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.

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 Explicit order 3/2 Runge-Kutta method for numerical solutions of stochastic differential equations by using Itô-Taylor expansion

2019 ◽  
Vol 17 (1) ◽  
pp. 1515-1525
Author(s):  
Yazid Alhojilan
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Taylor Expansion ◽  
Transport Theory ◽  
Numerical Solutions ◽  
Optimal Transport ◽  
Stochastic Integrals ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Runge Kutta

Abstract This paper aims to present a new pathwise approximation method, which gives approximate solutions of order $\begin{array}{} \displaystyle \frac{3}{2} \end{array}$ for stochastic differential equations (SDEs) driven by multidimensional Brownian motions. The new method, which assumes the diffusion matrix non-degeneracy, employs the Runge-Kutta method and uses the Itô-Taylor expansion, but the generating of the approximation of the expansion is carried out as a whole rather than individual terms. The new idea we applied in this paper is to replace the iterated stochastic integrals Iα by random variables, so implementing this scheme does not require the computation of the iterated stochastic integrals Iα. Then, using a coupling which can be found by a technique from optimal transport theory would give a good approximation in a mean square. The results of implementing this new scheme by MATLAB confirms the validity of the method.

scholarly journals Numerical Solution of Fuzzy Delay Differential Equations by Fifth Order Runge-Kutta Method

2021 ◽  
Vol 23 (11) ◽  
pp. 99-109
Author(s):  
T. Muthukumar ◽  
◽  
T. Jayakumar ◽  
D.Prasantha Bharathi ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Delay Differential Equations ◽  
Numerical Solutions ◽  
Kutta Method ◽  
Numerical Examples ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Delay Differential ◽  
Fifth Order

In this paper, we develop the numerical solutions of certain type called Fuzzy Delay Differential Equations(FDE) by using fifth order Runge-Kutta method for fuzzy differential equations. This method based on the seikkala derivative and finally we discuss the numerical examples to illustrate the theory.

Sign in / Sign up

Export Citation Format

Share Document