High Order Contractive Runge–Kutta Methods for Volterra Functional Differential Equations

2010 ◽  
Vol 47 (6) ◽  
pp. 4290-4325 ◽  
Author(s):  
Shoufu Li
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
High Order ◽  
Runge Kutta ◽  
Functional Differential

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.

Positive periodic solutions for singular high-order neutral functional differential equations

2018 ◽  
Vol 68 (2) ◽  
pp. 379-396 ◽  
Author(s):  
Fanchao Kong ◽  
Zhiguo Luo ◽  
Shiping Lu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Functional Differential Equations ◽  
High Order ◽  
Positive Periodic Solutions ◽  
Neutral Functional Differential Equation ◽  
Neutral Functional Differential Equations ◽  
Small Force ◽  
Functional Differential

Abstract In this paper, we establish new results on the existence of positive periodic solutions for the following high-order neutral functional differential equation (NFDE) $$\begin{array}{} (x(t)-cx(t-\sigma)) ^{(2m)}+f(x(t)) x'(t)+g(t,x(t-\delta))=e(t). \end{array}$$ The interesting thing is that g has a strong singularity at x = 0 and satisfies a small force condition at x = ∞, which is different from the corresponding ones known in the literature. Two examples are given to illustrate the effectiveness of our results.

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