Efficient algorithms for solving the fractional ordinary differential equations

Deng, Jingwei; Zhao, Lijing; Wu, Yue

doi:10.1016/j.amc.2015.07.048

Cited by 8 publications

(3 citation statements)

References 20 publications

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“…is a global operator (not local as in classical calculus) that presents "memory", with t > a and n−1 < α < n. With numerical methods emerged to simulate fractional system, who usually use the fractional derivative of Grünwald-Letnikov [11] [17] [31], was made it possible to implement fractional operators in controllers [3] [6] [23] [30] [32].…”

Section: Desired Positionmentioning

confidence: 99%

Evaluation of Fractional-Order Sliding Mode Control Applied to an Energy Harvesting System

Basilio

Oliveira

Ribeiro

et al. 2022

2022 16th International Workshop on Variable Structure Systems (VSS)

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HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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Section: Desired Positionmentioning

confidence: 99%

Evaluation of Fractional-Order Sliding Mode Control Applied to an Energy Harvesting System

Basilio

Oliveira

Ribeiro

et al. 2022

2022 16th International Workshop on Variable Structure Systems (VSS)

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“…Other numerical methods were suggested in [11,[26][27][28][29][30][31][32][33][34]. This paper focuses on the following three topics:…”

Section: Introductionmentioning

confidence: 99%

A Novel Numerical Method for Solving Nonlinear Fractional-Order Differential Equations and Its Applications

Lee,

Kim,

Jang

2024

Fractal Fract

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In this article, a considerably efficient predictor-corrector method (PCM) for solving Atangana–Baleanu Caputo (ABC) fractional differential equations (FDEs) is introduced. First, we propose a conventional PCM whose computational speed scales with quadratic time complexity O(N2) as the number of time steps N grows. A fast algorithm to reduce the computational complexity of the memory term is investigated utilizing a sum-of-exponentials (SOEs) approximation. The conventional PCM is equipped with a fast algorithm, and it only requires linear time complexity O(N). Truncation and global error analyses are provided, achieving a uniform accuracy order O(h2) regardless of the fractional order for both the conventional and fast PCMs. We demonstrate numerical examples for nonlinear initial value problems and linear and nonlinear reaction-diffusion fractional-order partial differential equations (FPDEs) to numerically verify the efficiency and error estimates. Finally, the fast PCM is applied to the fractional-order Rössler dynamical system, and the numerical results prove that the computational cost consumed to obtain the bifurcation diagram is significantly reduced using the proposed fast algorithm.

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“…Fractional order differential equations have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. We can find some applications in control, porous media, electromagnetic, etc., in reference to the monographs of Oldham and Spanier [17], Miller and Ross [16], Podlubny [18], Hilfer [9] and the papers of Stynes et al [19], El-Sayed [7], He [8], Lyu and Vong et al [15], Li and Yi et al [13], Benchohra et al [3,4], Agarwal et al [1] and the others [22,23,20,2,6,5,14] of the references therein.…”

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confidence: 99%

Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation

Sun¹,

Zeng²,

Song³

2020

Numerical Algebra, Control &Amp; Optimization

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In this paper, the existence and uniqueness of solution for a class of boundary value problems of nonlinear fractional order differential equations involving the Caputo fractional derivative are studied. The estimation of error between the approximate solution and the solution for such equation is presented by employing the quasilinear iterative method, and an example is given to demonstrate the application of our main result.

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Efficient algorithms for solving the fractional ordinary differential equations

Cited by 8 publications

References 20 publications

Evaluation of Fractional-Order Sliding Mode Control Applied to an Energy Harvesting System

Evaluation of Fractional-Order Sliding Mode Control Applied to an Energy Harvesting System

A Novel Numerical Method for Solving Nonlinear Fractional-Order Differential Equations and Its Applications

Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation

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