Numerical research of nonlinear system of fractional Volterra–Fredholm integral–differential equations via Block-Pulse functions and error analysis

Xie, Jiaquan; Yi, Mingxu

doi:10.1016/j.cam.2018.06.008

Cited by 24 publications

(13 citation statements)

References 14 publications

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“…where D α t denotes the fractional differential operator defined in the Caputo sense [26,27], m is the quality of the slider, c is the fractional damping coefficient, k is the spring stiffness coefficient, u 0 is the displacement at drive end, u is the slider displacement, and φ( _ u) is the frictional force. If the slider is driven at constant speed, setting the speed of the driving end as v, then the displacement u 0 of the driving end T can be expressed as…”

Section: Fractional Dynamic Model Of Mechanical System Hunting Phenommentioning

confidence: 99%

Numerical Vibration Displacement Solutions of Fractional Drawing Self‐Excited Vibration Model Based on Fractional Legendre Functions

et al. 2019

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In practice, due to the fact that the phenomenon of drawing self-excited vibration can be deemed as one of the hunting phenomena of the mechanical system, this study focuses on investigating the drawing self-excited vibration process through proposing the fractional differential equation model of hunting phenomenon of the mechanical system. The fractional Legendre functions together with their fractional differential operational matrices are used to numerically solve the model. In this way, the numerical solutions of vibration displacement of the model are obtained. At the end, the proposed model and algorithm are proved to be effective via analyzing the numerical results and phase position.

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Section: Fractional Dynamic Model Of Mechanical System Hunting Phenommentioning

confidence: 99%

Numerical Vibration Displacement Solutions of Fractional Drawing Self‐Excited Vibration Model Based on Fractional Legendre Functions

et al. 2019

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“…The nonlinearities or uncertainties will arise because of the time‐delays and external disturbances 6 . Recently, some numerical methods have been proposed to deal with the nonlinearities or uncertainties, such as Block‐Pulse function method, 7,8 integral‐differential method, 9 two‐dimensional analysis method, 10 and so on. Generally, the delay‐independent stability conditions are conservative because the time‐delays information is not considered in stability analysis of the system 11 .…”

Section: Introductionmentioning

confidence: 99%

RobustH_∞output feedback control for type‐2Takagi‐Sugenofuzzy systems with multiple time‐delays and disturbances: A descriptor redundancy approach

Zheng

Zhang

Sun

et al. 2021

Intl J Robust & Nonlinear

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The H∞ stability analysis and interval type‐2 Takagi‐Sugeno (T‐S) fuzzy control is studied for a class of interval type‐2 T‐S fuzzy systems. Firstly, the interval type‐2 T‐S fuzzy model is employed to approximate the system plant. Secondly, the delay‐dependent interval type‐2 T‐S fuzzy dynamic output feedback controller is designed. Thirdly, two classes of stability conditions in terms of linear matrix inequalities (LMIs) are derived, and the closed‐loop system is asymptotically stable and prescribed H∞ performance is achieved. The delay‐dependent interval type‐2 T‐S fuzzy dynamic output feedback controller is designed to relax the control design conditions. The descriptor redundancy method is employed for avoiding the crossing vectors between the system gain matrices and controller gain matrices. The delay‐dependent Lyapunov functional with descriptor redundancy matrix is introduced, and the upper and lower membership functions information is considered. Some slack matrices are introduced via the upper and lower membership functions information to solve the obtained LMIs. Finally, simulation examples are performed to show the effectiveness of the proposed methods.

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“…In 2016, Deif and Grace [13] developed a new iterative method to approximate the solution of a system of linear fractional differential integral equations. In 2019, Xie and Yi [14] developed a numerical method for solving a nonlinear system of fractional-order Volterra-Fredholm integro-differential equations based on block-pulse functions. In 2020, Saemi et al [15] developed a solution for the system of fractional-order Volterra-Fredholm integro-differential equations based on Müntz-Legendre wavelets.…”

Section: Introductionmentioning

confidence: 99%

Solving a System of Fractional-Order Volterra-Fredholm Integro-Differential Equations with Weakly Singular Kernels via the Second Chebyshev Wavelets Method

et al. 2021

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In this paper, by means of the second Chebyshev wavelet and its operational matrix, we solve a system of fractional-order Volterra–Fredholm integro-differential equations with weakly singular kernels. We estimate the functions by using the wavelet basis and then obtain the approximate solutions from the algebraic system corresponding to the main system. Moreover, the implementation of our scheme is presented, and the error bounds of approximations are analyzed. Finally, we evaluate the efficiency of the method through a numerical example.

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Numerical research of nonlinear system of fractional Volterra–Fredholm integral–differential equations via Block-Pulse functions and error analysis

Cited by 24 publications

References 14 publications

Numerical Vibration Displacement Solutions of Fractional Drawing Self‐Excited Vibration Model Based on Fractional Legendre Functions

Numerical Vibration Displacement Solutions of Fractional Drawing Self‐Excited Vibration Model Based on Fractional Legendre Functions

RobustH_∞output feedback control for type‐2Takagi‐Sugenofuzzy systems with multiple time‐delays and disturbances: A descriptor redundancy approach

Solving a System of Fractional-Order Volterra-Fredholm Integro-Differential Equations with Weakly Singular Kernels via the Second Chebyshev Wavelets Method

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Numerical research of nonlinear system of fractional Volterra–Fredholm integral–differential equations via Block-Pulse functions and error analysis

Cited by 24 publications

References 14 publications

Numerical Vibration Displacement Solutions of Fractional Drawing Self‐Excited Vibration Model Based on Fractional Legendre Functions

Numerical Vibration Displacement Solutions of Fractional Drawing Self‐Excited Vibration Model Based on Fractional Legendre Functions

RobustH∞output feedback control for type‐2Takagi‐Sugenofuzzy systems with multiple time‐delays and disturbances: A descriptor redundancy approach

Solving a System of Fractional-Order Volterra-Fredholm Integro-Differential Equations with Weakly Singular Kernels via the Second Chebyshev Wavelets Method

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RobustH_∞output feedback control for type‐2Takagi‐Sugenofuzzy systems with multiple time‐delays and disturbances: A descriptor redundancy approach