On the number of arbitrary parameters in the general solution to a weakly delayed planar linear discrete system with constant coefficients

2020 ◽  
Author(s):  
Hana Halfarová ◽  
Josef Diblík ◽  
Jan Šafařík
Keyword(s):  
General Solution ◽  
Discrete System ◽  
Linear Discrete System ◽  
Constant Coefficients

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.

scholarly journals General Solution Of Nth-Order Linear Ordinary Differential Equation With Constant Coefficients

2021 ◽  
Author(s):  
Rajnish Kumar Jha
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
General Solution ◽  
Linear Ordinary Differential Equation ◽  
Order Linear ◽  
Constant Coefficients

In this paper we present a formula for the general solution of Nth-order linear ordinary differential equation with constant coefficients as our main result. In this regard we also present two supporting results in this paper which reduce the order of the concerned differential equation by one and give the relation between the coefficients of the initial differential equation and the differential equation obtained. We also discuss about the complementary solution and homogeneous equations with regard to the main result described in this paper.

scholarly journals On a Decomposition Method in the Problem of Operation Speed for a Linear Discrete System with Bounded Control

2019 ◽  
Vol 09 (4) ◽  
pp. 157-161
Author(s):  
D.N. Ibragimov ◽  
E.E. Turchak
Keyword(s):  
Explicit Form ◽  
Discrete System ◽  
Decomposition Theorem ◽  
Two Dimensional ◽  
State Matrix ◽  
Bounded Control ◽  
Operation Speed ◽  
Linear Discrete System ◽  
Minimum Number ◽  
Main Decomposition

The article presents the problem of operation speed for a linear discrete system with bounded control. For the case when the minimum number of steps necessary for the system to reach zero significantly exceeds the dimension of the phase space, a method of decomposition into scalar and two-dimensional subsystems is developed, based on the reduction of the state matrix to normal Jordan form. Moreover, due to the developed algorithm for adding two polyhedrons with linear complexity, it is possible to construct sets of 0-controllability for two-dimensional subsystems in an explicit form. A description of the main tools for solving the problem of operation speed is also presented, as well as the statement of the decomposition problem. Further, some properties of polyhedrons in the plane are formulated and proved, on the basis of which an algorithm for calculating the set of vertices of the sum of two polyhedrons in R2 in explicit form is developed. In conclusion, the main decomposition theorem is formulated and proved. And on the basis of the developed methods, the solution to the problem of the optimal damping speed of a high-rise structure located in the zone of seismic activity was constructed.

scholarly journals A Simple Method of Identification for Linear Discrete System

1980 ◽  
Vol 16 (1) ◽  
pp. 55-62
Author(s):  
Aisuke KATAYAMA ◽  
Rin'nei WATANABE ◽  
Hideo TAMAMOTO
Keyword(s):  
Discrete System ◽  
Simple Method ◽  
Linear Discrete System

scholarly journals Linearization of Two Second-Order Ordinary Differential Equations via Fiber Preserving Point Transformations

2011 ◽  
Vol 2011 ◽  
pp. 1-21 ◽  
Author(s):  
Sakka Sookmee ◽  
Sergey V. Meleshko
Keyword(s):  
Linear System ◽  
General Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Necessary Conditions ◽  
Second Order ◽  
Point Transformations ◽  
Constant Coefficients ◽  
Linearizable System

The necessary form of a linearizable system of two second-order ordinary differential equations y1″=f1(x,y1,y2,y1′,y2′), y2″=f2(x,y1,y2,y1′,y2′) is obtained. Some other necessary conditions were also found. The main problem studied in the paper is to obtain criteria for a system to be equivalent to a linear system with constant coefficients under fiber preserving transformations. A linear system with constant coefficients is chosen because of its simplicity in finding the general solution. Examples demonstrating the procedure of using the linearization theorems are presented.

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