A short review on analytical methods for fractional equations with he’s fractional derivative

Wang, Yan; Zhang, Yufeng; Liu, Jian‐Gen; Muhammad, Iqbal

doi:10.2298/tsci160513036w

Cited by 14 publications

(23 citation statements)

References 12 publications

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“…The surfaces have been becoming the playground of many disciplines, such as physics, chemistry, material science, nanotechnology and modern textile science as well. Fractional calculus become reviving after about 300 years' silence to deal with unsmooth boundary conditions (Hu and He, 2016;Sing, 2017;Wang et al, 2017aWang et al, , 2017b, and nano reactors have been proposed to produce energy from "nothing" of the nanomaterials (El Naschie, 2015a, 2015b.…”

Section: Introductionmentioning

confidence: 99%

Geometrical potential and nanofiber membrane’s highly selective adsorption property

Tian¹,

2018

Adsorption Science & Technology

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The definition of a nanofiber is discussed, and the history of the development of bubble electrospinning is briefly elucidated, its main properties are emphasized, and the morphology of its products is summarized. A geometrical interpretation of the nanofiber membrane's adsorption is given as a natural intrinsic property of the geometrical potential of nanoscale structure. Nano-effect, Casimir effect, capillary effect, Earth's gravity difference, Newton's gravity, and lotus leaf's highly selective repulsion property can all be interpreted using the geometrical potential or boundary-induced force.

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Section: Introductionmentioning

confidence: 99%

Geometrical potential and nanofiber membrane’s highly selective adsorption property

Tian¹,

2018

Adsorption Science & Technology

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“…Substituting equation 21into equation 22and on discretizing t in 20 equally spaced points in the interval 0; 1 ½ , we get the fitness function (17). Initially, set a 1 ¼ a 2 ¼ a 3 ¼ a, T 0 and X 0 , and then process the SA algorithm, defined in the section SA algorithm.…”

Section: Application and Discussionmentioning

confidence: 99%

“…Objective function E X ð Þ ¼ E X 1;i ; X 2;i ; X 3;i ð Þ Initialize the initial temperature T 0 and initial guess X 0;i Set the final temperature T f and the max number of iterations j Define the cooling schedule T ! gT; 0 < g 1 while (T > T f and d < M) Drawn from a Boltzmann probability distribution Move randomly to a new location: Suppose that X is the optimal solution of the optimization problem (17), then by substituting it into equation (14), the approximate numerical solutions of equation 13is achieved.…”

Section: Sa Algorithmmentioning

confidence: 99%

“…Consequently, these novel aspects enrich the capabilities of fractional differential models by bringing diverse physical significances to light. [13][14][15] Hence, by means of different theories of fractional derivatives, the behaviours of many fractional differential equations have been studied and various techniques have been developed, [16][17][18][19] but still, there are many things that can be done in this area.…”

Section: Introductionmentioning

confidence: 99%

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Tracking the chaotic behaviour of fractional-order Chua’s system by Mexican hat wavelet-based artificial neural network

Khan

Hameed

Razzaq

et al. 2018

Journal of Low Frequency Noise, Vibration and Active Control

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In this paper, a process is devised systematically to scrutinize the scrolling chaotic behaviour of fractional-order Chua's system. The process is composed of fractional Laplace transformation, artificial neural network with Mexican hat wavelet as an activation function and simulated annealing. Sequentially, the parametric expansion of fractional Laplace transform is employed to convert the governing fractional system into an ordinary differential system. Next, artificial neural network and simulated annealing approximate and optimize the attained system and produce accurate solutions. The predictability and elaboration of double scrolling chaotic structures of fractional-order Chua's system are also studied using the Lyapunov exponent and fifth-fourth Runge-Kutta method. Moreover, the mean absolute error and root mean square error are measured for the convergence analysis of the proposed scheme. On the whole, the accurate approximate solutions, the phase plots of Lyapunov exponent spectrum and bifurcation maps of the dynamical evolution of fractional Chua's system are a triumph of this endeavour.

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“…This method was proposed by Hu and He 24 who also used the Fornberg-Whitham equation for all kinds of physical and engineering problems, 24 improvement of the Riemann-Liouville derivative, 25 and many other applications. 26,27 This method is different from a normal chaotic system where processing has to be done to make a determination. The time required for detection is less and diagnostic precision was better.…”

Section: Introductionmentioning

confidence: 99%

Intelligent real-time monitoring of Computer Numerical Control tool wear based on a fractional-order chaotic self-synchronization system

Chen

2018

Journal of Low Frequency Noise, Vibration and Active Control

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Tool wear and damage are unavoidable in machines which operate for long periods. The main objective of this study was the prediction and understanding of tool wear ahead of time. To achieve this a real-time monitoring system was needed which incidentally also improves product yield and efficiency. Traditionally, tool wear and working life have been estimated by past experience. To monitor the state of the tool, we used a method employing fractional-order chaotic selfsynchronization system matched with extension theory to analyze, extract, and measure tool vibration signals. The fractional-order Chen-Lee chaotic system was used to detect differences in micro vibrations, resulting from different degrees of tool wear, and these were introduced into the master and slave systems. The extreme changes produced by micro differences in the chaotic system, and matter element module design based on the extension theory, were used to accurately determine tool status. Chaos theory, wavelet packet analysis, and Fast Fourier transform were then used to compare the results. This method can reduce the number of sensors and the time required for real-time monitoring of expected life compared to previous determination methods where temperature and current diagnostics must be included.

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A short review on analytical methods for fractional equations with he’s fractional derivative

Cited by 14 publications

References 12 publications

Geometrical potential and nanofiber membrane’s highly selective adsorption property

Geometrical potential and nanofiber membrane’s highly selective adsorption property

Tracking the chaotic behaviour of fractional-order Chua’s system by Mexican hat wavelet-based artificial neural network

Intelligent real-time monitoring of Computer Numerical Control tool wear based on a fractional-order chaotic self-synchronization system

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