A probabilistic derivation of the stability condition of the difference equation for the diffusion equation

1983 ◽  
Vol 51 (5) ◽  
pp. 459-460 ◽  
Author(s):  
Bernard Kaplan
Keyword(s):  
Difference Equation ◽  
Diffusion Equation ◽  
Stability Condition ◽  
The Difference ◽  
The Stability

scholarly journals On the influence of the method of approximation of an unknown function on the stability of numerical methods for solving the anomalous diffusion equation

2019 ◽  
pp. 23-29
Author(s):  
Vladimir Andreevich Litvinov
Keyword(s):  
Diffusion Equation ◽  
Numerical Solution ◽  
Anomalous Diffusion ◽  
Caputo Derivative ◽  
Numerical Algorithms ◽  
Difference Approximation ◽  
Liouville Derivative ◽  
The Subject ◽  
The Difference ◽  
The Stability

The subject of the research is numerical algorithms for solving fractional partial differential equations. The object of the study is the stability of several algorithms for the numerical solution of the anomalous diffusion equation. Algorithms based on the difference representation of the fractional Riemann-Liuville derivative and the Caputo derivative for various orders of accuracy are considered. A comparison is made of the results of numerical calculations using the analyzed algorithms for a model problem with the exact solution of the anomalous diffusion equation for various orders of the fractional derivative with respect to the spatial coordinate. The results of the work were obtained on the basis of the analysis of the constructed difference schemes for stability, the conducted numerical experiments and a comparative analysis of the data obtained. The main conclusions of the study are the advantage of using the approximation of the fractional Caputo derivative compared to using the difference scheme for the fractional Riemann-Liouville derivative in the numerical solution of the anomalous diffusion equation. The paper also indicates the importance of choosing the method of difference approximation of the second derivative, which is a derivative of the Caputo.

scholarly journals On the Recursive Sequencexn=A+xn−kp/xn−1r

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
Fangkuan Sun ◽  
Xiaofan Yang ◽  
Chunming Zhang
Keyword(s):  
Difference Equation ◽  
Dynamic Behavior ◽  
Positive Solutions ◽  
Initial Conditions ◽  
Real Numbers ◽  
Positive Real ◽  
The Difference ◽  
The Stability

This paper studies the dynamic behavior of the positive solutions to the difference equationxn=A+xn−kp/xn−1r,n=1,2,…, whereA,p, andrare positive real numbers, and the initial conditions are arbitrary positive numbers. We establish some results regarding the stability and oscillation character of this equation forp∈(0,1).

scholarly journals Hyers-Ulam stability of hyperbolic Möbius difference equation

Filomat ◽  
2018 ◽  
Vol 32 (13) ◽  
pp. 4555-4575 ◽  
Author(s):  
Young Nam
Keyword(s):  
Fixed Point ◽  
Difference Equation ◽  
Line Segment ◽  
Initial Point ◽  
Logistic Equation ◽  
Complex Numbers ◽  
Ulam Stability ◽  
The Difference ◽  
The Stability ◽  
Repelling Fixed Point

Hyers-Ulam stability of the difference equation with the initial point z0 as follows zi+1 = azi+b/czi+d is investigated for complex numbers a,b,c and d where ad-bc = 1, c ? 0 and a+d ?R\[-2,2]. The stability of the sequence {zn}n?N0 holds if the initial point is in the exterior of a certain disk of which center is ?d/c . Furthermore, the region for stability can be extended to the complement of some neighborhood of the line segment between -d/c and the repelling fixed point of the map z ? az+b/cz+d. This result is the generalization of Hyers-Ulam stability of Pielou logistic equation.

scholarly journals New Method of Solving the Economic Complex Systems

2020 ◽  
Vol 2020 ◽  
pp. 1-26
Author(s):  
Ya-Juan Yang ◽  
Chung-Cheng Chen ◽  
Yen-Ting Chen
Keyword(s):  
Difference Equation ◽  
Economic System ◽  
Optimization Problems ◽  
Direct Method ◽  
Economic Optimization ◽  
The Difference ◽  
Long Division ◽  
The Stability ◽  
Controllability And Observability ◽  
Z Transformation

In this study, the authors first develop a direct method used to solve the linear nonhomogeneous time-invariant difference equation with the same number for inputs and outputs. Economic cybernetics is the crystallization for the integration of economics and cybernetics. It analyzes the stability, controllability, and observability of the economic system by establishing a system model and enables people to better understand the characteristics of the economic system and solve economic optimization problems. The economic model generally applies the discrete recurrence difference equation. The significant analytic approach for the difference equation is the z-transformation technique. The z-transformation state of the economic cybernetics state-space difference equation generally is a rational function with the same power for the numerator and the denominator. The proposed approach will take the place of the traditional methods without all annoying procedures involving the long division of some complicated polynomials, the expanded multiplication of many polynomial factors, the differentiation of some complicated polynomials, and the complex derivations of all partial fraction parameters. To highlight the novelty of this research, this study especially applies the proposed theorems originally belonging to engineering to the field of economic applications.

A New Look at the Stability Analysis of Delay-Differential Equations

2005 ◽  
Author(s):  
Tama´s Kalma´r-Nagy
Keyword(s):  
Difference Equation ◽  
Differential Equations ◽  
Characteristic Equation ◽  
Delay Differential Equations ◽  
Linear Delay ◽  
The Laplace Transform ◽  
Method Of Steps ◽  
The Difference ◽  
The Stability ◽  
Delay Differential

It is shown that the method of steps for linear delay-differential equations combined with the Laplace-transform can be used to determine the stability of the equation. The result of the method is an infinite dimensional difference equation whose stability corresponds to that of the transcendental characteristic equation. Truncations of this difference equation are used to construct numerical stability charts. The method is demonstrated on a first and second order delay equation. Correspondence between the transcendental characteristic equation and the difference equation is proved for the first order case.

scholarly journals Kolmogorov-Arnold-Moser Theory and Symmetries for a Polynomial Quadratic Second Order Difference Equation

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 790
Author(s):  
Tarek F. Ibrahim ◽  
Zehra Nurkanović
Keyword(s):  
Difference Equation ◽  
Equilibrium Points ◽  
Kam Theory ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Elliptic Equilibrium ◽  
Order Difference Equation ◽  
The Difference ◽  
The Stability ◽  
Theoretical Results

By using the Kolmogorov-Arnold-Moser (KAM) theory, we investigate the stability of two elliptic equilibrium points (zero equilibrium and negative equilibrium) of the difference equation t n + 1 = α t n + β t n 2 − t n − 1 , n = 0 , 1 , 2 , … , where are t − 1 , t 0 , α ∈ R , α ≠ 0 , β > 0 . By using the symmetries we find the periodic solutions with some periods. Finally, some numerical examples are given to verify our theoretical results.

scholarly journals A nonlinear difference equation with two parameters

1985 ◽  
Vol 26 (4) ◽  
pp. 430-451 ◽  
Author(s):  
A. Brown
Keyword(s):  
Difference Equation ◽  
General Problem ◽  
Iteration Process ◽  
Nonlinear Difference Equation ◽  
Cubic Equation ◽  
The Difference ◽  
The Stability ◽  
Image Position ◽  
Two Parameters ◽  
Parameter Problem

AbstractThe paper is mainly concerned with the difference equationwhere k and m are parameters, with k > 0. This equation arises from a method proposed for solving a cubic equation by iteration and represents a standardised form of the general problem. In using the above equation it is essential to know when the iteration process converges and this is discussed by means of the usual stability criterion. Critical values are obtained for the occurrence of solutions with period two and period three and the stability of these solutions is also examined. This was done by considering the changes as k increases, for a give value of m, which makes it effectively a one-parameter problem, and it turns out that the change with k can differ strongly from the usual behaviour for a one-parameter difference equation. For m = 2, for example it appears that the usual picture of stable 2-cycle solutions giving way to stable 4-cycle solutions is valid for smaller values of k but the situation is recersed for larger values of k where stable 4-cycle solutions precede stable 2-cycle solutions. Similar anomalies arise for the 3-cycle solutions.

scholarly journals A non-linear difference equation with two parameters. II

1985 ◽  
Vol 27 (2) ◽  
pp. 145-166 ◽  
Author(s):  
A. Brown
Keyword(s):  
Difference Equation ◽  
Linear Difference Equation ◽  
Anomalous Behaviour ◽  
Three Solutions ◽  
Unstable Solutions ◽  
The Difference ◽  
Set Up ◽  
The Stability ◽  
Image Position ◽  
Two Parameters

AbstractThe paper discusses solutions of period 4 for the difference equationwhere k and m are real parameters, with k > 0. For given values of k and m there are at most three solutions with period 4 and equations are set up to determine the elements of these solutions and the stability of each solution. Only real solutions are considered. The procedure that is used to find these solutions allows unstable solutions to be identified as well as stable solutions.In a previous paper, solutions of period 2 and period 3 were examined for this equation and there was evidence of anomalous behaviour in the way the stability intervals occurred. Some preliminary information about solutions of period 4 was mentioned in the discussion. The present paper provides more complete results, which confirm the anomalous behaviour and give a better idea of how the stability criterion changes for different families of solutions. These results are used to indicate the variety of behaviour that can be found for one-parameter systems by imposing suitable conditions on m and k.

scholarly journals Solutions of period four for a non-linear difference equation

1984 ◽  
Vol 26 (2) ◽  
pp. 146-164 ◽  
Author(s):  
A. Brown
Keyword(s):  
Difference Equation ◽  
Polynomial Equation ◽  
Linear Difference Equation ◽  
Simple Mathematical Model ◽  
Frequency Dependent Selection ◽  
Quartic Equation ◽  
Real Roots ◽  
The Difference ◽  
The Stability ◽  
Image Position

AbstractThe paper extends earlier work by using the factorisation method to discuss solutions of period four for the difference equationThis equation was suggested by R. M. May as a simple mathematical model for the effect of frequency-dependent selection in genetics. It is shown that for a given value of the parameter, a, the identification of solutions of period four can be reduced to finding real roots for a polynomial equation of degree eight. The appropriate values of xn follow from a quartic equation. By splitting up the problem in this way it becomes relatively straightforward to determine the critical values of a at which the various solutions of period four first appear and to discuss the stability of these solutions. Intervals of stability are tabulated in the paper.

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