Browsing by Author "Wu, Guo-Cheng"

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A New Application of the Fractional Logistic Map
(Editura Academiei Romane, 2016) Baleanu, Dumitru; Baleanu, Dumitru; Wu, Guo-Cheng; Zeng, Sheng-Da; 56389
The fractional chaotic map started to be applied in physics and engineering to properly treat some real-world phenomena. A shuffling method is proposed based on the fractional logistic map. The fractional difference order is used as a key. An image encryption scheme is designed by using the XOR operation and the security analysis is given. The obtained results demonstrate that the fractional difference order makes the encryption scheme highly secure.
A Novel Shuffling Technique Based on Fractional Chaotic Maps
(Elsevier Gmbh, 2018) Bai, Yun-Ru; Baleanu, Dumitru; Baleanu, Dumitru; Wu, Guo-Cheng; 56389
An image encryption technique based on the fractional logistic map is designed in this work. A novel shuffling technique is established by use of fractional chaotic signals. Then it is used to scramble pixel positions. The results are analyzed in comparison with the classical logistic map. Since the employed fractional chaotic map holds complicated dynamics behavior, the encryption result is highly secure. Moreover, by experimental and statistical analysis, we demonstrate that the encryption performance is better than the results in literature. (C) 2018 Published by Elsevier GmbH.
Analysis Of Fractional Non-Linear Diffusion Behaviors Based On Adomian Polynomials
(Vinca inst Nuclear Sci, 2017) Wu, Guo-Cheng; Baleanu, Dumitru; Baleanu, Dumitru; Luo, Wei-Hua; 56389
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.
Applications of Short Memory Fractional Differential Equations with Impulses
(2023) Baleanu, Dumitru; Wu, Guo-Cheng; Baleanu, Dumitru; 56389
Dynamical systems’ behavior is sometimes varied with some impulse and sudden changes in process. The dynamics of these systems can not be modeled by previous concepts of derivative or fractional derivatives any longer. The short memory concept is a solution and a better choice for fractional modeling of such processes. We apply short memory fractional differential equations for these systems. We propose collocation methods based on piecewise polynomials to approximate solutions of these equations. We provide various examples to demonstrate the application of the short memory derivative for impulse systems and efficiency of the presented numerical methods.
Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations
(Pergamon-Elsevier Science LTD, 2017) Baleanu, Dumitru; Wu, Guo-Cheng; Zeng, Sheng-Da; 56389
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.
Chaos synchronization of fractional chaotic maps based on the stability condition
(Elsevier Science Bv, 2016) Wu, Guo-Cheng; Baleanu, Dumitru; Baleanu, Dumitru; Xie, He-Ping; Chen, Fu-Lai
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.
Chaos synchronization of the discrete fractional logistic map
(Elsevier, 2014) Wu, Guo-Cheng; Baleanu, Dumitru; Baleanu, Dumitru; 56389
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.
Chaos Synchronization of the Fractional Rucklidge System Based On New Adomian Polynomials
(L and H Scientific Publishing, LLC, 2017) Baleanu, Dumitru; Baleanu, Dumitru; Huang, L. L.; 56389
The fractional Rucklidge system is a new kind of chaotic models which hold the feature of memory effects and can depict the long history interactions. A numerical formula is proposed by use of the fast Adomian polynomials. Chaotic behavior are discussed and the Poincare sections are given for various fractional cases. It's also applied in chaos synchronization of the fractional system.
Collocation methods for terminal value problems of tempered fractional differential equations
(Elsevier, 2020) Shiri, Babak; Baleanu, Dumitru; Wu, Guo-Cheng; Baleanu, Dumitru; 56389
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.
Discrete chaos in fractional delayed logistic maps
(Springer, 2015) Wu, Guo-Cheng; Baleanu, Dumitru; Baleanu, Dumitru
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.
Discrete Chaos in Fractional Sine and Standard Maps
(Elsevier, 2014) Wu, Guo-Cheng; Baleanu, Dumitru; Baleanu, Dumitru; Zeng, Sheng-Da; 56389
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.
Discrete fractional calculus for interval-valued systems
(Elsevier, 2021) Huang, Lan-Lan; Baleanu, Dumitru; Wu, Guo-Cheng; Baleanu, Dumitru; Wang, Hong-Yong; 56389
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.
Discrete fractional diffusion equation
(Springer, 2015) Wu, Guo-Cheng; Baleanu, Dumitru; Baleanu, Dumitru; Zeng, Sheng-Da; Deng, Zhen-Guo
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.
Discrete Fractional Diffusion Equation of Chaotic Order
(World Scientific Publ Co Pte Ltd, 2016) Wu, Guo-Cheng; Baleanu, Dumitru; Baleanu, Dumitru; Xie, He-Ping; Zeng, Sheng-Da; 56389
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.
Discrete fractional logistic map and its chaos
(Springer, 2014) Wu, Guo-Cheng; Baleanu, Dumitru; Baleanu, Dumitru; 56389
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.
Existence and Discrete Approximation for Optimization Problems Governed By Fractional Differential Equations
(Elsevier Science BV, 2018) Baleanu, Dumitru; Baleanu, Dumitru; Wu, Guo-Cheng; 56389
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.
Existence Results of Fractional Differential Equations with Riesz-Caputo Derivative
(Springer Heidelberg, 2017) Baleanu, Dumitru; Baleanu, Dumitru; Wu, Guo-Cheng; 56389
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.
Finite-Time Stability of Discrete Fractional Delay Systems: Gronwall Inequality and Stability Criterion
(Elsevier, 2018) Wu, Guo-Cheng; Baleanu, Dumitru; Baleanu, Dumitru; Zeng, Sheng-Da; 56389
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.
FractionaI Impulsive Differential Equations: Exact Solutions, Integral Equations and Short Memory Case
(Walter De Gruyter GMBH, 2019) Baleanu, Dumitru; Zeng, De-Qiang; Baleanu, Dumitru; 56389
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.
Fractional differential equations of Caputo-Katugampola type and numerical solutions
(Elsevier Science INC, 2017) Baleanu, Dumitru; Baleanu, Dumitru; Bai, Yunru; Wu, Guo-Cheng; 56389
This paper is concerned with a numerical method for solving generalized fractional differential equation of Caputo-Katugampola derivative. A corresponding discretization technique is proposed. Numerical solutions are obtained and convergence of numerical formulae is discussed. The convergence speed arrives at O(Delta T1-alpha). Numerical examples are given to test the accuracy. (C) 2017 Elsevier Inc. All rights reserved.