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Article
Publication date: 15 December 2020

Tarikul Islam and Armina Akter

Fractional order nonlinear evolution equations (FNLEEs) pertaining to conformable fractional derivative are considered to be revealed for well-furnished analytic solutions…

Abstract

Purpose

Fractional order nonlinear evolution equations (FNLEEs) pertaining to conformable fractional derivative are considered to be revealed for well-furnished analytic solutions due to their importance in the nature of real world. In this article, the autors suggest a productive technique, called the rational fractional (DξαG/G)-expansion method, to unravel the nonlinear space-time fractional potential Kadomtsev–Petviashvili (PKP) equation, the nonlinear space-time fractional Sharma–Tasso–Olver (STO) equation and the nonlinear space-time fractional Kolmogorov–Petrovskii–Piskunov (KPP) equation. A fractional complex transformation technique is used to convert the considered equations into the fractional order ordinary differential equation. Then the method is employed to make available their solutions. The constructed solutions in terms of trigonometric function, hyperbolic function and rational function are claimed to be fresh and further general in closed form. These solutions might play important roles to depict the complex physical phenomena arise in physics, mathematical physics and engineering.

Design/methodology/approach

The rational fractional (DξαG/G)-expansion method shows high performance and might be used as a strong tool to unravel any other FNLEEs. This method is of the form U(ξ)=i=0nai(DξαG/G)i/i=0nbi(DξαG/G)i.

Findings

Achieved fresh and further abundant closed form traveling wave solutions to analyze the inner mechanisms of complex phenomenon in nature world which will bear a significant role in the of research and will be recorded in the literature.

Originality/value

The rational fractional (DξαG/G)-expansion method shows high performance and might be used as a strong tool to unravel any other FNLEEs. This method is newly established and productive.

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Article
Publication date: 9 December 2020

Wei Meng, Qian Li, Bo Zeng and Yingjie Yang

The purpose of this paper is to unify the expression of fractional grey accumulating generation operator and the reducing generation operator, and build the FDGM(1,1…

Abstract

Purpose

The purpose of this paper is to unify the expression of fractional grey accumulating generation operator and the reducing generation operator, and build the FDGM(1,1) model with the unified fractional grey generation operator.

Design/methodology/approach

By systematically studying the properties of the fractional accumulating operator and the reducing operator, and analyzing the sensitivity of the order value, a unified expression of the fractional operators is given. The FDGM(1,1) model with the unified fractional grey generation operator is established. The relationship between the order value and the modeling error distribution is studied.

Findings

The expression of the fractional accumulating generation operator and the reducing generation operator can be unified to a simple expression. For −1<r < 1, the fractional grey generation operator satisfies the principle of new information priority. The DGM(1,1) model is a special case of the FDGM(1,1) model with r = 1.

Research limitations/implications

The sensitivity of the unified operator is verified through random numerical simulation method, and the theoretical proof was not yet possible.

Practical implications

The FDGM(1,1) model has a higher modeling accuracy and modeling adaptability than the DGM(1,1) by optimizing the order.

Originality/value

The expression of the fractional accumulating generation operator and the reducing generation operator is firstly unified. The FDGM(1,1) model with the unified fractional grey generation operator is firstly established. The unification of the fractional accumulating operator and the reducing operator improved the theoretical basis of grey generation operator.

Details

Grey Systems: Theory and Application, vol. ahead-of-print no. ahead-of-print
Type: Research Article
ISSN: 2043-9377

Keywords

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Article
Publication date: 12 August 2020

Amjid Ali, Teruya Minamoto, Umer Saeed and Mujeeb Ur Rehman

The purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear fractional differential equations involving ψ-Caputo derivative.

Abstract

Purpose

The purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear fractional differential equations involving ψ-Caputo derivative.

Design/methodology/approach

An operational matrix to find numerical approximation of ψ-fractional differential equations (FDEs) is derived. This study extends the method to nonlinear FDEs by using quasi linearization technique to linearize the nonlinear problems.

Findings

The error analysis of the proposed method is discussed in-depth. Accuracy and efficiency of the method are verified through numerical examples.

Research limitations/implications

The method is simple and a good mathematical tool for finding solutions of nonlinear ψ-FDEs. The operational matrix approach offers less computational complexity.

Originality/value

Engineers and applied scientists may use the present method for solving fractional models appearing in applications.

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Article
Publication date: 23 September 2020

Yu Bai, Lamei Huo and Yan Zhang

The purpose of this study is to investigate the unsteady stagnation-point flow and heat transfer of fractional Maxwell fluid towards a time power-law-dependent stretching…

Abstract

Purpose

The purpose of this study is to investigate the unsteady stagnation-point flow and heat transfer of fractional Maxwell fluid towards a time power-law-dependent stretching plate. Based on the characteristics of pressure in the boundary layer, the momentum equation with the fractional Maxwell model is firstly formulated to analyze unsteady stagnation-point flow. Furthermore, generalized Fourier’s law is considered in the energy equation and boundary condition of convective heat transfer.

Design/methodology/approach

The nonlinear fractional differential equations are solved by the newly developed finite difference scheme combined with L1-algorithm, whose convergence is verified by constructing a numerical example.

Findings

Some interesting results can be revealed. The larger fractional derivative parameter of velocity promotes the flow, while the smaller fractional derivative parameter of temperature accelerates the heat transfer. The temperature boundary layer is thicker than the velocity boundary layer, and the velocity enlarges as the stagnation parameter raises. This is because when Prandtl number < 1, the capacity of heat diffusion is greater than that of momentum diffusion. It is to be observed that all the temperature profiles first enhance a little and then reduce rapidly, which indicates the thermal retardation of Maxwell fluid.

Originality/value

The unsteady stagnation-point flow model of Maxwell fluid is extended from integral derivative to fractional derivative, which has more flexibility to describe viscoelastic fluid’s complex dynamic process and provide a theoretical basis for industrial processing.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. ahead-of-print no. ahead-of-print
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 28 November 2019

Guishu Liang and Yulan Yang

This paper aims to analyze soil electrical properties based on fractional calculus theory due to the fact that the frequency dependence of soil electrical parameters at…

Abstract

Purpose

This paper aims to analyze soil electrical properties based on fractional calculus theory due to the fact that the frequency dependence of soil electrical parameters at high frequencies exhibits a fractional effect. In addition, for the fractional-order formulation, this paper aims to provide a more accurate numerical algorithm for solving the fractional differential equations.

Design/methodology/approach

This paper analyzes the frequency-dependence of soil electrical properties based on fractional calculus theory. A collocation method based on the Puiseux series is proposed to solve fractional differential equations.

Findings

The algorithm proposed in this paper can be used to solve fractional differential equations of arbitrary order, especially for 0.5th-order equations, obtaining accurate numerical solutions. Calculating the impact response of the grounding electrode based on the fractional calculus theory can obtain a more accurate result.

Originality/value

This paper proposes an algorithm for solving fractional differential equations of arbitrary order, especially for 0.5th-order equations. Using fractional calculus theory to analyze the frequency-dependent effect of soil electrical properties, provides a new idea for ground-related transient calculation.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering , vol. 39 no. 2
Type: Research Article
ISSN: 0332-1649

Keywords

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Article
Publication date: 10 January 2020

Xiang Li, Zhijun Li and Zihao Wen

This paper aims to introduce a novel 4D hyperchaotic fractional-order system which can produce one-to-four-wing hyperchaotic attractors. In the study of chaotic systems…

Abstract

Purpose

This paper aims to introduce a novel 4D hyperchaotic fractional-order system which can produce one-to-four-wing hyperchaotic attractors. In the study of chaotic systems with variable-wing attractors, although some chaotic systems can generate one-to-four-wing attractors, none of them are hyperchaotic attractors, which is incomplete for the dynamic characteristics of chaotic systems.

Design/methodology/approach

A novel 4D fractional-order hyperchaotic system is proposed based on the classical three-dimensional Lü system. The complex and abundant dynamic behaviors of the fractional-order system are analyzed by phase diagrams, bifurcation diagrams and the corresponding Lyapunov exponents. In addition, SE and C0 algorithms are used to analyze the complexity of the fractional-order system. Then, the influence of order q on the system is also investigated. Finally, the circuit is implemented using physical components.

Findings

The most particular interest is that the system can generate one-to-four-wing hyperchaotic attractors with only one parameter variation. Then, the hardware circuit experimental results tally with the numerical simulations, which proves the validity and feasibility of the fractional-order hyperchaotic system. Besides, under different initial conditions, coexisting attractors can be obtained by changing the parameter d or the order q. Then, the complexity analysis of the system shows that the fractional-order chaotic system has higher complexity than the corresponding integer-order chaotic system.

Originality/value

The circuit structure of the fractional-order hyperchaotic system is simple and easy to implement, and one-to-four-wing hyperchaotic attractors can be observed in the circuit. To the best of the knowledge, this unique phenomenon has not been reported in any literature. It is of great reference value to analysis and circuit realization of fractional-order chaotic systems.

Details

Circuit World, vol. 46 no. 2
Type: Research Article
ISSN: 0305-6120

Keywords

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Article
Publication date: 12 August 2019

Umer Saeed and Muhammad Umair

The purpose of the paper is to extend the differential quadrature method (DQM) for solving time and space fractional non-linear partial differential equations on a…

Abstract

Purpose

The purpose of the paper is to extend the differential quadrature method (DQM) for solving time and space fractional non-linear partial differential equations on a semi-infinite domain.

Design/methodology/approach

The proposed method is the combination of the Legendre polynomials and differential quadrature method. The authors derived and constructed the new operational matrices for the fractional derivatives, which are used for the solutions of non-linear time and space fractional partial differential equations.

Findings

The fractional derivative of Lagrange polynomial is a big hurdle in classical DQM. To overcome this problem, the authors represent the Lagrange polynomial in terms of shifted Legendre polynomial. They construct a transformation matrix which transforms the Lagrange polynomial into shifted Legendre polynomial of arbitrary order. Then, they obtain the new weighting coefficients matrices for space fractional derivatives by shifted Legendre polynomials and use these in conversion of a non-linear fractional partial differential equation into a system of fractional ordinary differential equations. Convergence analysis for the proposed method is also discussed.

Originality/value

Many engineers can use the presented method for solving their time and space fractional non-linear partial differential equation models. To the best of the authors’ knowledge, the differential quadrature method has never been extended or implemented for non-linear time and space fractional partial differential equations.

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Article
Publication date: 1 February 2016

Wei Meng, Qian Li and Bo Zeng

The purpose of this paper is to derive the analytical expression of fractional order reducing generation operator (or inverse accumulating generating operation) and study…

Abstract

Purpose

The purpose of this paper is to derive the analytical expression of fractional order reducing generation operator (or inverse accumulating generating operation) and study its properties.

Design/methodology/approach

This disaggregation method includes three main steps. First, by utilizing Gamma function expanded for integer factorial, this paper expands one order reducing generation operator into integer order reducing generation operator and fractional order reducing generation operator, and gives the analytical expression of fractional order reducing generation operator. Then, studies the commutative law and exponential law of fractional order reducing generation operator. Lastly, gives several examples of fractional order reducing generation operator and verifies the commutative law and exponential law of fractional order reducing generation operator.

Findings

The authors pull the analytical expression of fractional order reducing generation operator and verify that fractional order reducing generation operator satisfies commutative law and exponential law.

Practical implications

Expanding the reducing generation operator would help develop grey prediction model with fractional order operators and widen the application fields of grey prediction models.

Originality/value

The analytical expression of fractional order reducing generation operator, properties of commutative law and exponential law for fractional order reducing generation operator are first studied.

Details

Grey Systems: Theory and Application, vol. 6 no. 1
Type: Research Article
ISSN: 2043-9377

Keywords

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Article
Publication date: 30 March 2010

Ahmet Yıldırım

This paper aims to present a general framework of the homotopy perturbation method (HPM) for analytic treatment of fractional partial differential equations in fluid…

Abstract

Purpose

This paper aims to present a general framework of the homotopy perturbation method (HPM) for analytic treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense.

Design/methodology/approach

Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation and fractional Klein‐Gordon equation are investigated to show the pertinent features of the technique.

Findings

HPM is a powerful and efficient technique in finding exact and approximate solutions for fractional partial differential equations in fluid mechanics. The implementation of the noise terms, if they exist, is a powerful tool to accelerate the convergence of the solution. The results so obtained reinforce the conclusions made by many researchers that the efficiency of the HPM and related phenomena gives it much wider applicability.

Originality/value

The essential idea of this method is to introduce a homotopy parameter, say p, which takes values from 0 to 1. When p = 0, the system of equations usually reduces to a sufficiently simplied form, which normally admits a rather simple solution. As p is gradually increased to 1, the system goes through a sequence of deformations, the solution for each of which is close to that at the previous stage of deformation.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 20 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 3 July 2017

Zaher Rahimi, Wojciech Sumelka and Xiao-Jun Yang

Recently, a new formulation has been introduced for non-local mechanics in terms of fractional calculus. Fractional calculus is a branch of mathematical analysis that…

Abstract

Purpose

Recently, a new formulation has been introduced for non-local mechanics in terms of fractional calculus. Fractional calculus is a branch of mathematical analysis that studies the differential operators of an arbitrary (real or complex) order and is used successfully in various fields such as mathematics, science and engineering. The purpose of this paper is to introduce a new fractional non-local theory which may be applicable in various simple or complex mechanical problems.

Design/methodology/approach

In this paper (by using fractional calculus), a fractional non-local theory based on the conformable fractional derivative (CFD) definition is presented, which is a generalized form of the Eringen non-local theory (ENT). The theory contains two free parameters: the fractional parameter which controls the stress gradient order in the constitutive relation and could be an integer and a non-integer and the non-local parameter to consider the small-scale effect in the micron and the sub-micron scales. The non-linear governing equation is solved by the Galerkin and the parameter expansion methods. The non-linearity of the governing equation is due to the presence of von-Kármán non-linearity and CFD definition.

Findings

The theory has been used to study linear and non-linear free vibration of the simply-supported (S-S) and the clamped-free (C-F) nano beams and then the influence of the fractional and the non-local parameters has been shown on the linear and non-linear frequency ratio.

Originality/value

A new parameter of the theory (the fractional parameter) makes the modeling more fixable – this model can conclude all of integer and non-integer operators and is not limited to special operators such as ENT. In other words, it allows us to use more sophisticated mathematics to model physical phenomena. On the other hand, in the comparison of classic fractional non-local theory, the theory applicable in various simple or complex mechanical problems may be used because of simpler forms of the governing equation owing to the use of CFD definition.

Details

Engineering Computations, vol. 34 no. 5
Type: Research Article
ISSN: 0264-4401

Keywords

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