Convergence Results for Discrete Trigonometric Collocation Methods with Product Integration in Hölder-Zygmund Spaces

1997 ◽  
Vol 16 (3) ◽  
pp. 689-708
Author(s):  
L. Schroderus
Keyword(s):  
Collocation Methods ◽  
Product Integration ◽  
Trigonometric Collocation ◽  
Convergence Results ◽  
Zygmund Spaces

PERIODIC SOLUTIONS OF DIFFERENTIAL ALGEBRAIC EQUATIONS WITH TIME DELAYS: COMPUTATION AND STABILITY ANALYSIS

2006 ◽  
Vol 16 (01) ◽  
pp. 67-84 ◽  
Author(s):  
TATYANA LUZYANINA ◽  
DIRK ROOSE
Keyword(s):  
Stability Analysis ◽  
Periodic Solutions ◽  
Time Delays ◽  
Differential Algebraic Equations ◽  
Collocation Methods ◽  
Algebraic Equations ◽  
Floquet Multipliers ◽  
Infinite Dimensional ◽  
Local Asymptotic Stability ◽  
Convergence Results

This paper concerns the computation and local stability analysis of periodic solutions to semi-explicit differential algebraic equations with time delays (delay DAEs) of index 1 and index 2. By presenting different formulations of delay DAEs, we motivate our choice of a direct treatment of these equations. Periodic solutions are computed by solving a periodic two-point boundary value problem, which is an infinite-dimensional problem for delay DAEs. We investigate two collocation methods based on piecewise polynomials: collocation at Radau IIA and Gauss–Legendre nodes. Using the obtained collocation equations, we compute an approximation to the Floquet multipliers which determine the local asymptotic stability of a periodic solution. Based on numerical experiments, we present orders of convergence for the computed solutions and Floquet multipliers and compare our results with known theoretical convergence results for initial value problems for delay DAEs. We end with examples on bifurcation analysis of delay DAEs.

Using the Modal and Trigonometric Collocation Methods in Rotor Dynamic Systems

2004 ◽  
Vol 126 (2) ◽  
pp. 229-234 ◽  
Author(s):  
Eduard Malenovsky´
Keyword(s):  
Dynamic Systems ◽  
Theoretical Basis ◽  
Collocation Methods ◽  
The Finite Element Method ◽  
Trigonometric Collocation ◽  
Steady State Responses ◽  
State Responses ◽  
Rotor Dynamic ◽  
Modal Method ◽  
Transient And Steady State

This article deals with the computational modeling of nonlinear rotor dynamic systems. The theoretical basis of the modal method, and combination with the method of dynamic compliances supplemented by the method of trigonometric collocation, is presented. The main analysis is focused on the solutions of transient and steady state responses. The algorithms for solving this range of problems are presented. The finite element method is the basis for both methods. The theoretical analysis is supplemented with a solution of an example model.

Stability of Implicit Difference Equations Generated by Parabolic Functional Differential Problems

2007 ◽  
Vol 7 (1) ◽  
pp. 68-82
Author(s):  
K. Kropielnicka
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Initial Boundary ◽  
Numerical Examples ◽  
Convergence Results ◽  
Nonlinear Parabolic ◽  
Implicit Difference Methods ◽  
Neumann Type ◽  
Difference Methods ◽  
Functional Differential

AbstractA general class of implicit difference methods for nonlinear parabolic functional differential equations with initial boundary conditions of the Neumann type is constructed. Convergence results are proved by means of consistency and stability arguments. It is assumed that given functions satisfy nonlinear estimates of Perron type with respect to functional variables. Differential equations with deviated variables and differential integral problems can be obtained from a general model by specializing given operators. The results are illustrated by numerical examples.

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