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21 – 30 of 314Haiyan Zhang, Muhammad Nadeem, Asim Rauf and Zhao Guo Hui
The purpose of this paper is to suggest the solution of time-fractional Fornberg–Whitham and time-fractional Fokker–Planck equations by using a novel approach.
Abstract
Purpose
The purpose of this paper is to suggest the solution of time-fractional Fornberg–Whitham and time-fractional Fokker–Planck equations by using a novel approach.
Design/methodology/approach
First, some basic properties of fractional derivatives are defined to construct a novel approach. Second, modified Laplace homotopy perturbation method (HPM) is constructed which yields to a direct approach. Third, two numerical examples are presented to show the accuracy of this derived method and graphically results showed that this method is very effective. Finally, convergence of HPM is proved strictly with detail.
Findings
It is not necessary to consider any type of assumptions and hypothesis for the development of this approach. Thus, the suggested method becomes very simple and a better approach for the solution of time-fractional differential equations.
Originality/value
Although many analytical methods for the solution of fractional partial differential equations are presented in the literature. This novel approach demonstrates that the proposed approach can be applied directly without any kind of assumptions.
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Talaat El-Sayed El-Danaf, Mfida Ali Zaki and Wedad Moenaaem
The purpose of this paper is to investigate the possibility of extension to the variational iteration and the Adomian decomposition methods for solving nonlinear Huxley equation…
Abstract
Purpose
The purpose of this paper is to investigate the possibility of extension to the variational iteration and the Adomian decomposition methods for solving nonlinear Huxley equation with time-fractional derivative.
Design/methodology/approach
Objectives achieved the main methods: the fractional derivative of f (x) in the Caputo sense is first stated. Second, the time-fractional Huxley equation is written in a differential operator form where the differential operator is in Caputo sense. After acting on both sides by the inverse operator of the fractional differential operator in Caputo sense, the Adomian's decomposition is then used to get the power series solution of the resulted time-fractional Huxley equation. Also, a second objective is achieved by applying the variational iteration method to get approximate solutions for the time-fractional Huxley equation.
Findings
There are some important findings to state and summarize here. First, the variational iteration method and the decomposition method provide the solutions in terms of convergent series with easily computable components for this considered problem. Second, it seems that the approximate solution of time-fractional Huxley equation using the decomposition method converges faster than the approximate solution using the variational iteration method. Third, the variational iteration method handles nonlinear equations without any need for the so-called Adomian polynomials. However, Adomian decomposition method provides the components of the exact solution, where these components should follow the summation given in Equation (21).
Originality/value
This paper presents new materials in terms of employing the variational iteration and the Adomian decomposition methods to solve the problem under consideration. It is expected that the results will give some insightful conclusions for the used techniques to handle similar fractional differential equations. This emphasizes the fact that the two methods are applicable to a broad class of nonlinear problems in fractional differential equations.
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Sunil Kumar, R.P. Chauhan, Shaher Momani and Samir Hadid
This paper aims to study the complex behavior of a dynamical system using fractional and fractal-fractional (FF) derivative operators. The non-classical derivatives are extremely…
Abstract
Purpose
This paper aims to study the complex behavior of a dynamical system using fractional and fractal-fractional (FF) derivative operators. The non-classical derivatives are extremely useful for investigating the hidden behavior of the systems. The Atangana–Baleanu (AB) and Caputo–Fabrizio (CF) derivatives are considered for the fractional structure of the model. Further, to add more complexity, the authors have taken the system with a CF fractal-fractional derivative having an exponential kernel. The active control technique is also considered for chaos control.
Design/methodology/approach
The systems under consideration are solved numerically. The authors show the Adams-type predictor-corrector scheme for the AB model and the Adams–Bashforth scheme for the CF model. The convergence and stability results are given for the numerical scheme. A numerical scheme for the FF model is also presented. Further, an active control scheme is used for chaos control and synchronization of the systems.
Findings
Simulations of the obtained solutions are displayed via graphics. The proposed system exhibits a very complex phenomenon known as chaos. The importance of the fractional and fractal order can be seen in the presented graphics. Furthermore, chaos control and synchronization between two identical fractional-order systems are achieved.
Originality/value
This paper mentioned the complex behavior of a dynamical system with fractional and fractal-fractional operators. Chaos control and synchronization using active control are also described.
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S. Sahoo, S. Saha Ray and S. Das
In this paper, the formulation and analytic solutions for fractional continuously variable order dynamic models, namely, fractional continuously mass-spring damper (continuously…
Abstract
Purpose
In this paper, the formulation and analytic solutions for fractional continuously variable order dynamic models, namely, fractional continuously mass-spring damper (continuously variable fractional order) systems, have been presented. The authors will demonstrate via two cases where the frictional damping given by fractional derivative, the order of which varies continuously – while the mass moves in a guide. Here, the continuously changing nature of the fractional-order derivative for dynamic systems has been studied for the first time. The solutions of the fractional continuously variable order mass-spring damper systems have been presented here by using a successive recursive method, and the closed form of the solutions has been obtained. By using graphical plots, the nature of the solutions has been discussed for the different cases of continuously variable fractional order of damping force for oscillator. The purpose of the paper is to formulate the continuously variable order mass-spring damper systems and find their analytical solutions by successive recursion method.
Design/methodology/approach
The authors have used the viscoelastic and viscous – viscoelastic dampers for describing the damping nature of the oscillating systems, where the order of the fractional derivative varies continuously.
Findings
By using the successive recursive method, here, the authors find the solution of the fractional continuously variable order mass-spring damper systems, and then obtain close-form solutions. The authors then present and discuss the solutions obtained in the cases with the continuously variable order of damping for an oscillator through graphical plots.
Originality/value
Formulation of fractional continuously variable order dynamic models has been described. Fractional continuous variable order mass-spring damper systems have been analysed. A new approach to find solutions of the aforementioned dynamic models has been established. Viscoelastic and viscous – viscoelastic dampers are described. The discussed damping nature of the oscillating systems has not been studied yet.
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Akbar Mohebbi, Mostafa Abbaszadeh and Mehdi Dehghan
The purpose of this paper is to show that the meshless method based on radial basis functions (RBFs) collocation method is powerful, suitable and simple for solving one and two…
Abstract
Purpose
The purpose of this paper is to show that the meshless method based on radial basis functions (RBFs) collocation method is powerful, suitable and simple for solving one and two dimensional time fractional telegraph equation.
Design/methodology/approach
In this method the authors first approximate the time fractional derivatives of mentioned equation by two schemes of orders O(τ3−α) and O(τ2−α), 1/2<α<1, then the authors will use the Kansa approach to approximate the spatial derivatives.
Findings
The results of numerical experiments are compared with analytical solution, revealing that the obtained numerical solutions have acceptance accuracy.
Originality/value
The results show that the meshless method based on the RBFs and collocation approach is also suitable for the treatment of the time fractional telegraph equation.
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The purpose of this paper is to explain why fractal, self‐similarity, and fractional Brownian motions are so pervasive in human systems.
Abstract
Purpose
The purpose of this paper is to explain why fractal, self‐similarity, and fractional Brownian motions are so pervasive in human systems.
Design/methodology/approach
The analysis involves mainly relative observation, Minkowskian observation, Euclidean observation, and fractional calculus.
Findings
It is shown that observation with informational invariance, which is a modeling of subjectivity, creates fractal, and self‐similarity.
Research limitations/implications
This result could have an application to the quantitative analysis of volatility in finance, for instance.
Practical implications
The paper supports the use of fractional dynamics to describe human systems.
Originality/value
The paper provides practical arguments that may explain why fractals are so pervasive in natural science, and mainly in systems involving human factors.
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– The purpose of this paper is to demonstrate how anomalous diffusion behaviors can be manifest in physically realizable phase change systems.
Abstract
Purpose
The purpose of this paper is to demonstrate how anomalous diffusion behaviors can be manifest in physically realizable phase change systems.
Design/methodology/approach
In the presence of heterogeneity the exponent in the diffusion time scale can become anomalous, exhibiting values that differ from the expected value of 1/2. Here the author investigates, through directed numerical simulation, the two-dimensional melting of a phase change material (PCM) contained in a pattern of cavities separated by a non-PCM matrix. Under normal circumstances we would expect that the progress of melting F(t) would exhibit the normal diffusion time exponent, i.e., F∼t1/2. The author’s intention is to investigate what features of the PCM cavity pattern might induce anomalous phase change, where the progress of melting has a time exponent different from n=1/2.
Findings
When the PCM cavity pattern has an internal length scale, i.e., when there is a sub-domain pattern which, when reproduced, gives us the full domain pattern, the direct simulation recovers the normal ∼t1/2 phase change behavior. When, however, there is no internal length scale, e.g., the pattern is a truncated fractal, an anomalous super diffusive behavior results with melting going as t n; n > 1/2. By studying a range of related fractal patterns, the author is able to relate the observed sub-diffusive exponent to the cavity pattern’s fractal dimension. The author also shows, how the observed behavior can be modeled with a non-local fractional diffusion treatment and how sub-diffusion phase change behavior (F∼t n; n < 1/2) results when the phase change nature of the materials in the cavity and matrix are inverted.
Research limitations/implications
Although the results clearly demonstrate under what circumstances anomalous phase change behavior can be practically produced, the question of an exact theoretical relationship between the cavity pattern geometry and the observed anomalous time exponent is not known.
Practical implications
The clear role of the influence of heterogeneity on heat flow behavior is illustrated. Suggesting that modeling heat and fluid flow in heterogeneous systems requires careful consideration.
Originality/value
The novel direct simulation of melting in a two-dimensional PCM cavity pattern provides a clear illustration of anomalous behavior in a classic heat and fluid flow system and by extension provides motivation to continue the numerical investigation of anomalous and non-local behaviors and fractional calculus tools.
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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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The purpose of this paper is to develop a scheme to study numerical solution of time fractional nonlinear evolution equations under initial conditions by reduced differential…
Abstract
Purpose
The purpose of this paper is to develop a scheme to study numerical solution of time fractional nonlinear evolution equations under initial conditions by reduced differential transform method.
Design/methodology/approach
The paper considers two models of special interest in physics with fractional‐time derivative of order, namely, the time fractional mKdV equation and time fractional convection diffusion equation with nonlinear source term.
Findings
The numerical results demonstrate the significant features, efficiency and reliability of the proposed method and the effects of different values are shown graphically.
Originality/value
The paper shows that the results obtained from the fractional analysis appear to be general.
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Xindong Zhang, Leilei Wei, Bo Tang and Yinnian He
In this article, the authors aim to present the homotopy analysis method (HAM) for obtaining the approximate solutions of space‐time fractional differential equations with initial…
Abstract
Purpose
In this article, the authors aim to present the homotopy analysis method (HAM) for obtaining the approximate solutions of space‐time fractional differential equations with initial conditions.
Design/methodology/approach
The series solution is developed and the recurrence relations are given explicitly. The initial approximation can be determined by imposing the initial conditions.
Findings
The comparison of the HAM results with the exact solutions is made; the results reveal that the HAM is very effective and simple. The HAM contains the auxiliary parameter h, which provides a simple way to adjust and control the convergence region of series solution. Numerical examples demonstrate the effect of changing homotopy auxiliary parameter h on the convergence of the approximate solution. Also, they illustrate the effect of the fractional derivative orders a and b on the solution behavior.
Originality/value
The idea can be used to find the numerical solutions of other fractional differential equations.
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