Design of Mittag–Leffler Kernel-Based Fractional-Order Digital Filter Using Fractional Delay Interpolation

By using fractional calculus and the summation by parts formula in this paper, the asymptotic behaviours of solutions of nonlinear neutral fractional delay pantograph equations with continuous arguments are investigated. The asymptotic estimates of solutions for the equation are obtained, which may imply asymptotic stability of solutions. In the end, a particular case is provided to illustrate the main result and the speed of the convergence of the obtained solutions.

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A fractional‐order generalized Taylor wavelet method for nonlinear fractional delay and nonlinear fractional pantograph differential equations

Mathematical Methods in the Applied Sciences ◽

10.1002/mma.7020 ◽

2020 ◽

Author(s):

Boonrod Yuttanan ◽

Mohsen Razzaghi ◽

Thieu N. Vo

Keyword(s):

Differential Equations ◽

Fractional Order ◽

Wavelet Method ◽

Fractional Delay

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An Overview of Fractional Order Signal Processing (FOSP) Techniques

Volume 5: 6th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B, and C ◽

10.1115/detc2007-34228 ◽

2007 ◽

Cited By ~ 10

Author(s):

YangQuan Chen ◽

Rongtao Sun ◽

Anhong Zhou

Keyword(s):

Signal Processing ◽

Fractional Order ◽

Fourier Transformation ◽

Moving Average ◽

Hurst Parameter ◽

Fractional Operator ◽

Order Linear ◽

Fundamental Properties ◽

Fractional Delay ◽

Fractional Delay Filter

This paper presents a brief overview of some existing fractional order signal processing (FOSP) techniques where the developments in the mathematical communities are introduced; relationship between the fractional operator and long-range dependence is demonstrated, and fundamental properties of each technique and some of its applications are summarized. Specifically, we presented a tutorial on 1) fractional order linear systems; 2) autoregressive fractional integrated moving average (ARFIMA); 3) 1/fαnoise; 4) Hurst parameter estimation; 5) fractional order Fourier transformation (FrFT); 6) fractional order linear transforms (Hartley, Sine, Cosine); 7) fractal; 8) fractional order splines; 9) fractional lower order moments (FLOM) and 10) fractional delay filter. Whenever possible, we indicate the connections between these FOSP techniques.

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On existence and stability results for nonlinear fractional delay differential equations

Boletim da Sociedade Paranaense de Matemática ◽

10.5269/bspm.v36i4.33603 ◽

2018 ◽

Vol 36 (4) ◽

pp. 55-75 ◽

Cited By ~ 9

Author(s):

Kishor D. Kucche ◽

Sagar T. Sutar

Keyword(s):

Differential Equations ◽

Fractional Order ◽

Delay Differential Equations ◽

Successive Approximation Method ◽

Existence And Uniqueness Results ◽

Uniqueness Results ◽

Existence And Stability ◽

Fractional Delay ◽

Delay Differential ◽

Rassias Stability

We establish existence and uniqueness results for fractional order delay differential equations. It is proved that successive approximation method can also be successfully applied to study Ulam--Hyers stability, generalized Ulam--Hyers stability, Ulam--Hyers--Rassias stability, generalized Ulam--Hyers--Rassias stability, $ \mathbb{E}_{\alpha}$--Ulam--Hyers stability and generalized $ \mathbb{E}_{\alpha}$--Ulam--Hyers stability of fractional order delay differential equations.

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Robust non-fragile fractional order PID controller for linear time invariant fractional delay systems

Journal of Process Control ◽

10.1016/j.jprocont.2014.07.001 ◽

2014 ◽

Vol 24 (9) ◽

pp. 1489-1494 ◽

Cited By ~ 14

Author(s):

Afshin Mesbahi ◽

Mohammad Haeri

Keyword(s):

Fractional Order ◽

Pid Controller ◽

Linear Time ◽

Delay Systems ◽

Linear Time Invariant ◽

Fractional Delay ◽

Time Invariant ◽

Fractional Order Pid ◽

Fractional Order Pid Controller

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Adoption of Vibration of Fuzzy Partial Fractional Order Equation in Monitoring of Noise Big Data under Infinite Impulse Response Digital Filter Algorithm

Fractals ◽

10.1142/s0218348x22400783 ◽

2021 ◽

Author(s):

Liwei Zhang ◽

Zhanling Zhang ◽

Ali Saleh Alshomrani ◽

Yinghui Zhang ◽

Lisha Wang

Keyword(s):

Big Data ◽

Impulse Response ◽

Fractional Order ◽

Digital Filter ◽

Order Equation ◽

Infinite Impulse Response

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Generalized fractional order Chebyshev wavelets for solving nonlinear fractional delay-type equations

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691319500140 ◽

2019 ◽

Vol 17 (03) ◽

pp. 1950014 ◽

Cited By ~ 1

Author(s):

Amir Saeed ◽

Umer Saeed

Keyword(s):

Differential Equation ◽

Fractional Order ◽

Operational Matrices ◽

Chebyshev Wavelets ◽

Method Of Steps ◽

Type Differential Equation ◽

Fractional Delay ◽

Delay Differential ◽

Nonlinear Delay ◽

Quasilinearization Technique

In this paper, we develop the generalized fractional order Chebyshev wavelets (GFCWs) from generalized fractional order of Chebyshev polynomials. The operational matrices for the presented wavelets are constructed and derived. We also proposed a technique by utilizing the GFCWs, the method of steps and quasilinearization technique for solving nonlinear fractional delay-type differential equations. According to the development, the method of step is used to transform the fractional nonlinear delay-type differential equation to a fractional nonlinear non-delay differential equation, and then apply the quasilinearization technique to discretize the obtained nonlinear equation. The GFCW method is utilized in each iteration of quasilinearization method for the improvement of solution. We perform the error analysis for the proposed technique. Procedure of implementation for the present method is also provided. Numerical simulation of some examples will be presented to demonstrate the benefits of computing with the present technique over existing methods in literature.

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