Trigonometrically fitted two-derivative Runge-Kutta methods for solving oscillatory differential equations

Fang, Yonglei; You, Xiong; Ming, Qinghe

doi:10.1007/s11075-013-9802-z

Cited by 26 publications

(17 citation statements)

References 17 publications

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“…Source : Yonglei et al [19]. For comparison purpose, in analyzing the numerical results, methods with the same order will be compared.…”

Section: Comparison With Fixed Step-size Methodsmentioning

confidence: 99%

A new embedded phase-fitted modified Runge-Kutta method for the numerical solution of oscillatory problems

Fawzi¹,

Senu²,

Ismail³

et al. 2016

ams

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In this work, a new way for constructing an efficiently modified Runge-Kutta (RK) method to solve first-order ordinary differential equations with oscillatory solutions is provided. The proposed method solves the first-order ODEs by first converting the second order ODEs to an equivalent first-order ODEs. The method of the embedded has algebraic orders five and four. The numerical results of the new method have been compared with those of existing methods and showed that the new method is more efficient.

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“…Source : Yonglei et al [19]. For comparison purpose, in analyzing the numerical results, methods with the same order will be compared.…”

Section: Comparison With Fixed Step-size Methodsmentioning

confidence: 99%

A new embedded phase-fitted modified Runge-Kutta method for the numerical solution of oscillatory problems

Fawzi¹,

Senu²,

Ismail³

et al. 2016

ams

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“…Moo et al [18] constructed two new phase-fitted and amplification-fitted RKN based on Garcia and Hairer methods of orders four and five, respectively. Recently, Fang et al [19] proposed a block hybrid trigonometrically fitted RKN methods for solving problem (1). Demba et al [20] derived two trigonometrically 2 Mathematical Problems in Engineering fitted RKN methods based on Garcia and Hairer methods of orders four and five, respectively, for periodic IVPs.…”

Section: Introductionmentioning

confidence: 99%

Exponentially Fitted and Trigonometrically Fitted Two-Derivative Runge-Kutta-Nyström Methods for Solving y′′(x)=fx,y

Mohamed

Senu

İbrahim

et al. 2018

Mathematical Problems in Engineering

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Two exponentially fitted and trigonometrically fitted explicit two-derivative Runge-Kutta-Nyström (TDRKN) methods are being constructed. Exponentially fitted and trigonometrically fitted TDRKN methods have the favorable feature that they integrate exactly second-order systems whose solutions are linear combinations of functions {exp( ), exp(− )} and {sin( ), cos( )} respectively, when ∈ R, the frequency of the problem. The results of numerical experiments showed that the new approaches are more efficient than existing methods in the literature.

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“…Motivated by the work of Chan and Tsai [ 18 ] and Fang et al [ 19 ] on the two-derivative Runge-Kutta methods (TDRK), the objective of this paper is to develop a novel type of exponentially fitted two-derivative Runge-Kutta (EFTDRK) methods for simulating genetic regulatory systems with an oscillatory structure. These new numerical integrators respect the limit cycle structure of the system and are expected to be more accurate than the traditional RK methods in the long-term integration of gene regulatory systems.…”

Section: Introductionmentioning

confidence: 99%

Exponentially Fitted Two-Derivative Runge-Kutta Methods for Simulation of Oscillatory Genetic Regulatory Systems

Chen

Zhang

et al. 2015

Computational and Mathematical Methods in Medicine

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Oscillation is one of the most important phenomena in the chemical reaction systems in living cells. The general purpose simulation algorithms fail to take into account this special character and produce unsatisfying results. In order to enhance the accuracy of the integrator, the second-order derivative is incorporated in the scheme. The oscillatory feature of the solution is captured by the integrators with an exponential fitting property. Three practical exponentially fitted TDRK (EFTDRK) methods are derived. To test the effectiveness of the new EFTDRK methods, the two-gene system with cross-regulation and the circadian oscillation of the period protein in Drosophila are simulated. Each EFTDRK method has the best fitting frequency which minimizes the global error. The numerical results show that the new EFTDRK methods are more accurate and more efficient than their prototype TDRK methods or RK methods of the same order and the traditional exponentially fitted RK method in the literature.

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Trigonometrically fitted two-derivative Runge-Kutta methods for solving oscillatory differential equations

Cited by 26 publications

References 17 publications

A new embedded phase-fitted modified Runge-Kutta method for the numerical solution of oscillatory problems

A new embedded phase-fitted modified Runge-Kutta method for the numerical solution of oscillatory problems

Exponentially Fitted and Trigonometrically Fitted Two-Derivative Runge-Kutta-Nyström Methods for Solving y′′(x)=fx,y

Exponentially Fitted Two-Derivative Runge-Kutta Methods for Simulation of Oscillatory Genetic Regulatory Systems

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