scholarly journals Analysis of the Time Fractional-Order Coupled Burgers Equations with Non-Singular Kernel Operators

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2326
Author(s):  
Noufe H. Aljahdaly ◽  
Ravi P. Agarwal ◽  
Rasool Shah ◽  
Thongchai Botmart
Keyword(s):  
Fractional Order ◽  
Decomposition Method ◽  
Fractional Derivatives ◽  
Burgers Equation ◽  
Singular Kernel ◽  
Exact Results ◽  
Burgers Equations ◽  
Kernel Operators ◽  
Natural Decomposition

In this article, we have investigated the fractional-order Burgers equation via Natural decomposition method with nonsingular kernel derivatives. The two types of fractional derivatives are used in the article of Caputo–Fabrizio and Atangana–Baleanu derivative. We employed Natural transform on fractional-order Burgers equation followed by inverse Natural transform, to achieve the result of the equations. To validate the method, we have considered a two examples and compared with the exact results.

Solving $$(1+n)$$-Dimensional Fractional Burgers Equation by Natural Decomposition Method

2020 ◽  
Vol 13 (4) ◽  
pp. 368-381
Author(s):  
M. Cherif ◽  
D. Ziane ◽  
A. K. Alomari ◽  
K. Belghaba
Keyword(s):  
Decomposition Method ◽  
Burgers Equation ◽  
Natural Decomposition

scholarly journals Fractional Whitham–Broer–Kaup Equations within Modified Analytical Approaches

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 125 ◽  
Author(s):  
Rasool Shah ◽  
Hassan Khan ◽  
Dumitru Baleanu
Keyword(s):  
Fractional Order ◽  
Traveling Wave ◽  
Fractional Derivatives ◽  
Traveling Wave Solution ◽  
Order System ◽  
Transform Method ◽  
Homotopy Analysis ◽  
Definition Of ◽  
Analytical Approaches ◽  
Natural Decomposition

The fractional traveling wave solution of important Whitham–Broer–Kaup equations was investigated by using the q-homotopy analysis transform method and natural decomposition method. The Caputo definition of fractional derivatives is used to describe the fractional operator. The obtained results, using the suggested methods are compared with each other as well as with the exact results of the problems. The comparison shows the best agreement of solutions with each other and with the exact solution as well. Moreover, the proposed methods are found to be accurate, effective, and straightforward while dealing with the fractional-order system of partial differential equations and therefore can be generalized to other fractional order complex problems from engineering and science.

scholarly journals APPLICATION OF THE CAPUTO-FABRIZIO FRACTIONAL DERIVATIVE WITHOUT SINGULAR KERNEL TO KORTEWEG-DE VRIES-BURGERS EQUATION∗

2016 ◽  
Vol 21 (2) ◽  
pp. 188-198 ◽  
Author(s):  
Emile Franc Doungmo Goufo
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Existence And Uniqueness ◽  
Burgers Equation ◽  
Continuous Solution ◽  
Singular Kernel ◽  
Fractional Differentiation ◽  
De Vries ◽  
Perturbation Parameters ◽  
Derivative Order

In order to bring a broader outlook on some unusual irregularities observed in wave motions and liquids’ movements, we explore the possibility of extending the analysis of Korteweg–de Vries–Burgers equation with two perturbation’s levels to the concepts of fractional differentiation with no singularity. We make use of the newly developed Caputo-Fabrizio fractional derivative with no singular kernel to establish the model. For existence and uniqueness of the continuous solution to the model, conditions on the perturbation parameters ν, µ and the derivative order α are provided. Numerical approximations are performed for some values of the perturbation parameters. This shows similar behaviors of the solution for close values of the fractional order α.

An efficient technique for two-dimensional fractional order biological population model

2020 ◽  
Vol 11 (01) ◽  
pp. 2050005 ◽  
Author(s):  
P. Veeresha ◽  
D. G. Prakasha
Keyword(s):  
Numerical Simulation ◽  
Differential Equations ◽  
Fractional Order ◽  
Population Model ◽  
Decomposition Method ◽  
Arbitrary Order ◽  
Two Dimensional ◽  
Decomposition Scheme ◽  
Biological Population ◽  
Natural Decomposition

In this paper, we find the solutions for two-dimensional biological population model having fractional order using fractional natural decomposition method (FNDM). The proposed method is a graceful blend of decomposition scheme with natural transform, and three examples are considered to validate and illustrate its efficiency. The nature of FNDM solution has been captured for distinct arbitrary order. In order to illustrate the proficiency and reliability of the considered scheme, the numerical simulation has been presented. The obtained results illuminate that the considered method is easy to apply and more effective to examine the nature of multi-dimensional differential equations of fractional order arisen in connected areas of science and technology.

scholarly journals Numerical Investigation of the Time-Fractional Whitham–Broer–Kaup Equation Involving without Singular Kernel Operators

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-21
Author(s):  
Kamsing Nonlaopon ◽  
Muhammad Naeem ◽  
Ahmed M. Zidan ◽  
Rasool Shah ◽  
Ahmed Alsanad ◽  
...  
Keyword(s):  
Fractional Order ◽  
Real World ◽  
Analytical Method ◽  
Laplace Transformation ◽  
Fractional Derivatives ◽  
Perturbation Technique ◽  
Homotopy Perturbation ◽  
Real World Problem ◽  
Kernel Operators ◽  
Homotopy Perturbation Technique

This paper aims to implement an analytical method, known as the Laplace homotopy perturbation transform technique, for the result of fractional-order Whitham–Broer–Kaup equations. The technique is a mixture of the Laplace transformation and homotopy perturbation technique. Fractional derivatives with Mittag-Leffler and exponential laws in sense of Caputo are considered. Moreover, this paper aims to show the Whitham–Broer–Kaup equations with both derivatives to see their difference in a real-world problem. The efficiency of both operators is confirmed by the outcome of the actual results of the Whitham–Broer–Kaup equations. Some problems have been presented to compare the solutions achieved with both fractional-order derivatives.

The properties of Bessel-type fractional derivatives and integrals

2016 ◽  
pp. 3973-3982
Author(s):  
V. R. Lakshmi Gorty
Keyword(s):  
Fractional Order ◽  
Real Line ◽  
Computer Algebra System ◽  
Fractional Derivatives ◽  
Graphical Representation ◽  
Fractional Integrals ◽  
The Real ◽  
Fractional Derivatives And Integrals ◽  
Half Axis ◽  
Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

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.

scholarly journals Vibration Equation of Fractional Order Describing Viscoelasticity and Viscous Inertia

Open Physics ◽  
2019 ◽  
Vol 17 (1) ◽  
pp. 850-856 ◽  
Author(s):  
Jun-Sheng Duan ◽  
Yun-Yun Xu
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Harmonic Excitation ◽  
Steady State Response ◽  
Vibration System ◽  
Derivative Operator ◽  
Vibration Equation ◽  
State Response ◽  
Frequency Relation

Abstract The steady state response of a fractional order vibration system subject to harmonic excitation was studied by using the fractional derivative operator ${}_{-\infty} D_t^\beta,$where the order β is a real number satisfying 0 ≤ β ≤ 2. We derived that the fractional derivative contributes to the viscoelasticity if 0 < β < 1, while it contributes to the viscous inertia if 1 < β < 2. Thus the fractional derivative can represent the “spring-pot” element and also the “inerterpot” element proposed in the present article. The viscosity contribution coefficient, elasticity contribution coefficient, inertia contribution coefficient, amplitude-frequency relation, phase-frequency relation, and influence of the order are discussed in detail. The results show that fractional derivatives are applicable for characterizing the viscoelasticity and viscous inertia of materials.

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