Numerical Approaches to Fractional Integrals and Derivatives: A Review

Cai, Min; Li, Changpin

doi:10.3390/math8010043

Cited by 41 publications

(25 citation statements)

References 99 publications

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“…In Reference [25] the authors solved linear and nonlinear FBVPs by a wavelet method. For an overview on numerical methods to solve fractional differential problems see, for instance, References [18,20,[26][27][28][29][30] and references therein.…”

Section: Introductionmentioning

confidence: 99%

On the Numerical Solution of Fractional Boundary Value Problems by a Spline Quasi-Interpolant Operator

Pitolli

2020

Axioms

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Boundary value problems having fractional derivative in space are used in several fields, like biology, mechanical engineering, control theory, just to cite a few. In this paper we present a new numerical method for the solution of boundary value problems having Caputo derivative in space. We approximate the solution by the Schoenberg-Bernstein operator, which is a spline positive operator having shape-preserving properties. The unknown coefficients of the approximating operator are determined by a collocation method whose collocation matrices can be constructed efficiently by explicit formulas. The numerical experiments we conducted show that the proposed method is efficient and accurate.

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Section: Introductionmentioning

confidence: 99%

On the Numerical Solution of Fractional Boundary Value Problems by a Spline Quasi-Interpolant Operator

Pitolli

2020

Axioms

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“…The solutions of equations (3.22) and (3.23) will be obtained numerically using approximations of the finite differences type [6,23]. For this, we consider a uniform discretization of the interval [0, t 0 ] with the points…”

Section: Numerical Solutions Of the Fractional Model Formentioning

confidence: 99%

Memory effects on the proliferative function in the cycle-specific of chemotherapy

Ahmed

Vieru

Zaman

2021

Math. Model. Nat. Phenom.

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A generalized mathematical model of the breast and ovarian cancer is developed by considering the fractional differential equations with Caputo time-fractional derivatives. The use of the fractional model shows that the time-evolution of the proliferating cell mass, the quiescent cell mass, and the proliferative function are significantly influenced by their history. Even if the classical model, based on the derivative of integer order has been studied in many papers, its analytical solutions are presented in order to make the comparison between the classical model and the fractional model. Using the finite difference method, numerical schemes to the Caputo derivative operator and Riemann-Liouville fractional integral operator are obtained. Numerical solutions to the fractional differential equations of the generalized mathematical model are determined for the chemotherapy scheme based on the function of "on-off" type. Numerical results, obtained with the Mathcad software, are discussed and presented in graphical illustrations. The presence of the fractional order of the time-derivative as a parameter of solutions gives important information regarding the proliferative function, therefore, could give the possible rules for more efficient chemotherapy.

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“…In this paper, we shall use the Riemann Liouville derivatives and integrals. The left hand sided Riemann Liouville derivative of a function u(t) at time t of any order α > 0 is defined as [21].…”

Section: Fractional Order Derivativesmentioning

confidence: 99%

“…The approximate numerical solutions can be obtained by reformulating the fractional derivatives and integrals as infinite integral solutions to integer-order ordinary differential equations [21]. For a function u(t) defined in some time interval [0,T], if we divide the time domain in N parts with uniform step length then the step size will be Δt = T N , and the ith grid point or instant is obtained by t i = iΔT for i = 0, 1, 2, … , N .…”

Section: Fractional Order Derivativesmentioning

confidence: 99%

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Variation of fraction in FOPID controller for vibration control of Euler–Bernoulli beam

et al. 2020

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Fractional-order derivatives and integrals have gained serious attention by the researchers in past few decades. It has been observed that fractional calculus is more suitable in modeling real life systems. In this paper, Fractional-Order PID (FOPID) controller is studied to suppress the vibration of cantilever beam modeled by Euler-Bernoulli model. Finite difference method is applied to analyze the behavior of the system at different fractional orders of PID controller, and the governing equations are simulated using MATLAB. The response of the system is evaluated at different fractionalorder control points, under constant PID parameters (K p , K i , and K d). Furthermore, the variation of frequency in transient response and 5% settling time is observed. Simulation results reveal that, though system with fractional order controller is more flexible now, the classical model of PID controller can also be properly tuned through PID constants to achieve the resultant response of the system. So, due to more simplified modelling, the classical model has an edge over the fractional order in the vibration control of cantilever beam.

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Numerical Approaches to Fractional Integrals and Derivatives: A Review

Cited by 41 publications

References 99 publications

On the Numerical Solution of Fractional Boundary Value Problems by a Spline Quasi-Interpolant Operator

On the Numerical Solution of Fractional Boundary Value Problems by a Spline Quasi-Interpolant Operator

Memory effects on the proliferative function in the cycle-specific of chemotherapy

Variation of fraction in FOPID controller for vibration control of Euler–Bernoulli beam

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