Numerical method of lines for evolution functional differential equations

2011 ◽  
Vol 19 (1) ◽  
Author(s):  
Z. Kamont ◽  
M. Netka
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Functional Differential Equations ◽  
Method Of Lines ◽  
Functional Differential

Numerical Analysis of Fractional Neutral Functional Differential Equations Based on Generalized Volterra-Integral Operators

2017 ◽  
Vol 12 (3) ◽  
Author(s):  
Xiao-Li Ding ◽  
Juan J. Nieto
Keyword(s):  
Differential Equation ◽  
Integral Operator ◽  
Differential Equations ◽  
Numerical Method ◽  
Functional Differential Equation ◽  
Functional Differential Equations ◽  
Waveform Relaxation ◽  
Neutral Functional Differential Equation ◽  
Neutral Functional Differential Equations ◽  
Functional Differential

We use waveform relaxation (WR) method to solve numerically fractional neutral functional differential equations and mainly consider the convergence of the numerical method with the help of a generalized Volterra-integral operator associated with the Mittag–Leffler function. We first give some properties of the integral operator. Using the proposed properties, we establish the convergence condition of the numerical method. Finally, we provide a new way to prove the convergence of waveform relaxation method for integer-order neutral functional differential equation, which is a special case of fractional neutral functional differential equation. Compared to the existing proof in the literature, our proof is concise and original.

Numerical Method of Hybrid Stochastic Functional Differential Equations with the Local Lipschitz Coefficients

2011 ◽  
Vol 267 ◽  
pp. 422-426
Author(s):  
Hua Yang ◽  
Feng Jiang ◽  
Jun Hao Hu
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Method ◽  
Lipschitz Condition ◽  
Numerical Solutions ◽  
Functional Differential Equations ◽  
Stochastic Functional Differential Equations ◽  
Local Lipschitz Condition ◽  
Functional Differential ◽  
Stochastic Functional

Recently, hybrid stochastic differential equations have received a great deal of attention. It is surprising that there are not any numerical schemes established for the hybrid stochastic functional differential equations. In this paper, the Euler—Maruyama method is developed, and the main aim is to show that the numerical solutions will converge to the true solutions under the local Lipschitz condition. The result obtained generalizes the earlier results.

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