MULTIDERIVATIVE HYBRID IMPLICIT RUNGE-KUTTA METHOD FOR SOLVING STIFF SYSTEM OF A FIRST ORDER DIFFERENTIAL EQUATION

2018 ◽  
Vol 106 (2) ◽  
pp. 543-562
Author(s):  
Olusheye A. Akinfenwa ◽  
Solomon A. Okunuga ◽  
Blessing I. Akinnukawe ◽  
Uthman O. Rufai ◽  
Ridwanulahi I. Abdulganiy
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Kutta Method ◽  
Stiff System ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta

scholarly journals On the integration processes of A. Huťa

1963 ◽  
Vol 3 (2) ◽  
pp. 202-206 ◽  
Author(s):  
J. C. Butcher
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Sixth Order ◽  
Order Accuracy ◽  
First Order ◽  
Runge Kutta ◽  
Image Position

Huta [1], [2] has given two processes for solving a first order differential equation to sixth order accuracy. His methods are each eight stage Runge-Kutta processes and differ mainly in that the later process has simpler coefficients occurring in it.

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.

Modeling on Falling Velocity of Sodiumtetraborate Aqueous Solution Drops before the Gelation of PVA-TiO2 Suspensions by the Runge-Kutta Method in Matlab 6.5

2008 ◽  
Vol 368-372 ◽  
pp. 1683-1685
Author(s):  
Cheng Long Yu ◽  
Xiu Feng Wang ◽  
Jun Xin Zhou ◽  
Hong Tao Jiang ◽  
Yan Wang
Keyword(s):  
Differential Equation ◽  
Aqueous Solution ◽  
Ordinary Differential Equation ◽  
Elevation Angle ◽  
Time Dependent ◽  
Quadratic Term ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Air Resistance ◽  
Runge Kutta

Numerical modeling on falling of sodiumtetraborate aqueous solution drops as the initiator before the gelation of PVA-TiO2 suspensions was conducted. Effect of time and elevation angle of the PVA-TiO2 suspensions on the falling velocity of the sodiumtetraborate aqueous solution drops was analyzed. An ordinary differential equation was given. Integration of the ordinary differential equation was fulfilled using the fourth-order Runge-Kutta method in Matlab 6.5. From the model, a two-order nonlinear effect of time on the velocity of the drops during falling is determined and the quadratic term -3.408t2 serves as the time dependent air resistance. The component of the falling velocity along the suspensions increases with the increasing of the elevation angle. However, for the component vertical to the suspensions, with elevation angle increasing, it decreases.

scholarly journals Nonlinear Parametric Resonance Behavior of Cables in a Long Cantilever Bridge for Sightseeing Platform

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Zengwei Guo ◽  
Pengfei Zi ◽  
Xuanbo He
Keyword(s):  
Differential Equation ◽  
Nonlinear Vibration ◽  
Damping Ratio ◽  
Element Model ◽  
Excitation Amplitude ◽  
Parametric Vibration ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Vibration Performance

In order to study the parametric vibration of stayed cables in a long cantilever bridge for a sightseeing platform, nonlinear parametric vibration equations of the stayed cables excited by the vibration of bridge deck and tower are derived. Then, a second-order differential equation is transformed into a first-order ordinary differential equation, which is solved by using the Runge–Kutta method. A finite element model of cables was also built to verify the solution of the Runge–Kutta method. Then, the inherent dynamic characteristics of the full structure and all the cables with different lengths were analyzed to discuss the potential risk of parametric vibration. The longest and shortest cables were taken as examples to explore their nonlinear vibration performance. The effects of damping ratio, excitation position, and amplitude on cables’ nonlinear vibration performance were investigated. The results show that it will be more efficient and convenient to use the Runge–Kutta method to calculate cables’ nonlinear vibration amplitude without loss of accuracy. In addition, short cables have more resonance zones compared to long cables. Especially, with the cable length shortening, the dominant frequencies of the dynamic response and its amplitude increase significantly, and the number of resonance zones also increases. However, excessive excitation amplitude will also cause multiple resonance zones in the cable. The parametric analysis results show that it is effective and efficient to mitigate the nonlinear vibration by adjusting the frequency relationship between the bridge and the cables, rather than by increasing the damping ratio.

Sign in / Sign up

Export Citation Format

Share Document