scholarly journals Stability Analysis of Additive Runge-Kutta Methods for Delay-Integro-Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-5
Author(s):  
Hongyu Qin ◽  
Zhiyong Wang ◽  
Fumin Zhu ◽  
Jinming Wen
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Runge Kutta

This paper is concerned with stability analysis of additive Runge-Kutta methods for delay-integro-differential equations. We show that if the additive Runge-Kutta methods are algebraically stable, the perturbations of the numerical solutions are controlled by the initial perturbations from the system and the methods.

scholarly journals Optimal rate of convergence for two classes of schemes to stochastic differential equations driven by fractional Brownian motions

2020 ◽  
Author(s):  
Jialin Hong ◽  
Chuying Huang ◽  
Xu Wang
Keyword(s):  
Differential Equations ◽  
Convergence Rate ◽  
Strong Convergence ◽  
Stochastic Differential Equations ◽  
Numerical Solutions ◽  
Brownian Motions ◽  
Fractional Brownian Motions ◽  
Optimal Rate Of Convergence ◽  
Runge Kutta ◽  
Strong Convergence Rate

Abstract This paper investigates numerical schemes for stochastic differential equations driven by multi-dimensional fractional Brownian motions (fBms) with Hurst parameter $H\in (\frac 12,1)$. Based on the continuous dependence of numerical solutions on the driving noises, we propose the order conditions of Runge–Kutta methods for the strong convergence rate $2H-\frac 12$, which is the optimal strong convergence rate for approximating the Lévy area of fBms. We provide an alternative way to analyse the convergence rate of explicit schemes by adding ‘stage values’ such that the schemes are interpreted as Runge–Kutta methods. Taking advantage of this technique the strong convergence rate of simplified step-$N$ Euler schemes is obtained, which gives an answer to a conjecture in Deya et al. (2012) when $H\in (\frac 12,1)$. Numerical experiments verify the theoretical convergence rate.

scholarly journals A COMPARATIVE EXPLORATION ON DIFFERENT NUMERICAL METHODS FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS

2020 ◽  
Vol 15 (12) ◽  
Author(s):  
Mohammad Asif Arefin
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Solutions ◽  
Euler Method ◽  
Euler’S Method ◽  
Runge Kutta Method ◽  
Accuracy Level ◽  
Runge Kutta ◽  
Matlab Programming

In this paper, the initial value problem of Ordinary Differential Equations has been solved by using different Numerical Methods namely Euler’s method, Modified Euler method, and Runge-Kutta method. Here all of the three proposed methods have to be analyzed to determine the accuracy level of each method. By using MATLAB Programming language first we find out the approximate numerical solution of some ordinary differential equations and then to determine the accuracy level of the proposed methods we compare all these solutions with the exact solution. It is observed that numerical solutions are in good agreement with the exact solutions and numerical solutions become more accurate when taken step sizes are very much small. Lastly, the error of each proposed method is determined and represents them graphically which reveals the superiority among all the three methods. We fund that, among the proposed methods Runge-Kutta 4th order method gives the accurate result and minimum amount of error.

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