scholarly journals Controllability of Flow-Conservation Transportation Networks with Fractional-Order Dynamics

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Wei Chen ◽  
Bo Zhou
Keyword(s):  
Theoretical Analysis ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Fractional Order ◽  
Location Problem ◽  
Transportation Networks ◽  
Qr Decomposition ◽  
Rank Condition ◽  
Numerical Examples ◽  
Flow Conservation

In this paper, we adapt the fractional derivative approach to formulate the flow-conservation transportation networks, which consider the propagation dynamics and the users’ behaviors in terms of route choices. We then investigate the controllability of the fractional-order transportation networks by employing the Popov-Belevitch-Hautus rank condition and the QR decomposition algorithm. Furthermore, we provide the exact solutions for the full controllability pricing controller location problem, which includes where to locate the controllers and how many controllers are required at the location positions. Finally, we illustrate two numerical examples to validate the theoretical analysis.

Bifurcations and Exact Solutions for a Class of MKdV Equations with the Conformable Fractional Derivative via Dynamical System Method

2020 ◽  
Vol 30 (01) ◽  
pp. 2050004 ◽  
Author(s):  
Jianli Liang ◽  
Longkun Tang ◽  
Yonghui Xia ◽  
Yi Zhang
Keyword(s):  
Fractional Derivative ◽  
Exact Solutions ◽  
Fractional Order ◽  
Traveling Wave ◽  
Nonlinear Differential Equations ◽  
Traveling Wave Solutions ◽  
Leibniz Rule ◽  
Wave Transformation ◽  
Conformable Fractional Derivative ◽  
Simplest Equation Method

In 2014, Khalil et al. [2014] proposed the conformable fractional derivative, which obeys chain rule and the Leibniz rule. In this paper, motivated by the monograph of Jibin Li [Li, 2013], we study the exact traveling wave solutions for a class of third-order MKdV equations with the conformable fractional derivative. Our approach is based on the bifurcation theory of planar dynamical systems, which is much different from the simplest equation method proposed in [Chen & Jiang, 2018]. By employing the traveling wave transformation [Formula: see text] [Formula: see text], we reduce the PDE to an ODE which depends on the fractional order [Formula: see text], then the analysis depends on the order [Formula: see text]. Moreover, as [Formula: see text], the exact solutions are consistent with the integer PDE. However, in all the existing papers, the reduced ODE is independent of the fractional order [Formula: see text]. It is believed that this method can be applicable to solve the other nonlinear differential equations with the conformable fractional derivative.

A Comparative Study of Integer and Noninteger Order Wavelets for Fractional Nonlinear Fredholm Integro-Differential Equations

2018 ◽  
Vol 13 (8) ◽  
Author(s):  
F. Mohammadi ◽  
J. A. Tenreiro Machado
Keyword(s):  
Differential Equations ◽  
Comparative Study ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Efficient Algorithm ◽  
Operational Matrix ◽  
Tau Method ◽  
Legendre Wavelets ◽  
Numerical Examples ◽  
Integer Order

This paper compares the performance of Legendre wavelets (LWs) with integer and noninteger orders for solving fractional nonlinear Fredholm integro-differential equations (FNFIDEs). The generalized fractional-order Legendre wavelets (FLWs) are formulated and the operational matrix of fractional derivative in the Caputo sense is obtained. Based on the FLWs, the operational matrix and the Tau method an efficient algorithm is developed for FNFIDEs. The FLWs basis leads to more efficient and accurate solutions of the FNFIDE than the integer-order Legendre wavelets. Numerical examples confirm the superior accuracy of the proposed method.

scholarly journals Analytical Method for Solving the Fractional Order Generalized KdV Equation by a Beta-Fractional Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Majid Bagheri ◽  
Ali Khani
Keyword(s):  
Fractional Derivative ◽  
Exact Solutions ◽  
Fractional Order ◽  
Kdv Equation ◽  
Time Derivative ◽  
Hyperbolic Function ◽  
Nonlinear Phenomena ◽  
Generalized Kdv Equation ◽  
Wave Solutions ◽  
Fractional Time Derivative

The present work is related to solving the fractional generalized Korteweg-de Vries (gKdV) equation in fractional time derivative form of order α . Some exact solutions of the fractional-order gKdV equation are attained by employing the new powerful expansion approach by using the beta-fractional derivative which is used to get many solitary wave solutions by changing various parameters. The obtained solutions include three classes of soliton wave solutions in terms of hyperbolic function, trigonometric function, and rational function solutions. The obtained solutions and the exact solutions are shown graphically, highlighting the effects of nonlinearity. Some of the nonlinear equations arise in fluid dynamics and nonlinear phenomena.

scholarly journals A solution method for integro-differential equations of conformable fractional derivative

2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 7-14 ◽  
Author(s):  
Mustafa Bayram ◽  
Veysel Hatipoglu ◽  
Sertan Alkan ◽  
Sebahat Das
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Fractional Derivatives ◽  
Solution Method ◽  
Approximate Solutions ◽  
Conformable Fractional Derivative ◽  
Conformable Derivative ◽  
Numerical Examples

The aim of this work is to determine an approximate solution of a fractional order Volterra-Fredholm integro-differential equation using by the Sinc-collocation method. Conformable derivative is considered for the fractional derivatives. Some numerical examples having exact solutions are approximately solved. The comparisons of the exact and the approximate solutions of the examples are presented both in tables and graphical forms.

scholarly journals Davey-Stewartson Equation with Fractional Coordinate Derivatives

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
H. Jafari ◽  
K. Sayevand ◽  
Yasir Khan ◽  
M. Nazari
Keyword(s):  
Fractional Derivative ◽  
Exact Solutions ◽  
Fractional Order ◽  
Homotopy Analysis Method ◽  
Fractional Derivatives ◽  
Stability And Convergence ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Good Agreement

We have used the homotopy analysis method (HAM) to obtain solution of Davey-Stewartson equations of fractional order. The fractional derivative is described in the Caputo sense. The results obtained by this method have been compared with the exact solutions. Stability and convergence of the proposed approach is investigated. The effects of fractional derivatives for the systems under consideration are discussed. Furthermore, comparisons indicate that there is a very good agreement between the solutions of homotopy analysis method and the exact solutions in terms of accuracy.

scholarly journals Analytical methods for non-linear fractional Kolmogorov-Petrovskii-Piskunov equation: Soliton solution and operator solution

2021 ◽  
pp. 102-102
Author(s):  
Bo Xu ◽  
Yufeng Zhang ◽  
Sheng Zhang
Keyword(s):  
Fractional Derivative ◽  
Exact Solutions ◽  
Fractional Order ◽  
Soliton Solution ◽  
Analytical Methods ◽  
Kpp Equation ◽  
Operator Representation ◽  
Operator Solution ◽  
Non Linear ◽  
Kink Soliton

Kolmogorov-Petrovskii-Piskunov (KPP) equation can be regarded as a generalized form of the Fitzhugh-Nagumo, Fisher and Huxley equations which have many applications in physics, chemistry and biology. In this paper, two fractional extended versions of the non-linear KPP equation are solved by analytical methods. Firstly, a new and more general fractional derivative is defined and some properties of it are given. Secondly, a solution in the form of operator representation of the non-linear KPP equation with the defined fractional derivative is obtained. Finally, some exact solutions including kink-soliton solution and other solutions of the non-linear KPP equation with Khalil et al.?s fractional derivative and variable coefficeints are obtained. It is shown that the fractional-order affects the propagation velocitie of the obtained kink-soliton solution.

A Numerical Method for Calculating Fractional Derivative

2015 ◽  
Vol 775 ◽  
pp. 426-430
Author(s):  
Jun Huang ◽  
Yong Jun Li ◽  
Fan Huang
Keyword(s):  
Fractional Derivative ◽  
Numerical Method ◽  
Fractional Order ◽  
High Precision ◽  
Good Stability ◽  
Order Derivative ◽  
Singularity Problem ◽  
Numerical Examples ◽  
Fractional Order Derivative

A numerical method is proposed for calculating the fractional order derivative and successfully resolving the integrand singularity problem based on Zhang-Shimizu algorithm. And then a method is developed to calculate the twice nonlinear fractional derivative, numerical examples demonstrate the numerical method with high precision and good stability.

scholarly journals Numerical Investigation of Fractional-Order Swift–Hohenberg Equations via a Novel Transform

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1263
Author(s):  
Kamsing Nonlaopon ◽  
Abdullah M. Alsharif ◽  
Ahmed M. Zidan ◽  
Adnan Khan ◽  
Yasser S. Hamed ◽  
...  
Keyword(s):  
Fluid Dynamics ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Numerical Investigation ◽  
Fractional Order ◽  
Thermal Convection ◽  
Formation Process ◽  
Decomposition Method ◽  
Numerical Examples

In this paper, the Elzaki transform decomposition method is implemented to solve the time-fractional Swift–Hohenberg equations. The presented model is related to the temperature and thermal convection of fluid dynamics, which can also be used to explain the formation process in liquid surfaces bounded along a horizontally well-conducting boundary. In the Caputo manner, the fractional derivative is described. The suggested method is easy to implement and needs a small number of calculations. The validity of the presented method is confirmed from the numerical examples. Illustrative figures are used to derive and verify the supporting analytical schemes for fractional-order of the proposed problems. It has been confirmed that the proposed method can be easily extended for the solution of other linear and non-linear fractional-order partial differential equations.

scholarly journals Shifted Genocchi Polynomials Operational Matrix for Solving Fractional Order Stiff System

2021 ◽  
Vol 2084 (1) ◽  
pp. 012023
Author(s):  
Abdulnasir Isah ◽  
Chang Phang
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Operational Matrix ◽  
High Accuracy ◽  
Stiff System ◽  
Numerical Examples ◽  
Genocchi Polynomials ◽  
Collocation Scheme

Abstract In this paper, we solve the fractional order stiff system using shifted Genocchi polynomials operational matrix. Different than the well known Genocchi polynomials, we shift the interval from [0, 1] to [1, 2] and name it as shifted Genocchi polynomials. Using the nice properties of shifted Genocchi polynomials which inherit from classical Genocchi polynomials, the shifted Genocchi polynomials operational matrix of fractional derivative will be derived. Collocation scheme are used together with the operational matrix to solve some fractional order stiff system. From the numerical examples, it is obvious that only few terms of shifted Genocchi polynomials is sufficient to obtain result in high accuracy.

scholarly journals Fractional-Order Colour Image Processing

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 457
Author(s):  
Manuel Henriques ◽  
Duarte Valério ◽  
Paulo Gordo ◽  
Rui Melicio
Keyword(s):  
Image Processing ◽  
Edge Detection ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Grey Scale ◽  
Colour Image ◽  
Colour Image Processing ◽  
Image Processing Algorithms ◽  
Processing Algorithms

Many image processing algorithms make use of derivatives. In such cases, fractional derivatives allow an extra degree of freedom, which can be used to obtain better results in applications such as edge detection. Published literature concentrates on grey-scale images; in this paper, algorithms of six fractional detectors for colour images are implemented, and their performance is illustrated. The algorithms are: Canny, Sobel, Roberts, Laplacian of Gaussian, CRONE, and fractional derivative.

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