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

Ahmad, Nur Amirah; Senu, Norazak; Ismail, Fudziah

doi:10.1155/2017/1871278

Cited by 7 publications

(4 citation statements)

References 25 publications

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“…In recent, many research are widely studied by researchers regarding the characteristics of solutions with frequency-dependent properties and the development of efficient methods to solve ODEs with exponential and oscillatory solutions to illustrate the model of application problems such as orbital mechanics, molecular dynamics and electronics, Van der Pol's equations, Kepler's problem in a dynamical system, Bessel equations and harmonic oscillator (Franco & Randez, 2018;Ahmad et al, 2020). Simos and Williams (1999) constructed exponentially and trigonometrically fitted Runge-Kutta methods with order three for solving the Schrödinger equation.…”

Section: Introductionmentioning

confidence: 99%

Efficient Frequency-Dependent Coefficients of Explicit Improved Two-Derivative Runge-Kutta Type Methods for Solving Third-Order IVPs

Chien¹,

Senu²,

Ahmadian³

et al. 2023

JST

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This study aims to propose sixth-order two-derivative improved Runge-Kutta type methods adopted with exponentially-fitting and trigonometrically-fitting techniques for integrating a special type of third-order ordinary differential equation in the form u^''' (t)=f(t,u(t),u^' (t)). The procedure of constructing order conditions comprised of a few previous steps, k-i for third-order two-derivative Runge-Kutta-type methods, has been outlined. These methods are developed through the idea of integrating initial value problems exactly with a numerical solution in the form of linear composition of the set functions e^ѡt and e^(-ѡt)for exponentially fitted and e^iѡt and e^(-iѡt) for trigonometrically-fitted with ѡ ϵ R. Parameters of two-derivative Runge-Kutta type method are adapted into principle frequency of exponential and oscillatory problems to construct the proposed methods. Error analysis of proposed methods is analysed, and the computational efficiency of proposed methods is demonstrated in numerical experiments compared to other existing numerical methods for integrating third-order ordinary differential equations with an exponential and periodic solution.

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Section: Introductionmentioning

confidence: 99%

Efficient Frequency-Dependent Coefficients of Explicit Improved Two-Derivative Runge-Kutta Type Methods for Solving Third-Order IVPs

Chien¹,

Senu²,

Ahmadian³

et al. 2023

JST

View full text Add to dashboard Cite

show abstract

“…In the development of methods with constant coefficients, one has to consider the algebraic order, the phase-lag order, the dissipation order, the strategy of minimizing the error constant and the size of interval of periodicity or the interval of absolute stability of the resulting methods (e.g., [1][2][3][4][5]). Meanwhile, development of methods with variable coefficients involves the usage of various techniques such as phase-fitting, amplification-fitting, trigonometric-fitting and exponential-fitting (e.g., [6][7][8]). These techniques are used to get methods specially adapted to the certain behavior or the structure of the solution of the problem, hence the methods with variable coefficients usually depend on the frequency of the problem.…”

Section: Introductionmentioning

confidence: 99%

Variable Step Exponentially Fitted Explicit Sixth-Order Hybrid Method with Four Stages for Spring-Mass and Other Oscillatory Problems

Samat

Ismail

2020

Symmetry

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For the numerical integration of differential equations with oscillatory solutions an exponentially fitted explicit sixth-order hybrid method with four stages is presented. This method is implemented using variable step-size while its derivation is accomplished by imposing each stage of the formula to integrate exactly { 1 , t , t 2 , … , t k , exp ( ± μ t ) } where the frequency μ is imaginary. The local error that is employed in the step-size selection procedure is approximated using an exponentially fitted explicit fourth-order hybrid method. Numerical comparisons of the new and existing hybrid methods for the spring-mass and other oscillatory problems are tabulated and discussed. The results show that the variable step exponentially fitted explicit sixth-order hybrid method outperforms the existing hybrid methods with variable coefficients for solving several problems with oscillatory solutions.

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“…Among these time steppers, very wide-spread are the ones with variable coefficients, such as phase-fitted and/or amplification-fitted, trigonometrically-fitted etc. (e.g., [19][20][21][22][23][24][25]). This approach requires a well-defined dominant frequency of the oscillations along the propagation coordinate and, when this is not the case, they under-perform.…”

Section: Introductionmentioning

confidence: 99%

Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients

et al. 2020

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Motivated by the limited work performed on the development of computational techniques for solving the nonlinear Schrödinger equation with time-dependent coefficients, we develop a modified Runge-Kutta pair with improved periodicity and stability characteristics. Additionally, we develop a modified step size control algorithm, which increases the efficiency of our pair and all other pairs included in the numerical experiments. The numerical results on the nonlinear Schrödinger equation with a periodic solution verified the superiority of the new algorithm in terms of efficiency.The new method also presents a good behaviour of the maximum absolute error and the global norm in time, even after a high number of oscillations.

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

Cited by 7 publications

References 25 publications

Efficient Frequency-Dependent Coefficients of Explicit Improved Two-Derivative Runge-Kutta Type Methods for Solving Third-Order IVPs

Efficient Frequency-Dependent Coefficients of Explicit Improved Two-Derivative Runge-Kutta Type Methods for Solving Third-Order IVPs

Variable Step Exponentially Fitted Explicit Sixth-Order Hybrid Method with Four Stages for Spring-Mass and Other Oscillatory Problems

Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients

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