Browsing by Author "Wu, Guo-Cheng"

Now showing 1 - 20 of 42

Citation - WoS: 12
Citation - Scopus: 12
Analysis of Fractional Non-Linear Diffusion Behaviors Based on Adomian Polynomials
(Vinca inst Nuclear Sci, 2017) Baleanu, Dumitru; Luo, Wei-Hua; Wu, Guo-Cheng
A time-fractional non-linear diffusion equation of two orders is considered to investigate strong non-linearity through porous media. An equivalent integral equation is established and Adomian polynomials are adopted to linearize non-linear terms. With the Taylor expansion of fractional order, recurrence formulae are proposed and novel numerical solutions are obtained to depict the diffusion behaviors more accurately. The result shows that the method is suitable for numerical simulation of the fractional diffusion equations of multi-orders.
Citation - WoS: 211
Citation - Scopus: 242
Chaos Analysis and Asymptotic Stability of Generalized Caputo Fractional Differential Equations
(Pergamon-elsevier Science Ltd, 2017) Wu, Guo-Cheng; Zeng, Sheng-Da; Baleanu, Dumitru
This paper investigates chaotic behavior and stability of fractional differential equations within a new generalized Caputo derivative. A semi-analytical method is proposed based on Adomian polynomials and a fractional Taylor series. Furthermore, chaotic behavior of a fractional Lorenz equation are numerically discussed. Since the fractional derivative includes two fractional parameters, chaos becomes more complicated than the one in Caputo fractional differential equations. Finally, Lyapunov stability is defined for the generalized fractional system. A sufficient condition of asymptotic stability is given and numerical results support the theoretical analysis. (C) Elsevier Ltd. All rights reserved.
Citation - WoS: 166
Citation - Scopus: 175
Chaos Synchronization of Fractional Chaotic Maps Based on the Stability Condition
(Elsevier Science Bv, 2016) Baleanu, Dumitru; Xie, He-Ping; Chen, Fu-Lai; Wu, Guo-Cheng
In the fractional calculus, one of the main challenges is to find suitable models which are properly described by discrete derivatives with memory. Fractional Logistic map and fractional Lorenz maps of Riemann-Liouville type are proposed in this paper. The general chaotic behaviors are investigated in comparison with the Caputo one. Chaos synchronization is designed according to the stability results. The numerical results show the method's effectiveness and fractional chaotic map's potential role for secure communication. (C) 2016 Published by Elsevier B.V.
Citation - WoS: 153
Citation - Scopus: 162
Chaos Synchronization of the Discrete Fractional Logistic Map
(Elsevier, 2014) Baleanu, Dumitru; Wu, Guo-Cheng
In this paper, master-slave synchronization for the fractional difference equation is studied with a nonlinear coupling method. The numerical simulation shows that the designed synchronization method can effectively synchronize the fractional logistic map. The Caputo-like delta derivative is adopted as the difference operator. (C) 2014 Elsevier B.V. All rights reserved.
Citation - WoS: 95
Citation - Scopus: 114
Collocation Methods for Terminal Value Problems of Tempered Fractional Differential Equations
(Elsevier, 2020) Wu, Guo-Cheng; Baleanu, Dumitru; Shiri, Babak
A class of tempered fractional differential equations with terminal value problems are investigated in this paper. Discretized collocation methods on piecewise polynomials spaces are proposed for solving these equations. Regularity results are constructed on weighted spaces and convergence order is studied. Several examples are supported the theoretical parts and compared with other methods. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.
Citation - WoS: 146
Citation - Scopus: 163
Discrete Chaos in Fractional Delayed Logistic Maps
(Springer, 2015) Baleanu, Dumitru; Wu, Guo-Cheng
Recently the discrete fractional calculus (DFC) started to gain much importance due to its applications to the mathematical modeling of realworld phenomena with memory effect. In this paper, the delayed logistic equation is discretized by utilizing the DFC approach and the related discrete chaos is reported. The Lyapunov exponent together with the discrete attractors and the bifurcation diagrams are given.
Citation - WoS: 134
Citation - Scopus: 147
Discrete Chaos in Fractional Sine and Standard Maps
(Elsevier, 2014) Baleanu, Dumitru; Zeng, Sheng-Da; Wu, Guo-Cheng
Fractional standard and sine maps are proposed by using the discrete fractional calculus. The chaos behaviors are then numerically discussed when the difference order is a fractional one. The bifurcation diagrams and the phase portraits are presented, respectively. (C) 2013 Elsevier B.V. All rights reserved.
Citation - WoS: 67
Citation - Scopus: 75
Discrete Fractional Calculus for Interval-Valued Systems
(Elsevier, 2021) Wu, Guo-Cheng; Baleanu, Dumitru; Wang, Hong-Yong; Huang, Lan-Lan
This study investigates linear fractional difference equations with respect to interval-valued functions. Caputo and Riemann-Liouville differences are defined. w-monotonicity is introduced and discrete Leibniz integral laws are provided. Then exact solutions of two linear equations are obtained by Picard's iteration. In comparison with the deterministic initial problems, the solutions are given in discrete Mittag-Leffler functions with and without delay, respectively. This paper provides a novel tool to understand fractional uncertainty problems on discrete time domains. (C) 2020 Elsevier B.V. All rights reserved.
Citation - WoS: 64
Citation - Scopus: 82
Discrete Fractional Diffusion Equation
(Springer, 2015) Baleanu, Dumitru; Zeng, Sheng-Da; Deng, Zhen-Guo; Wu, Guo-Cheng
The tool of the discrete fractional calculus is introduced to discrete modeling of diffusion problem. A fractional time discretization diffusion model is presented in the Caputo-like delta's sense. The numerical formula is given in form of the equivalent summation. Then, the diffusion concentration is discussed for various fractional difference orders. The discrete fractional model is a fractionization of the classical difference equation and can be more suitable to depict the random or discrete phenomena compared with fractional partial differential equations.
Citation - WoS: 22
Citation - Scopus: 26
Discrete Fractional Diffusion Equation of Chaotic Order
(World Scientific Publ Co Pte Ltd, 2016) Baleanu, Dumitru; Xie, He-Ping; Zeng, Sheng-Da; Wu, Guo-Cheng
Discrete fractional calculus is suggested in diffusion modeling in porous media. A variable-order fractional diffusion equation is proposed on discrete time scales. A function of the variable order is constructed by a chaotic map. The model shows some new random behaviors in comparison with other variable-order cases.
Citation - WoS: 464
Citation - Scopus: 528
Discrete Fractional Logistic Map and Its Chaos
(Springer, 2014) Baleanu, Dumitru; Wu, Guo-Cheng
A discrete fractional logistic map is proposed in the left Caputo discrete delta's sense. The new model holds discrete memory. The bifurcation diagrams are given and the chaotic behaviors are numerically illustrated.
Citation - WoS: 12
Citation - Scopus: 12
Existence and Discrete Approximation for Optimization Problems Governed by Fractional Differential Equations
(Elsevier Science Bv, 2018) Baleanu, Dumitru; Wu, Guo-Cheng; Bai, Yunru
We investigate a class of generalized differential optimization problems driven by the Caputo derivative. Existence of weak Caratheodory solution is proved by using Weierstrass existence theorem, fixed point theorem and Filippov implicit function lemma etc. Then a numerical approximation algorithm is introduced, and a convergence theorem is established. Finally, a nonlinear programming problem constrained by the fractional differential equation is illustrated and the results verify the validity of the algorithm. (C) 2017 Elsevier B.V. All rights reserved.
Citation - WoS: 40
Citation - Scopus: 40
Existence Results of Fractional Differential Equations With Riesz-Caputo Derivative
(Springer Heidelberg, 2017) Baleanu, Dumitru; Wu, Guo-Cheng; Chen, Fulai
This paper is concerned with a class of boundary value problems for fractional differential equations with the Riesz-Caputo derivative, which holds two-sided nonlocal effects. By means of a new fractional Gronwall inequalities and some fixed point theorems, we obtained some existence results of solutions. Three examples are given to illustrate the results.
Citation - WoS: 90
Citation - Scopus: 95
Finite-Time Stability of Discrete Fractional Delay Systems: Gronwall Inequality and Stability Criterion
(Elsevier, 2018) Baleanu, Dumitru; Zeng, Sheng-Da; Wu, Guo-Cheng
This study investigates finite-time stability of Caputo delta fractional difference equations. A generalized Gronwall inequality is given on a finite time domain. A finite-time stability criterion is proposed for fractional differential equations. Then the idea is extended to the discrete fractional case. A linear fractional difference equation with constant delays is considered and finite-time stable conditions are provided. One example is numerically illustrated to support the theoretical result. (c) 2017 Elsevier B.V. All rights reserved.
Citation - WoS: 29
Citation - Scopus: 30
Fractional Discrete-Time Diffusion Equation With Uncertainty: Applications of Fuzzy Discrete Fractional Calculus
(Elsevier Science Bv, 2018) Baleanu, Dumitru; Mo, Zhi-Wen; Wu, Guo-Cheng; Huang, Lan-Lan
This study provides some basics of fuzzy discrete fractional calculus as well as applications to fuzzy fractional discrete-time equations. With theories of r-cut set, fuzzy Caputo and Riemann-Liouville fractional differences are defined on a isolated time scale. Discrete Leibniz integral law is given by use of w-monotonicity conditions. Furthermore, equivalent fractional sum equations are established. Fuzzy discrete Mittag-Leffler functions are obtained by the Picard approximation. Finally, fractional discrete-time diffusion equations with uncertainty is investigated and exact solutions are expressed in form of two kinds of fuzzy discrete Mittag-Leffler functions. This paper suggests a discrete time tool for modeling discrete fractional systems with uncertainty. (C) 2018 Elsevier B.V. All rights reserved.
Citation - WoS: 112
Citation - Scopus: 120
Fractional Impulsive Differential Equations: Exact Solutions, Integral Equations and Short Memory Case
(Walter de Gruyter Gmbh, 2019) Zeng, De-Qiang; Baleanu, Dumitru; Wu, Guo-Cheng
Fractional impulsive differential equations are revisited first. Some fundamental solutions of linear cases are given in this study. One straightforward technique without using integral equation is adopted to obtain exact solutions which are given by use of piecewise functions. Furthermore, a class of short memory fractional differential equations is proposed and the variable case is discussed. Mittag-Leffler solutions with impulses are derived which both satisfy the equations and impulsive conditions, respectively.
Citation - WoS: 64
Citation - Scopus: 77
Image Encryption Technique Based on Fractional Chaotic Time Series
(Sage Publications Ltd, 2016) Baleanu, Dumitru; Lin, Zhen-Xiang; Wu, Guo-Cheng
Chaos in discrete fractional maps has been reported very recently. In this study, the chaotic time series of fractional order is used in the scrambling technique and a novel image encryption scheme is designed. The fractional difference order and the chaotic coefficient play crucial roles in controlling chaotic behaviors. The encrypted results are analyzed, which shows that the encryption scheme is highly secure.
Citation - WoS: 163
Citation - Scopus: 177
Jacobian Matrix Algorithm for Lyapunov Exponents of the Discrete Fractional Maps
(Elsevier, 2015) Baleanu, Dumitru; Wu, Guo-Cheng
The Jacobian matrix algorithm is often used to calculate the Lyapunov exponents of the chaotic systems. This study extends the algorithm to discrete fractional cases. The tangent maps with memory effect are presented. The Lyapunov exponents of one and two dimensional fractional logistic maps are calculated. The positive ones are used to distinguish the chaotic areas of the maps. (C) 2014 Elsevier B.V. All rights reserved.
Citation - WoS: 80
Citation - Scopus: 79
Lattice Fractional Diffusion Equation in Terms of a Riesz-Caputo Difference
(Elsevier, 2015) Baleanu, Dumitru; Deng, Zhen-Guo; Zeng, Sheng-Da; Wu, Guo-Cheng
A fractional difference is defined by the use of the right and the left Caputo fractional differences. The definition is a two-sided operator of Riesz type and introduces back and forward memory effects in space difference. Then, a fractional difference equation method is suggested for anomalous diffusion in discrete finite domains. A lattice fractional diffusion equation is proposed and the numerical simulation of the diffusion process is discussed for various difference orders. The result shows that the Riesz difference model is particularly suitable for modeling complicated dynamical behaviors on discrete media. (C) 2015 Elsevier B.V. All rights reserved.
Citation - WoS: 4
Citation - Scopus: 5
Lattice Fractional Diffusion Equation of Random Order
(Wiley, 2017) Baleanu, Dumitru; Xie, He-Ping; Zeng, Sheng-Da; Wu, Guo-Cheng
The discrete fractional calculus is used to fractionalize difference equations. Simulations of the fractional logistic map unravel that the chaotic solution is conveniently obtained. Then a Riesz fractional difference is defined for fractional partial difference equations on discrete finite domains. A lattice fractional diffusion equation of random order is proposed to depict the complicated random dynamics and an explicit numerical formulae is derived directly from the Riesz difference. Copyright (C) 2015 John Wiley & Sons, Ltd.