scholarly journals Quasilinearization of the Initial Value Problem for Difference Equations with “Maxima”

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
S. Hristova ◽  
A. Golev ◽  
K. Stefanova
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Initial Value Problem ◽  
Linear Difference Equation ◽  
Successive Approximations ◽  
Time Interval ◽  
Initial Value ◽  
Past Time ◽  
The Given ◽  
Unknown Solution

The object of investigation of the paper is a special type of difference equations containing the maximum value of the unknown function over a past time interval. These equations are adequate models of real processes which present state depends significantly on their maximal value over a past time interval. An algorithm based on the quasilinearization method is suggested to solve approximately the initial value problem for the given difference equation. Every successive approximation of the unknown solution is the unique solution of an appropriately constructed initial value problem for a linear difference equation with “maxima,” and a formula for its explicit form is given. Also, each approximation is a lower/upper solution of the given mixed problem. It is proved the quadratic convergence of the successive approximations. The suggested algorithm is realized as a computer program, and it is applied to an example, illustrating the advantages of the suggested scheme.

scholarly journals Monotone-Iterative Method for the Initial Value Problem with Initial Time Difference for Differential Equations with “Maxima”

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
S. Hristova ◽  
A. Golev
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Upper And Lower Solutions ◽  
Successive Approximations ◽  
Initial Time ◽  
Time Interval ◽  
Monotone Iterative Method ◽  
Initial Value ◽  
Monotone Iterative ◽  
The Given

The object of investigation of the paper is a special type of functional differential equations containing the maximum value of the unknown function over a past time interval. An improved algorithm of the monotone-iterative technique is suggested to nonlinear differential equations with “maxima.” The case when upper and lower solutions of the given problem are known at different initial time is studied. Additionally, all initial value problems for successive approximations have both initial time and initial functions different. It allows us to construct sequences of successive approximations as well as sequences of initial functions, which are convergent to the solution and to the initial function of the given initial value problem, respectively. The suggested algorithm is realized as a computer program, and it is applied to several examples, illustrating the advantages of the suggested scheme.

scholarly journals Note on some representations of general solutions to homogeneous linear difference equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Stevo Stević ◽  
Bratislav Iričanin ◽  
Witold Kosmala ◽  
Zdeněk Šmarda
Keyword(s):  
General Solution ◽  
Difference Equation ◽  
Difference Equations ◽  
Fibonacci Sequence ◽  
Linear Difference Equation ◽  
Linear Difference Equations ◽  
Second Order Difference Equation ◽  
Constant Coefficients ◽  
Order Difference Equation ◽  
Homogeneous Linear

Abstract It is known that every solution to the second-order difference equation $x_{n}=x_{n-1}+x_{n-2}=0$ x n = x n − 1 + x n − 2 = 0 , $n\ge 2$ n ≥ 2 , can be written in the following form $x_{n}=x_{0}f_{n-1}+x_{1}f_{n}$ x n = x 0 f n − 1 + x 1 f n , where $f_{n}$ f n is the Fibonacci sequence. Here we find all the homogeneous linear difference equations with constant coefficients of any order whose general solution have a representation of a related form. We also present an interesting elementary procedure for finding a representation of general solution to any homogeneous linear difference equation with constant coefficients in terms of the coefficients of the equation, initial values, and an extension of the Fibonacci sequence. This is done for the case when all the roots of the characteristic polynomial associated with the equation are mutually different, and then it is shown that such obtained representation also holds in other cases. It is also shown that during application of the procedure the extension of the Fibonacci sequence appears naturally.

Sharp explicit oscillation conditions for difference equations with several delays

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Kirill M. Chudinov
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Linear Difference Equation ◽  
First Order ◽  
Variable Delays ◽  
All Solutions ◽  
Oscillation Test ◽  
Several Delays ◽  
Lower Limits

Abstract We consider explicit sufficient conditions for all solutions of a first-order linear difference equation with several variable delays and non-negative coefficients to be oscillatory. The conditions have the form of inequalities bounding below the upper and lower limits of the sums of coefficients over a subset of the discrete semiaxis. Our main results are oscillation tests based on a new principle for composing the estimated sums of coefficients. We also give some results in the form of examples, including a counterexample to a wrong oscillation test cited in several recent papers.

scholarly journals Existence, uniqueness and continuability of solutions of impulsive differential-difference equations

1999 ◽  
Vol 12 (3) ◽  
pp. 293-300 ◽  
Author(s):  
D. D. Bainov ◽  
I. M. Stamova
Keyword(s):  
Difference Equations ◽  
Initial Value Problem ◽  
Sufficient Conditions ◽  
Initial Value

We consider an initial value problem for impulsive differential-difference equations, and obtain sufficient conditions for the existence, uniqueness, and continuability of solutions of such problem.

scholarly journals Perron-Type Criterion for Linear Difference Equations with Distributed Delay

2007 ◽  
Vol 2007 ◽  
pp. 1-12
Author(s):  
Jehad O. Alzabut ◽  
Thabet Abdeljawad
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Trivial Solution ◽  
Distributed Delay ◽  
Linear Difference Equation ◽  
Asymptotically Stable ◽  
Linear Difference Equations ◽  
Type Criterion

It is shown that if a linear difference equation with distributed delay of the formΔx(n)=∑k=−d0Δkζ(n+1,k−1)x(n+k−1),n≥1, satisfies a Perron condition then its trivial solution is uniformly asymptotically stable.

A Numerical Algorithm to Initial Value Problem of Caputo Fractional-Order Differential Equation

2015 ◽  
Author(s):  
Lu Bai ◽  
Dingyü Xue
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Error Analysis ◽  
Numerical Solution ◽  
Fractional Order ◽  
Numerical Algorithm ◽  
Initial Value Problem ◽  
Order Differential Equation ◽  
Fractional Order Differential Equation ◽  
Initial Value

A numerical algorithm is presented to solve the initial value problem of linear and nonlinear Caputo fractional-order differential equations. Firstly, nonzero initial value problem should be transformed into zero initial value problem. Error analysis has been done to polynomial algorithm, the reason has been found why the calculation error of the algorithm is large. A new algorithm called exponential function algorithm is proposed based on the analysis. The obtained fractional-order differential equation is transformed into difference equation. If the differential equation is linear, the obtained difference equation is explicit, the numerical solution can be solved directly; otherwise, the obtained difference equation is implicit, the predictor of the numerical solution can be obtained with extrapolation algorithm, substitute the predictor into the equation, the corrector can be solved. Error analysis has been done to the new algorithm, the algorithm is of first order.

scholarly journals ρ-PROJECTIVE AND σ-INVOLUNTARY VARIATIONAL INEQUALITIES AND IMPROVED PROJECTION METHOD FOR PROJECTED DYNAMICAL SYSTEM

2021 ◽  
Vol 9 (2) ◽  
pp. 18-24
Author(s):  
Siddharth Mitra ◽  
Prasanta Kumar Das
Keyword(s):  
Dynamical System ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Gradient Method ◽  
Initial Value Problem ◽  
Equivalence Theorem ◽  
Variational Inequality Problems ◽  
Initial Value ◽  
Type Method ◽  
The Given

Purpose of study: To introduce the concept of projective and involuntary variational inequality problems of order  and  respectively. To study the equivalence theorem between these problems. To study the projected dynamical system using self involutory variational inequality problems. Methodology: Improved extra gradient method is used. Main Finding: Using a self-solvable improved extra gradient method we solve the variational inequalities. The algorithm of the projected dynamical system is provided using the RK-4 method whose equilibrium point solves the involutory variational inequality problems. Application of this study: Runge-Kutta type method of order 2 and 4 is used for the initial value problem with the given projected dynamical system with the help of self involutory variational inequality problems. The originality of this study:  The concept of self involutory variational inequality problems, projective and involuntary variational inequality problems of order  and  respectively are newly defined.

scholarly journals Meromorphic Solutions of Linear Difference Equations with Polynomial Coefficients

2017 ◽  
Vol 59 (1) ◽  
pp. 159-168
Author(s):  
Y. Zhang ◽  
Z. Gao ◽  
H. Zhang
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Necessary Conditions ◽  
Meromorphic Solution ◽  
Linear Difference Equation ◽  
Meromorphic Solutions ◽  
Linear Difference Equations ◽  
Polynomial Coefficients ◽  
Difference Analogue ◽  
Borel Exceptional Values

AbstractWe study the growth of the transcendental meromorphic solution f(z) of the linear difference equation:where q(z), p0(z), ..., pn-(z) (n ≥ 1) are polynomials such that p0(z)pn(z) ≢ 0, and obtain some necessary conditions guaranteeing that the order of f(z) satisfies σ(f) ≥ 1 using a difference analogue of the Wiman-Valiron theory. Moreover, we give the form of f(z) with two Borel exceptional values when two of p0(z), ..., pn(z) have the maximal degrees.

scholarly journals Some roughness results concerning reducibility for linear difference equations

1988 ◽  
Vol 11 (4) ◽  
pp. 793-804 ◽  
Author(s):  
Garyfalos Papaschinopoulos
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Exponential Dichotomy ◽  
Coefficient Matrix ◽  
Linear Difference Equation ◽  
Almost Periodic ◽  
Linear Difference Equations ◽  
Joint Property

In this paper we prove first that the exponential dichotomy of linear difference equations is “rough”. Moreover we prove that if the coefficient matrix of a linear difference equation is almost periodic, then the Joint property of having an exponential dichotomy with a projectionPand being reducible withPby an almost periodic kinematics similarity is “rough”.

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