A two-dimensional Chebyshev wavelets approach for solving the Fokker-Planck equations of time and space fractional derivatives type with variable coefficients

Xie, Jiaquan; Yao, Zhibin; Gui, Hu; Zhao, Fuqiang; Li, Dongyang

doi:10.1016/j.amc.2018.03.040

Cited by 21 publications

(17 citation statements)

References 26 publications

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“…Definition The fractional Caputo derivative of a function f ( t ) is defined as .

(_{c} D_{t}^{α} f) (t) = \frac{1}{Γ (n - α)} \int_{0}^{t} {((), t - τ)}^{n - α - 1} f^{((), n)} (τ) italicd τ, n - 1 < α \leq n,

…”

Section: Fractional Calculusmentioning

confidence: 99%

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Numerical scheme for solving system of fractional partial differential equations with Volterra‐type integral term through two‐dimensional block‐pulse functions

Xie

Ren

et al. 2019

Numerical Methods Partial

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In the current study, an approximate scheme is established for solving the fractional partial differential equations (FPDEs) with Volterra integral terms via two‐dimensional block‐pulse functions (2D‐BPFs). According to the definitions and properties of 2D‐BPFs, the original problem is transformed into a system of linear algebra equations. By dispersing the unknown variables for these algebraic equations, the numerical solutions can be obtained. Besides, the proof of the convergence of this system is given. Finally, several numerical experiments are presented to test the feasibility and effectiveness of the proposed method.

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“…Definition The fractional Caputo derivative of a function f ( t ) is defined as .

(_{c} D_{t}^{α} f) (t) = \frac{1}{Γ (n - α)} \int_{0}^{t} {((), t - τ)}^{n - α - 1} f^{((), n)} (τ) italicd τ, n - 1 < α \leq n,

…”

Section: Fractional Calculusmentioning

confidence: 99%

“…The most common definition of fractional order is the Riemann-Liouville integral, where the fractional operator I ( > 0) of a function, is defined as [18,19]…”

Section: Fractional Calculus Definition 21mentioning

confidence: 99%

Numerical scheme for solving system of fractional partial differential equations with Volterra‐type integral term through two‐dimensional block‐pulse functions

Xie

Ren

et al. 2019

Numerical Methods Partial

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“…So far, various numerical methods are presented to solve fractional differential equations. These methods include wavelets method [11,12], Chebyshev and Legendre polynomials [13,14], and collocation method [15][16][17][18][19]. In [20], N. I. Mahmudov utilized an approximate method to study partial-approximate controllability of semilinear nonlocal fractional evolution equations.…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulation of Fractional Control System Using Chebyshev Polynomials

Jun

Xie

2018

Mathematical Problems in Engineering

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“…Podlubny [17] first proposed the model of fractional order proportional-integral-differential controller. The basic knowledge of the definition, properties, Laplace transformation, and application of fractional order calculus is introduced in [18][19][20][21][22]. Compared with the integer , the expression of fractional controller has two parameters , , which increases the adjustment range of parameters.…”

Section: Introductionmentioning

confidence: 99%

Study on the Control Method of Mine-Used Bolter Manipulator Based on Fractional Order Algorithm and Input Shaping Technology

Jun

Huang

2018

Mathematical Problems in Engineering

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Mine-used bolter is the main equipment to solve the imbalance of excavation and anchor in well mining, and the manipulator is the main working mechanism of mine bolt drilling rig. The manipulator positioning requires high rapidity and stability. For this reason, this paper proposes a composite control method of “input shaping + fractional order PDμ control”. According to the mathematical model of the valve-controlled cylinder, the fractional-order controller PDμ is developed. At the same time, the input shaping is used to feed forward the accurate positioning and path planning of the manipulator, which not only improves the robustness of the system, but also shortens the stability time of the system and restrains the maximum amplitude of the system vibration. In this paper, the control effects of fractional order PDμ controller and integer order PD controller are compared. The results show that the maximum amplitude of the control system is reduced by 75% and the stabilization time is reduced by 60% after using the fractional order PDμ controller, which fully reflects the superiority of the fractional order controller in response speed, adjusting time, and steady-state accuracy. Finally, the control effects of “input shaping + fractional order PDμ control” and fractional order PDμ controller on the stability of the system are compared. The maximum amplitude of the system was reduced by 50% by using “input shaping + fractional order PDμ control”. Numerical simulation confirms the feasibility and effectiveness of the composite control method. This composite control method provides theoretical support for the precise positioning of the manipulator, and the high stability and high safety of the manipulator also expand the application scope and depth of the composite control method.

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A two-dimensional Chebyshev wavelets approach for solving the Fokker-Planck equations of time and space fractional derivatives type with variable coefficients

Cited by 21 publications

References 26 publications

Numerical scheme for solving system of fractional partial differential equations with Volterra‐type integral term through two‐dimensional block‐pulse functions

Numerical scheme for solving system of fractional partial differential equations with Volterra‐type integral term through two‐dimensional block‐pulse functions

Numerical Simulation of Fractional Control System Using Chebyshev Polynomials

Study on the Control Method of Mine-Used Bolter Manipulator Based on Fractional Order Algorithm and Input Shaping Technology

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