On the analytical development of incomplete Riemann–Liouville fractional calculus

Fernandez, Arran; Ustaoğlu, Ceren; Özarslan, Mehmet Ali

doi:10.3906/mat-2101-64

Cited by 4 publications

(3 citation statements)

References 1 publication

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“…where u(x) is a continuous function in the domain x ä [0, 1] and the fractional derivative is considered as Riemann Liouvelli fractional derivative [38,39]. The boundary conditions are u(0) = 1, and u(1) = 0.…”

Section: Formalismmentioning

confidence: 99%

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Application of Haar wavelet to shear-wave equation and corresponding fractional differential equation

Zephania

Harisankar²,

Sil³

2023

Phys. Scr.

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Shear waves find applications in several branches of science, such as geophysics, earth science, medical science etc. The Haar wavelet (HW) scheme is employed to solve the governing equation of the horizontal component of the shear wave (SH). The solutions of SH waves obtained from HW are compared with the exact solutions and some of the available results from approximation methods, such as the homotopy perturbation method (HPM) and wavelet Galerkin method with Daubechies wavelet (WG). HW solutions are found to be more accurate than WG at points away from the resonance and at the proximity of the resonance. HW yields solutions with higher accuracy than HPM solutions. The SH wave equation is also studied using the concept of fractional calculus by introducing arbitrary parameter $\alpha$, especially in the vicinity of the resonance with the values of $\alpha$ around one. The solutions are found to be damped oscillatory for $ \alpha <1 $, and diverging oscillatory for $\alpha > 1 $, respectively. The solutions are insensitive to small variations $ \alpha $ at and around the resonance point corresponding to the ODE. At a point far from the resonance, the solution with $\alpha \approx 1 $ matches nicely with those for $ \alpha \ne 1 $. The amplitude of the solution for $ \alpha= 1 $ becomes very large at a point very close to the resonance. In contrast, amplitudes of the solutions for $\alpha \ne 1$ remain the same in the vicinity of the resonance, including it. Therefore, if necessary, the parameter $ \alpha $ may be the control to avoid resonance.

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Section: Formalismmentioning

confidence: 99%

“…In order to solve the fractional differential equations like equation (5), we use the Riemann-Liouville fractional integral operator of order γ > 0 of function f (x) is defined as [38,39,41]:…”

Section: Haar Waveletmentioning

confidence: 99%

Application of Haar wavelet to shear-wave equation and corresponding fractional differential equation

Zephania

Harisankar²,

Sil³

2023

Phys. Scr.

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“…Proceeding from the generalisations of the beta function expressed above, various generalisations of Equation (1.1) have been introduced and investigated by many authors (see [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]).…”

Section: Introductionmentioning

confidence: 99%

Some generalised extended incomplete beta functions and applications

Yağcı

Şahіn

Kıymaz

et al. 2022

Journal of New Results in Science

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This paper introduces generalised incomplete beta functions defined by the generalised beta function. Firstly, we provide some of the generalised beta function's basic properties, such as integral representations, summation formulas, Mellin transform, and beta distribution. We then present several fundamental properties, such as integral representations, summation formulas, and recurrence relations with the help of the generalised incomplete beta functions.

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Application of fractional-order nonlinear equations in coordinated control of multi-agent systems

Jia,

Cui,

Patro

et al. 2023

Nonlinear Engineering

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In order to solve the coordinated operation of voltage and frequency of microgrids and achieve effective distribution of output power, we propose the application of fractional-order nonlinear equations in coordination. This method designs a distributed impulse coordinated control strategy, to achieve the coordinated operation of the system. The distributed coordinated control structure and mathematical model are established, and the distributed two-level coordinated control strategy of the microgrid system is adopted. Aiming at the secondary control problem of microgrids, a distributed coordinated control protocol is designed. The results showed that after adding the distributed second-level coordination control of frequency at 3 s, the output active power of the four distributed power sources in the microgrid model is maintained to an evenly divided state after about 1 s. The output voltage and frequency utilizing the microgrid’s decentralized power supply quickly reach the ideal reference value, within the allowable error range, the output of the system can achieve coordinated control, and the active power can be distributed proportionally.

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On the analytical development of incomplete Riemann–Liouville fractional calculus

Cited by 4 publications

References 1 publication

Application of Haar wavelet to shear-wave equation and corresponding fractional differential equation

Application of Haar wavelet to shear-wave equation and corresponding fractional differential equation

Some generalised extended incomplete beta functions and applications

Application of fractional-order nonlinear equations in coordinated control of multi-agent systems

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