scholarly journals Numerical Solutions of Nonlinear Ordinary Differential Equations by Using Adaptive Runge-Kutta Method

2019 ◽  
Vol 17 ◽  
pp. 147-154
Author(s):  
Abhinandan Chowdhury ◽  
Sammie Clayton ◽  
Mulatu Lemma
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Solutions ◽  
Adaptive Methods ◽  
Integration Step ◽  
Nonlinear Ordinary Differential Equations ◽  
Kutta Method ◽  
Step Size ◽  
Runge Kutta Method ◽  
Runge Kutta

We present a study on numerical solutions of nonlinear ordinary differential equations by applying Runge-Kutta-Fehlberg (RKF) method, a well-known adaptive Runge-kutta method. The adaptive Runge-kutta methods use embedded integration formulas which appear in pairs. Typically adaptive methods monitor the truncation error at each integration step and automatically adjust the step size to keep the error within prescribed limit. Numerical solutions to different nonlinear initial value problems (IVPs) attained by RKF method are compared with corresponding classical Runge-Kutta (RK4) approximations in order to investigate the computational superiority of the former. The resulting gain in efficiency is compatible with the theoretical prediction. Moreover, with the aid of a suitable time-stepping scheme, we show that the RKF method invariably requires less number of steps to arrive at the right endpoint of the finite interval where the IVP is being considered.

scholarly journals DEGENERATE MATRIX METHOD FOR SOLVING NONLINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS

1998 ◽  
Vol 3 (1) ◽  
pp. 45-56
Author(s):  
T. Cîrulis ◽  
O. Lietuvietis
Keyword(s):  
Nonlinear Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Matrix Method ◽  
Iteration Procedure ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Systems Of Differential Equations ◽  
Runge Kutta ◽  
Degenerate Matrix

Degenerate matrix method for numerical solving nonlinear systems of ordinary differential equations is considered. The method is based on an application of special degenerate matrix and usual iteration procedure. The method, which is connected with an implicit Runge‐Kutta method, can be simply realized on computers. An estimation for the error of the method is given.

scholarly journals Study on Banded Implicit Runge–Kutta Methods for Solving Stiff Differential Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
M. Y. Liu ◽  
L. Zhang ◽  
C. F. Zhang
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Coefficient Matrix ◽  
Kutta Method ◽  
Stiff Differential Equations ◽  
Banded Matrix ◽  
Fully Implicit ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Cost

The implicit Runge–Kutta method with A-stability is suitable for solving stiff differential equations. However, the fully implicit Runge–Kutta method is very expensive in solving large system problems. Although some implicit Runge–Kutta methods can reduce the cost of computation, their accuracy and stability are also adversely affected. Therefore, an effective banded implicit Runge–Kutta method with high accuracy and high stability is proposed, which reduces the computation cost by changing the Jacobian matrix from a full coefficient matrix to a banded matrix. Numerical solutions and results of stiff equations obtained by the methods involved are compared, and the results show that the banded implicit Runge–Kutta method is advantageous to solve large stiff problems and conducive to the development of simulation.

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