scholarly journals Analytical and numerical aspect of coincidence point problem of quasi-contractive operators

2019 ◽  
Vol 105 (119) ◽  
pp. 101-121
Author(s):  
Faik Gürsoy ◽  
Müzeyyen Ertürk ◽  
Abdul Khan ◽  
Vatan Karakaya
Keyword(s):  
Convergence Rate ◽  
Strong Convergence ◽  
Rate Of Convergence ◽  
Metric Space ◽  
Iteration Method ◽  
Coincidence Point ◽  
Point Problem ◽  
Convex Metric Space ◽  
Numerical Examples ◽  
Strong Convergence Rate

We propose a new Jungck-S iteration method for a class of quasi-contractive operators on a convex metric space and study its strong convergence, rate of convergence and stability. We also provide conditions under which convergence of this method is equivalent to Jungck-Ishikawa iteration method. Some numerical examples are provided to validate the theoretical findings obtained herein. Our results are refinement and extension of the corresponding ones existing in the current literature.

scholarly journals Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen–Cahn equation

2018 ◽  
Vol 39 (4) ◽  
pp. 2096-2134 ◽  
Author(s):  
Charles-Edouard Bréhier ◽  
Jianbo Cui ◽  
Jialin Hong
Keyword(s):  
Convergence Rate ◽  
Strong Convergence ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
A Priori Estimates ◽  
A Priori ◽  
Auxiliary Problem ◽  
Splitting Scheme ◽  
Cahn Equation ◽  
Strong Convergence Rate

Abstract This article analyses an explicit temporal splitting numerical scheme for the stochastic Allen–Cahn equation driven by additive noise in a bounded spatial domain with smooth boundary in dimension $d\leqslant 3$. The splitting strategy is combined with an exponential Euler scheme of an auxiliary problem. When $d=1$ and the driving noise is a space–time white noise we first show some a priori estimates of this splitting scheme. Using the monotonicity of the drift nonlinearity we then prove that under very mild assumptions on the initial data this scheme achieves the optimal strong convergence rate $\mathcal{O}(\delta t^{\frac 14})$. When $d\leqslant 3$ and the driving noise possesses some regularity in space we study exponential integrability properties of the exact and numerical solutions. Finally, in dimension $d=1$, these properties are used to prove that the splitting scheme has a strong convergence rate $\mathcal{O}(\delta t)$.

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.

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