair of explicit Runge–Kutta–Nyström methods with four stages using phase-fitted and amplification-fitted techniques is developed in this paper. The new adapted pair can exactly integrate (except round-off errors) the common test: y″=−w2y. The local truncation error of the new method is derived, and we show that the order of convergence is maintained. The stability analysis is addressed, and we demonstrate that the developed method is absolutely stable, and thus appropriate for solving stiff problems. The numerical experiments show a better performance of the new embedded pair in comparison with other existing RKN pairs of similar characteristics.

Numerical Method for The Time Fractional Fokker-Planck Equation

2012 ◽  
Vol 4 (06) ◽  
pp. 848-863 ◽  
Author(s):  
Xue-Nian Cao ◽  
Jiang-Li Fu ◽  
Hu Huang
Keyword(s):  
Theoretical Analysis ◽  
Numerical Method ◽  
Numerical Experiments ◽  
Truncation Error ◽  
Planck Equation ◽  
Local Truncation Error ◽  
Order Of Accuracy ◽  
Fokker Planck Equation ◽  
The Stability ◽  
Fokker Planck

AbstractIn this paper, a new numerical algorithm for solving the time fractional Fokker-Planck equation is proposed. The analysis of local truncation error and the stability of this method are investigated. Theoretical analysis and numerical experiments show that the proposed method has higher order of accuracy for solving the time fractional Fokker-Planck equation.

scholarly journals Efficient Two-Derivative Runge-Kutta-Nyström Methods for Solving General Second-Order Ordinary Differential Equations y′′(x)=f(x,y,y′)

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
T. S. Mohamed ◽  
N. Senu ◽  
Z. B. Ibrahim ◽  
N. M. A. Nik Long
Keyword(s):  
Special Class ◽  
Numerical Experiments ◽  
Stability Property ◽  
Second Derivative ◽  
New Methods ◽  
Nyström Methods ◽  
Three Stages ◽  
Runge Kutta ◽  
The Stability ◽  
Third Derivative

This paper proposes and investigates a special class of explicit Runge-Kutta-Nyström (RKN) methods for problems in the form y′′(x)=f(x,y,y′) including third derivatives and denoted as STDRKN. The methods involve one evaluation of second derivative and many evaluations of third derivative per step. In this study, methods with two and three stages of orders four and five, respectively, are presented. The stability property of the methods is discussed. Numerical experiments have clearly shown the accuracy and the efficiency of the new methods.

scholarly journals The Stability of the Aggregate Loss Distribution

Risks ◽  
2018 ◽  
Vol 6 (3) ◽  
pp. 91 ◽  
Author(s):  
Riccardo Gatto
Keyword(s):  
Stability Analysis ◽  
Survival Function ◽  
Saddlepoint Approximation ◽  
Computational Formula ◽  
Important Indicator ◽  
The Common ◽  
Infinitesimal Perturbation ◽  
Aggregate Loss ◽  
Aggregate Loss Distribution ◽  
The Stability

In this article we introduce the stability analysis of a compound sum: it consists of computing the standardized variation of the survival function of the sum resulting from an infinitesimal perturbation of the common distribution of the summands. Stability analysis is complementary to the classical sensitivity analysis, which consists of computing the derivative of an important indicator of the model, with respect to a model parameter. We obtain a computational formula for this stability from the saddlepoint approximation. We apply the formula to the compound Poisson insurer loss with gamma individual claim amounts and to the compound geometric loss with Weibull individual claim amounts.

scholarly journals Numerical Study on Phase-Fitted and Amplification-Fitted Diagonally Implicit Two Derivative Runge-Kutta Method for Periodic IVPS

2021 ◽  
Vol 50 (6) ◽  
pp. 1799-1814
Author(s):  
Norazak Senu ◽  
Nur Amirah Ahmad ◽  
Zarina Bibi Ibrahim ◽  
Mohamed Othman
Keyword(s):  
Fourth Order ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
Numerical Study ◽  
Kutta Method ◽  
Initial Value ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability

A fourth-order two stage Phase-fitted and Amplification-fitted Diagonally Implicit Two Derivative Runge-Kutta method (PFAFDITDRK) for the numerical integration of first-order Initial Value Problems (IVPs) which exhibits periodic solutions are constructed. The Phase-Fitted and Amplification-Fitted property are discussed thoroughly in this paper. The stability of the method proposed are also given herewith. Runge-Kutta (RK) methods of the similar property are chosen in the literature for the purpose of comparison by carrying out numerical experiments to justify the accuracy and the effectiveness of the derived method.

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.

A new high algebraic order efficient finite difference method for the solution of the Schrödinger equation

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4999-5012 ◽  
Author(s):  
Ming Dong ◽  
Theodore Simos
Keyword(s):  
Truncation Error ◽  
Interval Of Periodicity ◽  
Phase Lag ◽  
Local Truncation Error ◽  
Step Method ◽  
Dinger Equation ◽  
Algebraic Order ◽  
Test Equation ◽  
The Stability ◽  
Scalar Test Equation

The development of a new five-stages symmetric two-step method of fourteenth algebraic order with vanished phase-lag and its first, second, third and fourth derivatives is analyzed in this paper. More specifically: (1) we will present the development of the new method, (2) we will determine the local truncation error (LTE) of the new proposed method, (3) we will analyze the local truncation error based on the radial time independent Schr?dinger equation, (4) we will study the stability and the interval of periodicity of the new proposed method based on a scalar test equation with frequency different than the frequency of the scalar test equation used for the phase-lag analysis, (5) we will test the efficiency of the new obtained method based on its application on the coupled differential equations arising from the Schr?dinger equation.

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