Order of Convergence of Linear Multistep Methods for Functional Differential Equations

1975 ◽  
Vol 12 (6) ◽  
pp. 876-886 ◽  
Author(s):  
Bruce A. Chartres ◽  
Robert S. Stepleman
Keyword(s):  
Differential Equations ◽  
Order Of Convergence ◽  
Multistep Methods ◽  
Functional Differential Equations ◽  
Linear Multistep Methods ◽  
Functional Differential

scholarly journals Classical Theory of Linear Multistep Methods for Volterra Functional Differential Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Yunfei Li ◽  
Shoufu Li
Keyword(s):  
Differential Equations ◽  
Numerical Experiments ◽  
Multistep Methods ◽  
Functional Differential Equations ◽  
Linear Multistep Methods ◽  
Interpolation Theory ◽  
Initial Value ◽  
Classical Stability ◽  
Functional Differential ◽  
Delay Differential

Based on the linear multistep methods for ordinary differential equations (ODEs) and the canonical interpolation theory that was presented by Shoufu Li who is exactly the second author of this paper, we propose the linear multistep methods for general Volterra functional differential equations (VFDEs) and build the classical stability, consistency, and convergence theories of the methods. The methods and theories presented in this paper are applicable to nonneutral, nonstiff, and nonlinear initial value problems in ODEs, Volterra delay differential equations (VDDEs), Volterra integro-differential equations (VIDEs), Volterra delay integro-differential equations (VDIDEs), etc. At last, some numerical experiments verify the correctness of our theories.

On application of the i-smooth analysis methodology to elaboration of numerical methods for solving functional differential equations

2021 ◽  
pp. 26-34
Author(s):  
Arkadii V. Kim
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Order Of Convergence ◽  
Functional Differential Equations ◽  
Convergence Conditions ◽  
Neutral Differential Equations ◽  
Infinite Dimensional ◽  
Analysis Methodology ◽  
Different Types ◽  
Functional Differential

The article discusses a number of aspects of the application of i -smooth analysis in the development of numerical methods for solving functional differential equations (FDE). The principle of separating finite- and infinite-dimensional components in the structure of numerical schemes for FDE is demonstrated with concrete examples, as well as the usage of different types of prehistory interpolation, those by Lagrange and Hermite. A general approach to constructing Runge–Kutta-like numerical methods for nonlinear neutral differential equations is presented. Convergence conditions are obtained and the order of convergence of such methods is established.

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