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 The Use of Phase Lag and Amplification Error Derivatives for the Construction of a Modified Runge-Kutta-Nyström Method

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
D. F. Papadopoulos ◽  
T. E. Simos
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Numerical Results ◽  
Phase Lag ◽  
Nyström Method ◽  
Nystrom Method ◽  
Modified Method ◽  
Algebraic Order ◽  
Runge Kutta

A new modified Runge-Kutta-Nyström method of fourth algebraic order is developed. The new modified RKN method is based on the fitting of the coefficients, due to the nullification not only of the phase lag and of the amplification error, but also of their derivatives. Numerical results indicate that the new modified method is much more efficient than other methods derived for solving numerically the Schrödinger equation.

scholarly journals Numerical Solution of Second-Order Fuzzy Differential Equation Using Improved Runge-Kutta Nystrom Method

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Faranak Rabiei ◽  
Fudziah Ismail ◽  
Ali Ahmadian ◽  
Soheil Salahshour
Keyword(s):  
Differential Equation ◽  
Computational Cost ◽  
Second Order ◽  
Fuzzy Differential Equation ◽  
Nyström Method ◽  
Numerical Examples ◽  
Time Consumption ◽  
Nystrom Method ◽  
Fuzzy Function ◽  
Runge Kutta

We develop the Fuzzy Improved Runge-Kutta Nystrom (FIRKN) method for solving second-order fuzzy differential equations (FDEs) based on the generalized concept of higher-order fuzzy differentiability. The scheme is two-step in nature and requires less number of stages which leads to less number of function evaluations in comparison with the existing Fuzzy Runge-Kutta Nystrom method. Therefore, the new method has a lower computational cost which effects the time consumption. We assume that the fuzzy function and its derivative are Hukuhara differentiable. FIRKN methods of orders three, four, and five are derived with two, three, and four stages, respectively. The numerical examples are given to illustrate the efficiency of the methods.

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