scholarly journals Exp-function method using modified Riemann-Liouville derivative for Burger's equations of fractional-order

2013 ◽  
pp. 19 ◽  
Author(s):  
Qazi Mahmood Ul Hassan ◽  
Syed Tauseef Mohyud-Din
Keyword(s):  
Fractional Order ◽  
Function Method ◽  
Liouville Derivative ◽  
Burger's Equations

scholarly journals Correction: FRACTIONAL ORDER BARBALAT’S LEMMA AND ITS APPLICATIONS IN THE STABILITY OF FRACTIONAL ORDER NONLINEAR SYSTEMS

2017 ◽  
Vol 22 (4) ◽  
pp. 503-513 ◽  
Author(s):  
Fei Wang ◽  
Yongqing Yang
Keyword(s):  
Nonlinear Systems ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Caputo Fractional Derivative ◽  
Barbalat’S Lemma ◽  
Liouville Derivative ◽  
Liouville Fractional Derivative ◽  
The Stability ◽  
The Relationship ◽  
Theoretical Results

This paper investigates fractional order Barbalat’s lemma and its applications for the stability of fractional order nonlinear systems with Caputo fractional derivative at first. Then, based on the relationship between Caputo fractional derivative and Riemann-Liouville fractional derivative, fractional order Barbalat’s lemma with Riemann-Liouville derivative is derived. Furthermore, according to these results, a set of new formulations of Lyapunov-like lemmas for fractional order nonlinear systems are established. Finally, an example is presented to verify the theoretical results in this paper.

Solitary Wave and other Solutions of Nonlinear Space-Time Fractional Differential Equation Systems

2018 ◽  
pp. 515-522
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Expansion Method ◽  
Travelling Wave ◽  
Boussinesq System ◽  
Liouville Derivative ◽  
Fractional Differential ◽  
Fractional System ◽  
Partial Fractional Differential Equation

In this study, we have successfully found some travelling wave solutions of the variant Boussinesq system and fractional system of two-dimensional Burgers' equations of fractional order by using the -expansion method. These exact solutions contain hyperbolic, trigonometric and rational function solutions. The fractional complex transform is generally used to convert a partial fractional differential equation (FDEs) with modified Riemann-Liouville derivative into ordinary differential equation. We showed that the considered transform and method are very reliable, efficient and powerful in solving wide classes of other nonlinear fractional order equations and systems.

Analytical Approach for the Space-time Nonlinear Partial Differential Fractional Equation

2014 ◽  
Vol 15 (7-8) ◽  
Author(s):  
Ahmet Bekir ◽  
Özkan Güner
Keyword(s):  
Function Method ◽  
Analytical Approach ◽  
Fractional Derivatives ◽  
Space Time ◽  
Variable Method ◽  
Partial Differential ◽  
Functional Variable ◽  
Liouville Derivative ◽  
Functional Variable Method ◽  
Fractional Equation

AbstractIn this paper, the fractional derivatives in the sense of modified Riemann–Liouville derivative and the functional variable method, exp-function method and

scholarly journals Fractional Describing Function Analysis of PWPF Modulator

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Xinsheng Wang ◽  
Danwei Wang ◽  
Senqiang Zhu ◽  
Eng Kee Poh
Keyword(s):  
Phase Shift ◽  
Fractional Order ◽  
Pulse Width ◽  
Function Method ◽  
Function Analysis ◽  
Pulse Frequency ◽  
Describing Function ◽  
Nonlinear Properties ◽  
Describing Function Method ◽  
Effective Analysis

Pulse-width pulse-frequency (PWPF) modulators are widely used in spacecraft thruster control. Their dynamic characteristic is still lack of effective analysis tools. This paper presents a fractional describing function method to describe the frequency characteristics of PWPF. A frequency-dependent gain and phase shift are clearly described by fractional-order expression, and the fractional-order behaviors depict the nonlinear properties of PWPF modulators. This fractional describing function method can also be applied to other kinds of modulators.

scholarly journals Optimal Control Problem Investigation for Linear Time-Invariant Systems of Fractional Order with Lumped Parameters Described by Equations with Riemann-Liouville Derivative

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
V. A. Kubyshkin ◽  
S. S. Postnov
Keyword(s):  
Fractional Order ◽  
Caputo Derivative ◽  
Linear Time ◽  
Fractional Order Systems ◽  
Linear Time Invariant ◽  
Liouville Derivative ◽  
Lumped Parameters ◽  
Linear Time Invariant Systems ◽  
Time Invariant ◽  
Problem Setting

This paper studies two optimal control problems for linear time-invariant systems of fractional order with lumped parameters whose dynamics is described by equations which contain Riemann-Liouville derivative. The first problem is to find control with minimal norm and the second one is to find control with minimal control time at given restriction for control norm. The problem setting with nonlocal initial conditions is considered which differs from other known settings for integer-order systems and fractional-order systems described in terms of equations with Caputo derivative. Admissible controls are allowed to belong to the class of functions which arep-integrable on half segment. The basic investigation approach is the moment method. The correctness and solvability of moment problem are validated for considered problem setting for the system of arbitrary dimension. It is shown that corresponding conditions are analogous to those derived for systems which are described in terms of equations with Caputo derivative. For several particular cases of one- and two-dimensional systems the posed problems are solved explicitly. The dependencies of basic values from derivative index and control time are analyzed. The comparison is performed of obtained results with known results for analogous integer-order systems and fractional-order systems which are described by equations with Caputo derivative.

scholarly journals Asymptotic synchronization of fractional order non-identical complex dynamical networks with Parameter Uncertainties

2021 ◽  
Author(s):  
S Aadhithiyan ◽  
R. Raja ◽  
Bo Kou ◽  
G Selvam ◽  
Michal Niezabitowski ◽  
...  
Keyword(s):  
Complex Networks ◽  
Fractional Order ◽  
Lyapunov Method ◽  
Direct Lyapunov Method ◽  
Parameter Uncertainties ◽  
Complex Dynamic ◽  
Fractional Order Calculus ◽  
Response System ◽  
Liouville Derivative ◽  
Asymptotic Synchronization

This article specically deals with the asymptotic synchronization of non-identical complex dynamic fractional order networks with uncertainty. Initially, by using the Riemann-Liouville derivative, we developed a model for the general non-identical complex network, and based on the properties of fractional order calculus and the direct Lyapunov method in fractional order, we proposed that drive and response system if nonidentical complex networks ensuring asymp-totic synchronization by using neoteric control. Second, taking into account the uncertainties of non-identical complex networks in state matrices and evaluating theirrequirements forasymptotic synchronization. In addition, to explain the eectiveness of the proposed approach, two numerical simulations are given.

scholarly journals Oscillation on a Class of Differential Equations of Fractional Order

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Tongbo Liu ◽  
Bin Zheng ◽  
Fanwei Meng
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Riccati Transformation ◽  
Liouville Derivative ◽  
Inequality Technique

Based on Riccati transformation and certain inequality technique, some new oscillatory criteria are established for the solutions of a class of sequential differential equations with fractional order defined in the modified Riemann-Liouville derivative. The oscillatory criteria established are of new forms compared with the existing results so far in the literature. For illustrating the validity of the results established, we present some examples for them.

Sign in / Sign up

Export Citation Format

Share Document