Lee Khai Chien, Norazak Senu, Ali Ahmadian and Siti Nur Iqmal Ibrahim
Pertanika Journal of Science & Technology, Volume 31, Issue 2, March 2023
Keywords: Exponentially-fitted, third-order ordinary differential equations, trigonometrically-fitted, two-derivative runge-kutta methods
Published on: 20 March 2023
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 eѡt and e-ѡtfor exponentially fitted and eiѡt 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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