scholarly journals Regions of Absolute Stability

1979 ◽  
Vol 8 (102) ◽  
Author(s):  
Jørgen Sand ◽  
Ole Østerby
Keyword(s):  
Differential Equations ◽  
Computer Science ◽  
Ordinary Differential Equations ◽  
Large Class ◽  
Absolute Stability ◽  
Stability Regions ◽  
New Information ◽  
Pade Approximations ◽  
Backward Differentiation Formulae ◽  
The Stability

At the Department of Computer Science a system has been developed for plotting regions of absolute stability for a large class of formulae and methods for solving systems of ordinary differential equations. This report is a pictorial guide through the stability regions of a number of well-known formulae thereby showing the capabilities of our programs, and hopefully also giving some new information about the methods. In an appendix we give coefficients for Adams, Nystrom, generalized Milne-Simpson and backward differentiation formulae up to order 12 (resp. 11) and coefficients for Pade approximations to the exponential up to degree 6.

scholarly journals 3-Point Block Methods for Direct Integration of General Second-Order Ordinary Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
J. O. Ehigie ◽  
S. A. Okunuga ◽  
A. B. Sofoluwe
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Absolute Stability ◽  
Second Order ◽  
Stability And Convergence ◽  
Block Methods ◽  
Good Region ◽  
The Stability ◽  
The Individual ◽  
Region Of Absolute Stability

A Multistep collocation techniques is used in this paper to develop a 3-point explicit and implicit block methods, which are suitable for generating solutions of the general second-order ordinary differential equations of the form . The derivation of both explicit and implicit block schemes is given for the purpose of comparison of results. The Stability and Convergence of the individual methods of the block schemes are investigated, and the methods are found to be 0-stable with good region of absolute stability. The 3-point block schemes derived are tested on standard mechanical problems, and it is shown that the implicit block methods are superior to the explicit ones in terms of accuracy.

Stability regions for multi-parameter systems of periodic second order ordinary differential equations

1979 ◽  
Vol 84 (3-4) ◽  
pp. 249-257 ◽  
Author(s):  
Patrick J. Browne ◽  
B. D. Sleeman
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Second Order ◽  
Parameter System ◽  
Stability Regions ◽  
The Stability ◽  
Image Position

SynopsisThis paper studies the stability regions associated with the multi-parameter systemwhere the functions qr(xr), ars(xr) are periodic and the system is subjected to periodic or semi-periodic boundary conditions.

scholarly journals A New Eighth Order Runge-Kutta Family Method

2019 ◽  
Vol 11 (2) ◽  
pp. 190
Author(s):  
Séka Hippolyte ◽  
Assui Kouassi Richard
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stability Region ◽  
Eighth Order ◽  
Stability Regions ◽  
New Family ◽  
Runge Kutta ◽  
The Stability

In this article, a new family of Runge-Kutta methods of 8^th order for solving ordinary differential equations is discovered and depends on the parameters b_8 and a_10;5. For b8 = 49/180 and a10;5 = 1/9, we find the Cooper-Verner method [1]. We show that the stability region depends only on coefficient a_10;5. We compare the stability regions according to the values of a_10;5 with respect to the stability region of Cooper-Verner.

Stability in terms of two measures for population growth models with impulsive perturbations

2020 ◽  
Vol 13 (06) ◽  
pp. 2050051
Author(s):  
Zhinan Xia ◽  
Qianlian Wu ◽  
Dingjiang Wang
Keyword(s):  
Differential Equations ◽  
Population Growth ◽  
Ordinary Differential Equations ◽  
Trivial Solution ◽  
Growth Models ◽  
Two Measures ◽  
Impulsive Perturbations ◽  
The Stability ◽  
Generalized Ordinary Differential Equations ◽  
Population Growth Models

In this paper, we establish some criteria for the stability of trivial solution of population growth models with impulsive perturbations. The working tools are based on the theory of generalized ordinary differential equations. Here, the conditions concerning the functions are more general than the classical ones.

Stability of Solutions of Ordinary Differential Equations with Respect to a Closed Set

1968 ◽  
Vol 20 ◽  
pp. 720-726
Author(s):  
T. G. Hallam ◽  
V. Komkov
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Compact Set ◽  
Stability Of Solutions ◽  
Closed Set ◽  
The Stability ◽  
Stability Results

The stability of the solutions of an ordinary differential equation will be discussed here. The purpose of this note is to compare the stability results which are valid with respect to a compact set and the stability results valid with respect to an unbounded set. The stability of sets is a generalization of stability in the sense of Liapunov and has been discussed by LaSalle (5; 6), LaSalle and Lefschetz (7, p. 58), and Yoshizawa (8; 9; 10).

scholarly journals Ultimate Boundedness of the Systems Governed by Stochastic Differential Equations

1972 ◽  
Vol 47 ◽  
pp. 111-144 ◽  
Author(s):  
Yoshio Miyahara
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stochastic Differential Equations ◽  
Functional Differential Equations ◽  
Ultimate Boundedness ◽  
Functional Differential ◽  
The Stability

The stability of the systems given by ordinary differential equations or functional-differential equations has been studied by many mathematicians. The most powerful tool in this field seems to be the Liapunov’s second method (see, for example [6]).

scholarly journals An Accurate Block Hybrid Collocation Method for Third Order Ordinary Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Lee Ken Yap ◽  
Fudziah Ismail ◽  
Norazak Senu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Collocation Method ◽  
Film Flow ◽  
Direct Solution ◽  
Thin Film Flow ◽  
Block Method ◽  
Third Order ◽  
Block Form ◽  
The Stability

The block hybrid collocation method with two off-step points is proposed for the direct solution of general third order ordinary differential equations. Both the main and additional methods are derived via interpolation and collocation of the basic polynomial. These methods are applied in block form to provide the approximation at five points concurrently. The stability properties of the block method are investigated. Some numerical examples are tested to illustrate the efficiency of the method. The block hybrid collocation method is also implemented to solve the nonlinear Genesio equation and the problem in thin film flow.

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