A Comparative Study of the Fractional-Order System of Burgers Equations

This paper is related to the fractional view analysis of coupled Burgers equations, using innovative analytical techniques. The fractional analysis of the proposed problems has been done in terms of the Caputo-operator sense. In the current methodologies, first, we applied the Elzaki transform to the targeted problem. The Adomian decomposition method and homotopy perturbation method are then implemented to obtain the series form solution. After applying the inverse transform, the desire analytical solution is achieved. The suggested procedures are verified through specific examples of the fractional Burgers couple systems. The current methods are found to be effective methods having a close resemblance with the actual solutions. The proposed techniques have less computational cost and a higher rate of convergence. The proposed techniques are, therefore, beneficial to solve other systems of fractional-order problems.

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Analytical Solutions of (2+Time Fractional Order) Dimensional Physical Models, Using Modified Decomposition Method

Applied Sciences ◽

10.3390/app10010122 ◽

2019 ◽

Vol 10 (1) ◽

pp. 122 ◽

Cited By ~ 4

Author(s):

Hassan Khan ◽

Umar Farooq ◽

Rasool Shah ◽

Dumitru Baleanu ◽

Poom Kumam ◽

...

Keyword(s):

Fractional Order ◽

Decomposition Method ◽

Adomian Decomposition Method ◽

Analytical Techniques ◽

Form Solution ◽

Physical Models ◽

Fractional Partial Differential Equations ◽

Analytical Technique ◽

Adomian Decomposition ◽

Closed Contact

In this article, a new analytical technique based on an innovative transformation is used to solve (2+time fractional-order) dimensional physical models. The proposed method is the hybrid methodology of Shehu transformation along with Adomian decomposition method. The series form solution is obtained by using the suggested method which provides the desired rate of convergence. Some numerical examples are solved by using the proposed method. The solutions of the targeted problems are represented by graphs which have confirmed closed contact between the exact and obtained solutions of the problems. Based on the novelty and straightforward implementation of the method, it is considered to be one of the best analytical techniques to solve linear and non-linear fractional partial differential equations.

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The Analysis of Fractional-Order Helmholtz Equations via a Novel Approach

10.20944/preprints202103.0256.v1 ◽

2021 ◽

Author(s):

Pongsakorn Sunthrayuth ◽

Zeyad Al-Zhour ◽

Yu-Ming Chu

Keyword(s):

Fractional Order ◽

Present Method ◽

Computational Cost ◽

Analytical Techniques ◽

Form Solution ◽

Effective Technique ◽

Helmholtz Equations ◽

Inverse Transform ◽

Novel Approach ◽

Fractional Analysis

This paper is related to the fractional view analysis of Helmholtz equations, using innovative analytical techniques. The fractional analysis of the proposed problems has been done in terms of Caputo-operator sense. In the current methodology, first, we applied the r-Laplace transform to the targeted problem. The iterative method is then implemented to obtain the series form solution. After using the inverse transform of the r-Laplace, the desire analytical solution is achieved. The suggested procedure is verified through specific examples of the fractional Helmholtz equations. The present method is found to be an effective technique having a closed resemblance with the actual solutions. The proposed technique has less computational cost and a higher rate of convergence. The suggested methods are therefore very useful to solve other systems of fractional order problems.

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Analytical Investigation of Noyes–Field Model for Time-Fractional Belousov–Zhabotinsky Reaction

Complexity ◽

10.1155/2021/3248376 ◽

2021 ◽

Vol 2021 ◽

pp. 1-21

Author(s):

Mohammed Kbiri Alaoui ◽

Rabia Fayyaz ◽

Adnan Khan ◽

Rasool Shah ◽

Mohammed S. Abdo

Keyword(s):

Fractional Order ◽

Homotopy Perturbation Method ◽

Adomian Decomposition Method ◽

Analytical Techniques ◽

Analytical Investigation ◽

Physical Problems ◽

Adomian Decomposition ◽

The Given ◽

Belousov Zhabotinsky Reaction ◽

Belousov Zhabotinskii Reaction

In this article, we find the solution of time-fractional Belousov–Zhabotinskii reaction by implementing two well-known analytical techniques. The proposed methods are the modified form of the Adomian decomposition method and homotopy perturbation method with Yang transform. In Caputo manner, the fractional derivative is used. The solution we obtained is in the form of series which helps in investigating the analytical solution of the time-fractional Belousov–Zhabotinskii (B-Z) system. To verify the accuracy of the proposed methods, an illustrative example is taken, and through graphs, the solution is shown. Also, the fractional-order and integer-order solutions are compared with the help of graphs which are easy to understand. It has been verified that the solution obtained by using the given approaches has the desired rate of convergence to the exact solution. The proposed technique’s principal benefit is the low amount of calculations required. It can also be used to solve fractional-order physical problems in a variety of domains.

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An Efficient Analytical Technique, for The Solution of Fractional-Order Telegraph Equations

Mathematics ◽

10.3390/math7050426 ◽

2019 ◽

Vol 7 (5) ◽

pp. 426 ◽

Cited By ~ 6

Author(s):

Hassan Khan ◽

Rasool Shah ◽

Poom Kumam ◽

Dumitru Baleanu ◽

Muhammad Arif

Keyword(s):

Fractional Order ◽

Decomposition Method ◽

Adomian Decomposition Method ◽

Analytical Techniques ◽

Telegraph Equation ◽

High Rate ◽

Analytical Technique ◽

Decomposition Procedure ◽

Adomian Decomposition ◽

Telegraph Equations

In the present article, fractional-order telegraph equations are solved by using the Laplace-Adomian decomposition method. The Caputo operator is used to define the fractional derivative. Series form solutions are obtained for fractional-order telegraph equations by using the proposed method. Some numerical examples are presented to understand the procedure of the Laplace-Adomian decomposition method. As the Laplace-Adomian decomposition procedure has shown the least volume of calculations and high rate of convergence compared to other analytical techniques, the Laplace-Adomian decomposition method is considered to be one of the best analytical techniques for solving fractional-order, non-linear partial differential equations—particularly the fractional-order telegraph equation.

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Complex Dynamics and Hard Limiter Control of a Fractional-Order Buck-Boost System

Mathematical Problems in Engineering ◽

10.1155/2021/5572840 ◽

2021 ◽

Vol 2021 ◽

pp. 1-16

Author(s):

Bo Yan ◽

Shaojie Wang ◽

Shaobo He

Keyword(s):

Fractional Order ◽

Complex Dynamics ◽

Adomian Decomposition Method ◽

Nonlinear Circuits ◽

Fractional Order System ◽

Order System ◽

Control Analysis ◽

Adomian Decomposition ◽

Current Limiter ◽

And Control

Chaos and control analysis for the fractional-order nonlinear circuits is a recent hot topic. In this study, a fractional-order model is deduced from a Buck-Boost converter, and its discrete solution is obtained based on the Adomian decomposition method (ADM). Chaotic dynamic characteristics of the fractional-order system are investigated by the bifurcation diagram, 0-1 test, spectral entropy (SE) algorithm, and NIST test. Meanwhile, the control of the fractional-order Buck-Boost model is discussed through two different ways, namely, the intensity feedback and the hard limiter control. Specifically, the hard limiter control can be realized using a current limiter in the circuit, where the current limiter device is applied to control the branch current. The results show that the proposed fractional-order system has complex dynamic behaviors and potential application values in the engineering field.

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An Analytical Technique, Based on Natural Transform to Solve Fractional-Order Parabolic Equations

Entropy ◽

10.3390/e23081086 ◽

2021 ◽

Vol 23 (8) ◽

pp. 1086

Author(s):

Ravi P. Agarwal ◽

Fatemah Mofarreh ◽

Rasool Shah ◽

Waewta Luangboon ◽

Kamsing Nonlaopon

Keyword(s):

Closed Form ◽

Fractional Order ◽

Parabolic Equations ◽

Closed Form Solution ◽

Adomian Decomposition Method ◽

Form Solution ◽

Order Problem ◽

Analytical Technique ◽

Closed Form Solutions ◽

Adomian Decomposition

This research article is dedicated to solving fractional-order parabolic equations using an innovative analytical technique. The Adomian decomposition method is well supported by natural transform to establish closed form solutions for targeted problems. The procedure is simple, attractive and is preferred over other methods because it provides a closed form solution for the given problems. The solution graphs are plotted for both integer and fractional-order, which shows that the obtained results are in good contact with the exact solution of the problems. It is also observed that the solution of fractional-order problems are convergent to the solution of integer-order problem. In conclusion, the current technique is an accurate and straightforward approximate method that can be applied to solve other fractional-order partial differential equations.

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Dynamics and Complexity Analysis of Fractional-Order Chaotic Systems with Line Equilibrium Based on Adomian Decomposition

Complexity ◽

10.1155/2020/5710765 ◽

2020 ◽

Vol 2020 ◽

pp. 1-13

Author(s):

Heng Chen ◽

Tengfei Lei ◽

Su Lu ◽

WenPeng Dai ◽

Lijun Qiu ◽

...

Keyword(s):

Lyapunov Exponent ◽

Fractional Order ◽

Complexity Analysis ◽

Adomian Decomposition Method ◽

Order System ◽

Largest Lyapunov Exponent ◽

Initial Value ◽

Adomian Decomposition ◽

Line Equilibrium ◽

Lyapunov Exponent Spectrum

In this paper, the Adomian decomposition method (ADM) is applied to solve the fractional-order system with line equilibrium. The dynamics of the system is analyzed by means of the Lyapunov exponent spectrum, bifurcations, chaotic attractor, and largest Lyapunov exponent diagram. At the same time, through the Lyapunov exponent spectrum and bifurcation graph of the system under the change of the initial value, the influence of fractional order q on the system state can be observed. That is, integer-order systems do not have the phenomenon of attractors coexistence, while fractional-order systems have it.

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FRACTIONAL-ORDER CHAOS: A NOVEL FOUR-WING ATTRACTOR IN COUPLED LORENZ SYSTEMS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127409024785 ◽

2009 ◽

Vol 19 (10) ◽

pp. 3329-3338 ◽

Cited By ~ 12

Author(s):

DONATO CAFAGNA ◽

GIUSEPPE GRASSI

Keyword(s):

Fractional Order ◽

Time Domain ◽

Chaotic Attractor ◽

Decomposition Method ◽

Chaotic Dynamics ◽

Adomian Decomposition Method ◽

Order System ◽

Adomian Decomposition ◽

Fractional System ◽

Order Four

By using a time-domain approach, this paper analyzes the chaotic dynamics of the fractional-order system obtained by coupling two fractional Lorenz systems. The work exploits the Adomian decomposition method (ADM), shows that the proposed fractional system with order as low as 2.88 exhibits a novel four-wing chaotic attractor. The existence of chaos is also confirmed by the application of the "0-1 test" for chaos. Finally, an analysis of the system equilibria is conducted, which shows that the fractional-order four-wing attractor is truly the counterpart of the recently introduced integer-order four-wing attractor.

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Implementation of fractional optimal control problems in real-world applications

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2020-0088 ◽

2020 ◽

Vol 23 (6) ◽

pp. 1783-1796

Author(s):

Neelam Singha

Keyword(s):

Optimal Control ◽

Fractional Order ◽

Adomian Decomposition Method ◽

Necessary Optimality Conditions ◽

Order Model ◽

Fractional Optimal Control ◽

Adomian Decomposition ◽

Real World Applications ◽

Fractional Optimal Control Problems ◽

And Control

Abstract In this article, we aim to analyze a mathematical model of tumor growth as a problem of fractional optimal control. The considered fractional-order model describes the interaction of effector-immune cells and tumor cells, including combined chemo-immunotherapy. We deduce the necessary optimality conditions together with implementing the Adomian decomposition method on the suggested fractional-order optimal control problem. The key motive is to perform numerical simulations that shall facilitate us in understanding the behavior of state and control variables. Further, the graphical interpretation of solutions effectively validates the applicability of the present analysis to investigate the growth of cancer cells in the presence of medical treatment.

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