The stability of the positive solution for a fractional SIR model

2016 ◽  
Vol 10 (01) ◽  
pp. 1750014 ◽  
Author(s):  
Yingjia Guo
Keyword(s):  
Differential Equation ◽  
Positive Solution ◽  
Stability Theory ◽  
Lyapunov Stability Theory ◽  
Sir Model ◽  
Differentiable Functions ◽  
Classical Calculus ◽  
Global Positive Solution ◽  
Fractional Differential ◽  
The Stability

In order to deal with non-differentiable functions, a modification of the Riemann–Liouville definition is recently proposed which appears to provide a framework for a fractional calculus which is quite parallel with classical calculus. Based on these new results, we study on the fractional SIR model in this paper. First, we give the general solution of the fractional differential equation. And then a unique global positive solution of the SIR model is obtained. Furthermore, we get the Lyapunov stability theory. By using this stability theory, the asymptotic stability of the positive solution is analyzed.

scholarly journals Research on Computer Modeling of Fractional Differential Equation Applied Mathematics

2021 ◽  
Vol 2083 (3) ◽  
pp. 032036
Author(s):  
Linlin Su
Keyword(s):  
Differential Equation ◽  
Time Series ◽  
Hopf Bifurcation ◽  
Equilibrium Solution ◽  
Applied Mathematics ◽  
Bifurcation Diagrams ◽  
Financial Systems ◽  
Fractional Differential ◽  
The Stability ◽  
Evolution Behavior

Abstract This paper qualitatively analyzes the stability of the equilibrium solution of a class of fractional chaotic financial systems and the conditions for the occurrence of Hopf bifurcation, and uses the Adams-Bashford-Melton predictive-correction finite difference method to pass the analysis Bifurcation diagrams, phase diagrams, and time series diagrams are used to simulate the complex evolution behavior of the system.

scholarly journals Some Properties of Approximate Solutions of Linear Differential Equations

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 806 ◽  
Author(s):  
Ginkyu Choi Soon-Mo Choi ◽  
Jaiok Jung ◽  
Roh
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Approximate Solutions ◽  
Small Error ◽  
Second Order ◽  
Differentiable Functions ◽  
Constant Coefficients ◽  
Classical Integral ◽  
Linear Differential ◽  
The Stability

In this paper, we will consider the Hyers-Ulam stability for the second order inhomogeneous linear differential equation, u ′ ′ ( x ) + α u ′ ( x ) + β u ( x ) = r ( x ) , with constant coefficients. More precisely, we study the properties of the approximate solutions of the above differential equation in the class of twice continuously differentiable functions with suitable conditions and compare them with the solutions of the homogeneous differential equation u ′ ′ ( x ) + α u ′ ( x ) + β u ( x ) = 0 . Several mathematicians have studied the approximate solutions of such differential equation and they obtained good results. In this paper, we use the classical integral method, via the Wronskian, to establish the stability of the second order inhomogeneous linear differential equation with constant coefficients and we will compare our result with previous ones. Specially, for any desired point c ∈ R we can have a good approximate solution near c with very small error estimation.

scholarly journals Uniqueness of Iterative Positive Solution to Nonlinear Fractional Differential Equations with Negatively Perturbed Term

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Jingjing Tan ◽  
Meixia Li ◽  
Aixia Pan
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Convergence Rate ◽  
Boundary Value Problem ◽  
Positive Solution ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Iterative Scheme ◽  
Approximation Error ◽  
Fractional Differential

We prove that there are unique positive solutions for a new kind of fractional differential equation with a negatively perturbed term boundary value problem. Our methods rely on an iterative algorithm which requires constructing an iterative scheme to approximate the solution. This allows us to calculate the estimation of the convergence rate and the approximation error.

Stability Analysis on S Plane Control of Underwater Vehicles

2013 ◽  
Vol 437 ◽  
pp. 716-721 ◽  
Author(s):  
Yue Ming Li ◽  
Ying Hao Zhang ◽  
Guo Cheng Zhang ◽  
Zhong Hui Hu
Keyword(s):  
Stability Analysis ◽  
Lyapunov Stability ◽  
Stability Theory ◽  
Lyapunov Stability Theory ◽  
Underwater Vehicles ◽  
Velocity Control ◽  
Position Controller ◽  
The Stability

This paper addresses the stability analysis on S Plane Control in terms of both position and velocity control. Employing Lyapunov stability theory and T-passivity theory, this paper proves the stability of the position controller based on S Plane Control, and on this ground, the stability analysis of the velocity controller based on S Plane Control is done. Finally, the S Plane Control results obtained from the sea trials are given.

scholarly journals Existence of Positive Solution for BVP of Nonlinear Fractional Differential Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Jun-Rui Yue ◽  
Jian-Ping Sun ◽  
Shuqin Zhang
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Positive Solution ◽  
Fractional Differential Equation ◽  
Nonlinear Fractional Differential Equation ◽  
Krasnoselskii Fixed Point Theorem ◽  
Fractional Differential

We consider the following boundary value problem of nonlinear fractional differential equation:(CD0+αu)(t)=f(t,u(t)),  t∈[0,1],  u(0)=0,   u′(0)+u′′(0)=0,  u′(1)+u′′(1)=0, whereα∈(2,3]is a real number, CD0+αdenotes the standard Caputo fractional derivative, andf:[0,1]×[0,+∞)→[0,+∞)is continuous. By using the well-known Guo-Krasnoselskii fixed point theorem, we obtain the existence of at least one positive solution for the above problem.

scholarly journals Positive Solution of a Nonlinear Fractional Differential Equation Involving Caputo Derivative

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Changyou Wang ◽  
Haiqiang Zhang ◽  
Shu Wang
Keyword(s):  
Differential Equation ◽  
Positive Solution ◽  
Fractional Differential Equation ◽  
Caputo Derivative ◽  
Nonlinear Term ◽  
Upper And Lower Solutions ◽  
Initial Value ◽  
Nonlinear Fractional Differential Equation ◽  
Control Functions ◽  
Fractional Differential

This paper is concerned with a nonlinear fractional differential equation involving Caputo derivative. By constructing the upper and lower control functions of the nonlinear term without any monotone requirement and applying the method of upper and lower solutions and the Schauder fixed point theorem, the existence and uniqueness of positive solution for the initial value problem are investigated. Moreover, the existence of maximal and minimal solutions is also obtained.

scholarly journals Boundary value problem for nonlinear fractional differential equations of variable order via Kuratowski MNC technique

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Amar Benkerrouche ◽  
Dumitru Baleanu ◽  
Mohammed Said Souid ◽  
Ali Hakem ◽  
Mustafa Inc
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Measure Of Noncompactness ◽  
Research Study ◽  
Boundary Value ◽  
Existence Of Solution ◽  
Variable Order ◽  
Kuratowski Measure Of Noncompactness ◽  
Fractional Differential ◽  
The Stability

AbstractIn the present research study, for a given multiterm boundary value problem (BVP) involving the Riemann-Liouville fractional differential equation of variable order, the existence properties are analyzed. To achieve this aim, we firstly investigate some specifications of this kind of variable-order operators, and then we derive the required criteria to confirm the existence of solution and study the stability of the obtained solution in the sense of Ulam-Hyers-Rassias (UHR). All results in this study are established with the help of the Darbo’s fixed point theorem (DFPT) combined with Kuratowski measure of noncompactness (KMNC). We construct an example to illustrate the validity of our observed results.

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