$B$-Convergence Theory of Runge--Kutta Methods for Stiff Volterra Functional Differential Equations with Infinite Integration Interval

2015 ◽  
Vol 53 (6) ◽  
pp. 2570-2583 ◽  
Author(s):  
Shoufu Li ◽  
Yunfei Li
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Convergence Theory ◽  
Integration Interval ◽  
Runge Kutta ◽  
Functional Differential

scholarly journals Numerical Solutions of Stochastic Functional Differential Equations

2003 ◽  
Vol 6 ◽  
pp. 141-161 ◽  
Author(s):  
Xuerong Mao
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Functional Differential Equations ◽  
Convergence Theory ◽  
Stochastic Functional Differential Equations ◽  
True Solution ◽  
Linear Growth Condition ◽  
Mean Square Convergence ◽  
Functional Differential ◽  
Stochastic Functional

AbstractIn this paper, the strong mean square convergence theory is established for the numerical solutions of stochastic functional differential equations (SFDEs) under the local Lipschitz condition and the linear growth condition. These two conditions are generally imposed to guarantee the existence and uniqueness of the true solution, so the numerical results given here were obtained under quite general conditions.

RUNGE–KUTTA METHODS FOR RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS

2005 ◽  
Vol 15 (08) ◽  
pp. 1203-1251 ◽  
Author(s):  
STEFANO MASET ◽  
LUCIO TORELLI ◽  
ROSSANA VERMIGLIO
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Polynomial Function ◽  
Functional Differential Equations ◽  
Order Conditions ◽  
Polynomial Functions ◽  
Retarded Functional Differential Equations ◽  
Runge Kutta ◽  
Functional Differential ◽  
Order Four

We introduce Runge–Kutta (RK) methods for Retarded Functional Differential Equations (RFDEs). With respect to RK methods (A, b, c) for Ordinary Differential Equations the weights vector b ∈ ℝs and the coefficients matrix A ∈ ℝs×s are replaced by ℝs-valued and ℝs×s-valued polynomial functions b(·) and A(·) respectively. Such methods for RFDEs are different from Continuous RK (CRK) methods where only the weights vector is replaced by a polynomial function. We develop order conditions and construct explicit methods up to the convergence order four.

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