scholarly journals Second Order Equations in Functional Spaces: Qualitative and Discrete Well-Posedness

2015 ◽  
Vol 2015 ◽  
pp. 1-63 ◽  
Author(s):  
A. Ashyralyev ◽  
J. Pastor ◽  
S. Piskarev ◽  
H. A. Yurtsever
Keyword(s):  
Approximate Solution ◽  
Difference Equations ◽  
Second Order ◽  
Difference Schemes ◽  
Well Posedness ◽  
Present Survey ◽  
The Stability ◽  
Well Posed ◽  
Second Order Equations

The present survey contains the recent results on the local and nonlocal well-posed problems for second order differential and difference equations. Results on the stability of differential problems for second order equations and of difference schemes for approximate solution of the second order problems are presented.

REGULARIZED DIFFERENCE SCHEMES FOR EVOLUTIONARY SECOND ORDER EQUATIONS

1992 ◽  
Vol 02 (03) ◽  
pp. 295-315 ◽  
Author(s):  
ALEKSANDR A. SAMARSKII ◽  
PETER N. VABISHCHEVICH
Keyword(s):  
Initial Conditions ◽  
Type Equation ◽  
Second Order ◽  
Difference Schemes ◽  
Inversion Method ◽  
Extension Problem ◽  
Elliptic Type ◽  
The Stability ◽  
Well Posed ◽  
Second Order Equations

The questions of approximate solution of unstable problems for evolutionary second order equations are discussed in this paper. The classical Cauchy problem for elliptic type equation is a significant example of such problem. Incorrectness of this problem (the Hadamard example) is due to instability of the solution towards small perturbations of the initial conditions. The extension problem of the solutions of well-posed elliptic problems beyond the calculation region boundary is also discussed. The stability of corresponding difference schemes is investigated by basing on general theory of ρ-stability. The principle of the regularization of three-layer difference schemes is developed for the unstable problems. It is shown that the regularized difference schemes correspond to some modification of quasi-inversion method.

scholarly journals Examination of Particle Tails

2008 ◽  
Vol 2008 ◽  
pp. 1-10 ◽  
Author(s):  
Tim Blackwell ◽  
Dan Bratton
Keyword(s):  
Power Law ◽  
Difference Equations ◽  
Particle Swarm ◽  
Second Order ◽  
Particle Swarm Optimisation ◽  
Position Distribution ◽  
Length Scales ◽  
Stochastic Difference Equations ◽  
First Order ◽  
Second Order Equations

The tail of the particle swarm optimisation (PSO) position distribution at stagnation is shown to be describable by a power law. This tail fattening is attributed to particle bursting on all length scales. The origin of the power law is concluded to lie in multiplicative randomness, previously encountered in the study of first-order stochastic difference equations, and generalised here to second-order equations. It is argued that recombinant PSO, a competitive PSO variant without multiplicative randomness, does not experience tail fattening at stagnation.

Stability of Solutions of Differential-operator and Operator-difference Equations in the Sense of Perturbation of Operators

2006 ◽  
Vol 6 (3) ◽  
pp. 269-290 ◽  
Author(s):  
B. S. Jovanović ◽  
S. V. Lemeshevsky ◽  
P. P. Matus ◽  
P. N. Vabishchevich
Keyword(s):  
Differential Operator ◽  
Difference Equations ◽  
Hyperbolic Equations ◽  
Second Order ◽  
Difference Schemes ◽  
Order Differential Operator ◽  
Stability Of Solutions ◽  
Operator Equations ◽  
The Difference ◽  
Coefficient Stability

Abstract Estimates of stability in the sense perturbation of the operator for solving first- and second-order differential-operator equations have been obtained. For two- and three-level operator-difference schemes with weights similar estimates hold. Using the results obtained, we construct estimates of the coefficient stability for onedimensional parabolic and hyperbolic equations as well as for the difference schemes approximating the corresponding differential problems.

scholarly journals A Note on the Second Order of Accuracy Stable Difference Schemes for the Nonlocal Boundary Value Hyperbolic Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-29 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Ozgur Yildirim
Keyword(s):  
Hyperbolic Equation ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Second Order ◽  
Difference Schemes ◽  
Stability Estimates ◽  
Order Of Accuracy ◽  
Hyperbolic Problem ◽  
Definite Operator ◽  
The Stability

The second order of accuracy absolutely stable difference schemes are presented for the nonlocal boundary value hyperbolic problem for the differential equations in a Hilbert spaceHwith the self-adjoint positive definite operatorA. The stability estimates for the solutions of these difference schemes are established. In practice, one-dimensional hyperbolic equation with nonlocal boundary conditions and multidimensional hyperbolic equation with Dirichlet conditions are considered. The stability estimates for the solutions of these difference schemes for the nonlocal boundary value hyperbolic problem are established. Finally, a numerical method proposed and numerical experiments, analysis of the errors, and related execution times are presented in order to verify theoretical statements.

On the Stability of Some Linear Nonautonomous Random Systems

1968 ◽  
Vol 35 (1) ◽  
pp. 7-12 ◽  
Author(s):  
E. F. Infante
Keyword(s):  
Random Systems ◽  
Second Order ◽  
Stability Conditions ◽  
Almost Sure Stability ◽  
Stability Properties ◽  
The Stability ◽  
Second Order Equations

A theorem and two corollaries for the almost sure stability of linear nonautonomous random systems are presented. These results are applied to the study of the stability properties of some often encountered second-order equations and the obtained stability conditions are compared to previously known criteria.

scholarly journals Special cases of critical linear difference equations

2021 ◽  
pp. 1-17
Author(s):  
Jan Jekl
Keyword(s):  
Difference Equations ◽  
Adjoint Equation ◽  
Second Order ◽  
Linear Difference Equations ◽  
Order Linear ◽  
Special Cases ◽  
Even Order ◽  
Nonnegative Coefficients ◽  
Sturm Liouville ◽  
Second Order Equations

In this paper, we investigate even-order linear difference equations and their criticality. However, we restrict our attention only to several special cases of the general Sturm–Liouville equation. We wish to investigate on such cases a possible converse of a known theorem. This theorem holds for second-order equations as an equivalence; however, only one implication is known for even-order equations. First, we show the converse in a sense for one term equations. Later, we show an upper bound on criticality for equations with nonnegative coefficients as well. Finally, we extend the criticality of the second-order linear self-adjoint equation for the class of equations with interlacing indices. In this way, we can obtain concrete examples aiding us with our investigation.

scholarly journals On the Solutions of Four Second-Order Nonlinear Difference Equations

2019 ◽  
Author(s):  
İnci Okumuş ◽  
Yüksel Soykan
Keyword(s):  
Difference Equations ◽  
Second Order ◽  
Global Behavior ◽  
Nonlinear Difference Equations ◽  
Rational Difference Equations ◽  
The Stability

This paper deals with the form, the stability character, the periodicity and the global behavior of solutions of the following four rational difference equations x_{n+1} = ((±1)/(x_{n}(x_{n-1}±1)-1)) x_{n+1} = ((±1)/(x_{n}(x_{n-1}∓1)+1)).

scholarly journals On well-posedness of the nonlocal boundary value problem for parabolic difference equations

2004 ◽  
Vol 2004 (2) ◽  
pp. 273-286 ◽  
Author(s):  
A. Ashyralyev ◽  
I. Karatay ◽  
P. E. Sobolevskii
Keyword(s):  
Boundary Value Problem ◽  
Difference Equations ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Stability Estimates ◽  
Well Posedness ◽  
Nonlocal Boundary Value Problem ◽  
The Difference ◽  
Type Boundary ◽  
The Stability

We consider the nonlocal boundary value problem for difference equations(uk−uk−1)/τ+Auk=φk,1≤k≤N,Nτ=1, andu0=u[λ/τ]+φ,0<λ≤1, in an arbitrary Banach spaceEwith the strongly positive operatorA. The well-posedness of this nonlocal boundary value problem for difference equations in various Banach spaces is studied. In applications, the stability and coercive stability estimates in Hölder norms for the solutions of the difference scheme of the mixed-type boundary value problems for the parabolic equations are obtained. Some results of numerical experiments are given.

Sign in / Sign up

Export Citation Format

Share Document