Generalized mittag-leffler function and generalized fractional calculus operators

“…Some researchers found interesting the analytical results of the linear fractional-order differential equations are represented by the Mittag-Leffler function, which exhibits a power-law asymptotic behavior [12]. Therefore, the fractional calculus is being widely used to analyze the random signals with power-law size distributions or power-law decay of correlations [13,14].…”

Section: Fractal and Fractional Gaussian Noisementioning

confidence: 99%

A Novel Method for Control Performance Assessment with Fractional Order Signal Processing and Its Application to Semiconductor Manufacturing

et al. 2018

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Abstract:The significant task for control performance assessment (CPA) is to review and evaluate the performance of the control system. The control system in the semiconductor industry exhibits a complex dynamic behavior, which is hard to analyze. This paper investigates the interesting crossover properties of Hurst exponent estimations and proposes a novel method for feature extraction of the nonlinear multi-input multi-output (MIMO) systems. At first, coupled data from real industry are analyzed by multifractal detrended fluctuation analysis (MFDFA) and the resultant multifractal spectrum is obtained. Secondly, the crossover points with spline fit in the scale-law curve are located and then employed to segment the entire scale-law curve into several different scaling regions, in which a single Hurst exponent can be estimated. Thirdly, to further ascertain the origin of the multifractality of control signals, the generalized Hurst exponents of the original series are compared with shuffled data. At last, non-Gaussian statistical properties, multifractal properties and Hurst exponents of the process control variables are derived and compared with different sets of tuning parameters. The results have shown that CPA of the MIMO system can be better employed with the help of fractional order signal processing (FOSP).

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“…then the evolution problem 12) with an analytic function g(t) as boundary condition, admits an operational solution…”

Section: Theorem 32 Consider the Following Initial Value Problem (Ivp)mentioning

confidence: 99%

Analytic solutions of fractional integro-differential equations of Volterra type with variable coefficients

Tomovski

Garra

2013

fcaa

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In this paper we study explicit solutions of fractional integro-differential equations with variable coefficients involving Prabhakar-type operators. Analytic solutions to equations involving Prabhakar operators and Laguerre derivatives are obtained by means of operational methods.Cauchy type problems for fractional integro-differential equations of Volterra type with generalized Riemann-Liouville derivative operator, which contain generalized Mittag-Leffler function in the kernel are also considered. Using the Laplace transform method, explicit solutions of the fractional integro-differential equations of Volterra type with variable coefficients, proposed by Srivastava and Tomovski in [28] are established in terms of the multinomial Wright function.MSC 2010 : Primary 26A33; Secondary 33C20, 33E12

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Generalized mittag-leffler function and generalized fractional calculus operators

Cited by 450 publications

References 12 publications

Generalized Langevin Equation and the Prabhakar Derivative

Generalized Langevin Equation and the Prabhakar Derivative

A Novel Method for Control Performance Assessment with Fractional Order Signal Processing and Its Application to Semiconductor Manufacturing

Analytic solutions of fractional integro-differential equations of Volterra type with variable coefficients

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