Global Stability of Fractional Order Coupled Systems with Impulses via a Graphic Approach

Abstract

Based on the graph theory and stability theory of dynamical system, this paper studies the stability of the trivial solution of a coupled fractional-order system. Some sufficient conditions are obtained to guarantee the global stability of the trivial solution. Finally, a comparison between fractional-order system and integer-order system ends the paper.

Solutions for a Singular Hadamard-Type Fractional Differential Equation by the Spectral Construct Analysis

“…is Green's function of equation (5). Now, let xðtÞ = −D t β zðtÞ, then the Hadamard-type fractional differential equation (1) reduces to the following convenient form:…”

“…In fluid mechanics, when a fluid is subjected to a severe impact to form a fracture, singular points or singular domains also follow the fracture. Normally, at singular points and domains, the extreme behaviour such as blow-up phenomena [2,3], impulsive influence [4][5][6][7][8][9], and chaotic system [10][11][12][13], often leads to some difficulties for people in understanding and predicting the corresponding natural problems. Hence, the study of singularity for complex systems governed by differential equations [14][15][16][17][18][19][20][21][22][23][24][25][26][27] is important and interesting in deepening the understanding of the internal laws of dynamic system.…”

An Iterative Algorithm for Solving n‐Order Fractional Differential Equation with Mixed Integral and Multipoint Boundary Conditions

“…Based on the wide range applications of calculus, in recent years, the study for various differential equations has become a frontier issue of nonlinear field and many mathematical methods and techniques, such as iterative techniques [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17], dual approach and perturbed techniques [18][19][20][21][22][23], fixed-point theorems , lower-upper solution method [51][52][53], variational method [54][55][56][57][58][59][60][61][62][63][64][65][66][67][68], numerical methods and stability analysis [69][70][71][72][73][74][75][76]…”

Extremal Solutions for a Class of Tempered Fractional Turbulent Flow Equations in a Porous Medium

“…As a powerful tool of modeling the above phenomena, in recent years, the fractional calculus theory has been perfected gradually by many researchers, and various different types of fractional derivatives were studied, such as Riemann-Liouville derivatives [16,, Hadamardtype derivatives [63][64][65][66][67][68][69][70][71], Katugampola-Caputo derivatives [72], conformable derivatives [73][74][75][76], Caputo-Fabrizio derivatives [77,78], Hilfer derivatives [79][80][81][82], and tempered fractional derivatives [83]. ese works also enlarged and enriched the application of the fractional calculus in impulsive theories [84][85][86][87][88][89], chaotic system [90][91][92][93], and resonance phenomena [94][95][96]. Among them, by using the fixed point theorem of the mixed monotone operator, Zhang et al [9] established the result of uniqueness of the positive solution for the Riemann-Liouville-type turbulent flow in a porous medium:…”