General Linear Methods for the Numerical Solution of Functional-Differential Equations
Copyright policy )General Linear Methods for the Numerical Solution of Functional-Differential Equations
This paper is concerned with the numerical solution of initial value problems for delay differential equations with the form: \( x' = f(t, x_t(\cdot)), t\in [\alpha, \beta]\), \(x(t)= \varphi(t)\), \(t \in [\alpha - \tau, \alpha]\) where \( \tau>0\) is a given constant and \( x_t( \cdot)= \{ x(t+s)\); \(\alpha - \tau \leq t+s \leq t \}\) is the history of the phase vector at time \(t\). By introducing suitable assumptions on \(f\), the author develops the well known relations between the concepts of consistency, stability and convergence for linear step by step methods on uniform grids. The theory is applied to explicit Runge-Kutta methods and to a general class of linear methods \( u_{n+1} = S_n u_n + h \Phi (t_n, u_n,h)\) which are assumed to be strongly stable giving necessary and sufficient conditions for convergence of order \(p\) in terms of consistency conditions.
- Ural State Law University Russian Federation
- Ural State Agrarian University Russian Federation
delay differential equations, convergence, Runge-Kutta methods, consistency, linear discretization methods, stability, Numerical approximation of solutions of functional-differential equations, Numerical methods for initial value problems involving ordinary differential equations, functional-differential equations, Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations, initial value problems, one-step method, Stability and convergence of numerical methods for ordinary differential equations
delay differential equations, convergence, Runge-Kutta methods, consistency, linear discretization methods, stability, Numerical approximation of solutions of functional-differential equations, Numerical methods for initial value problems involving ordinary differential equations, functional-differential equations, Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations, initial value problems, one-step method, Stability and convergence of numerical methods for ordinary differential equations
selected citations These citations are derived from selected sources.This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).16 popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.Average influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).Top 10% impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.Average
Citations provided by BIP!Popularity provided by BIP!