scholarly journals Exact Solution of a Linear Difference Equation in a Finite Number of Steps

2018 ◽  
Vol 14 (1) ◽  
pp. 7560-7563
Author(s):  
Sergey Mikhailovich Skovpen ◽  
Albert Saitovich Iskhakov
Keyword(s):  
Exact Solution ◽  
Difference Equation ◽  
Finite Number ◽  
Algebraic System ◽  
Linear Difference Equation ◽  
Algebraic Equations ◽  
Necessary And Sufficient Condition ◽  
Linear Algebraic Equations ◽  
Iterative Equations ◽  
Necessary And Sufficient

An exact solution of a linear difference equation in a finite number of steps has been obtained. This refutes the conventional wisdom that a simple iterative method for solving a system of linear algebraic equations is approximate. The nilpotency of the iteration matrix is the necessary and sufficient condition for getting an exact solution. The examples of iterative equations providing an exact solution to the simplest algebraic system are presented.

scholarly journals Instability of second-order nonhomogeneous linear difference equations with real-valued coefficients

2021 ◽  
Vol 37 (3) ◽  
pp. 489-495
Author(s):  
MASAKAZU ONITSUKA ◽  
◽  
Keyword(s):  
Difference Equation ◽  
Linear Difference Equation ◽  
Second Order ◽  
Ulam Stability ◽  
Necessary And Sufficient Condition ◽  
Linear Difference Equations ◽  
Real Numbers ◽  
Step Size ◽  
Order Linear ◽  
Necessary And Sufficient

In J. Comput. Anal. Appl. (2020), pp. 152--165, the author dealt with Hyers--Ulam stability of the second-order linear difference equation $\Delta_h^2x(t)+\alpha \Delta_hx(t)+\beta x(t) = f(t)$ on $h\mathbb{Z}$, where $\Delta_hx(t) = (x(t+h)-x(t))/h$ and $h\mathbb{Z} = \{hk|\,k\in\mathbb{Z}\}$ for the step size $h>0$; $\alpha$ and $\beta$ are real numbers; $f(t)$ is a real-valued function on $h\mathbb{Z}$. The purpose of this paper is to clarify that the second-order linear difference equation has no Hyers--Ulam stability when the step size $h>0$ and the coefficients $\alpha$ and $\beta$ satisfy suitable conditions. Finally, a necessary and sufficient condition for Hyers--Ulam stability is obtained.

scholarly journals New efficiency algorithm for solving fractional order differential-algebraic system

2016 ◽  
Vol 5 (3) ◽  
pp. 152
Author(s):  
Sameer Hasan ◽  
Eman Namah
Keyword(s):  
Exact Solution ◽  
Fractional Order ◽  
Lagrange Multiplier ◽  
Iteration Method ◽  
Algebraic System ◽  
Analytic Solution ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Differential Algebraic System ◽  
Fractional Differential

This work provided the evolution of the algorithm for analytic solution of system of fractional differential-algebraic equations (FDAEs).The algorithm referred to good effective method for combination the Laplace Iteration method with general Lagrange multiplier (LLIM). Through this method we have reached excellent results in comparison with exact solution as we illustrated in our examples.

scholarly journals Bounded solutions of a difference equation with finite number of jumps of operator coefficient

2020 ◽  
Vol 12 (1) ◽  
pp. 165-172
Author(s):  
A. Chaikovs'kyi ◽  
O. Lagoda
Keyword(s):  
Difference Equation ◽  
Finite Number ◽  
Existence And Uniqueness ◽  
Bounded Solution ◽  
Sufficient Conditions ◽  
Operator Coefficient ◽  
Bounded Solutions ◽  
Piecewise Constant ◽  
Variable Operator ◽  
Necessary And Sufficient

We study the problem of existence of a unique bounded solution of a difference equation with variable operator coefficient in a Banach space. There is well known theory of such equations with constant coefficient. In that case the problem is solved in terms of spectrum of the operator coefficient. For the case of variable operator coefficient correspondent conditions are known too. But it is too hard to check the conditions for particular equations. So, it is very important to give an answer for the problem for those particular cases of variable coefficient, when correspondent conditions are easy to check. One of such cases is the case of piecewise constant operator coefficient. There are well known sufficient conditions of existence and uniqueness of bounded solution for the case of one jump. In this work, we generalize these results for the case of finite number of jumps of operator coefficient. Moreover, under additional assumption we obtained necessary and sufficient conditions of existence and uniqueness of bounded solution.

scholarly journals Asymptotic stability of a full term linear difference equation with two parameters

2015 ◽  
Vol 63 (1) ◽  
pp. 283-290
Author(s):  
Petr Tomášek
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Sufficient Conditions ◽  
Linear Difference Equation ◽  
Full Term ◽  
Asymptotically Stable ◽  
Term Linear ◽  
Asymptotic Stability Region ◽  
Two Parameters ◽  
Necessary And Sufficient

Abstract The paper introduces an efficient form of necessary and sufficient conditions for a special full term linear difference equation with two real parameters to be asymptotically stable. The result is obtained utilizing the Schur Cohn criterion. The asymptotic stability region in the parameters plane is also illustrated in the paper.

scholarly journals Some results concerning frames in Banach spaces

2007 ◽  
Vol 38 (3) ◽  
pp. 267-276 ◽  
Author(s):  
S. K. Kaushik
Keyword(s):  
Finite Number ◽  
Banach Spaces ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Linearly Independent ◽  
Banach Frame ◽  
Minimal Sequence ◽  
Definition Of ◽  
Necessary And Sufficient

A necessary and sufficient condition for the associated sequence of functionals to a complete minimal sequence to be a Banach frame has been given. We give the definition of a weak-exact Banach frame, and observe that an exact Banach frame is weak-exact. An example of a weak-exact Banach frame which is not exact has been given. A necessary and sufficient condition for a Banach frame to be a weak-exact Banach frame has been obtained. Finally, a necessary condition for the perturbation of a retro Banach frame by a finite number of linearly independent vectors to be a retro Banach frame has been given.

Automorphism Groups of Algebras of Finite Type

1972 ◽  
Vol 24 (6) ◽  
pp. 1065-1069 ◽  
Author(s):  
Matthew Gould
Keyword(s):  
Automorphism Group ◽  
Finite Number ◽  
Finite Type ◽  
Permutation Group ◽  
Universal Algebra ◽  
Automorphism Groups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

By “algebra” we shall mean a finitary universal algebra, that is, a pair 〈A; F〉 where A and F are nonvoid sets and every element of F is a function, defined on A, of some finite number of variables. Armbrust and Schmidt showed in [1] that for any finite nonvoid set A, every group G of permutations of A is the automorphism group of an algebra defined on A and having only one operation, whose rank is the cardinality of A. In [6], Jónsson gave a necessary and sufficient condition for a given permutation group to be the automorphism group of an algebra, whereupon Plonka [8] modified Jonsson's condition to characterize the automorphism groups of algebras whose operations have ranks not exceeding a prescribed bound.

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