The Fractional Orthogonal Difference with Applications

Diekema, Enno

doi:10.3390/math3020487

Cited by 4 publications

(7 citation statements)

References 13 publications

(23 reference statements)

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“…Thus N = 2 − 14 3 + 126 25 − 66 35 −1 = 525 256 , so recovering the formula for ω [4] 1 (x). The kernel function k(x) is obtained from ω(x) as its dth derivative (Equation (1.4)).…”

Section: Appendix B: Example Solutionsupporting

confidence: 54%

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Higher Accuracy Order in Differentiation-by-Integration

Liptaj

2021

Mathematical Modelling and Analysis

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In this text explicit forms of several higher precision order kernel functions (to be used in the differentiation-by-integration procedure) are given for several derivative orders. Also, a system of linear equations is formulated which allows to construct kernels with an arbitrary precision for an arbitrary derivative order. A computer study is realized and it is shown that numerical differentiation based on higher precision order kernels performs much better (w.r.t. errors) than the same procedure based on the usual Legendre-polynomial kernels. Presented results may have implications for numerical implementations of the differentiation-by-integration method.

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“…Thus N = 2 − 14 3 + 126 25 − 66 35 −1 = 525 256 , so recovering the formula for ω [4] 1 (x). The kernel function k(x) is obtained from ω(x) as its dth derivative (Equation (1.4)).…”

Section: Appendix B: Example Solutionsupporting

confidence: 54%

“…The appropriate tools and methods for the discretization of the DbI procedure have already been presented in several texts (see e.g. [4]).…”

Section: Summary Conclusion and Outlookmentioning

confidence: 99%

Higher Accuracy Order in Differentiation-by-Integration

Liptaj

2021

Mathematical Modelling and Analysis

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“…Since the trajectories are linear and the width of the beam is not affected during propagation, the propagation of the two Gaussian rays is uniform. The beam always acquires a symmetric linear phase during its propagation in the inverse space, which is demonstrated in equation (3). We can infer that this phase will affect the beam profile particularly for a transverse displacement in space.…”

Section: Numerical Results For a Two-wave Mixing Configurationmentioning

confidence: 89%

“…One of the advantages of fractional calculation is its application to different scalar magnitudes ranging from the nano-metric scale [1] to the macro-metric scale [2]. It should be noted that the use of fractional calculus has been employed from practical applications of filters obtained from orthogonal polynomials and the Butterworth filter [3], through studies of viscoelastic behavior with experimental adjustments [4], numerical studies on transport affected by hydraulic conditions at a distance in natural geological deposits [5], up to the modeling of electrical properties of biological systems [6].…”

Section: Introductionmentioning

confidence: 99%

Optical phase-change in plasmonic nanoparticles by a two-wave mixing

et al. 2019

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Described herein is the optical phase-change and photothermal phenomena induced by a two-wave mixing configuration in a nonlinear optical material. For experimental analysis of these findings, the third-order nonlinear optical response exhibited by bimetallic Au–Pt nanoparticles embedded in a TiO2 matrix was studied. A vectorial two-wave mixing technique was conducted to explore the automatic generation of polarized self-diffraction by the participation of the optical Kerr effect in the samples. Fractional derivative calculations allowed us to identify the physical mechanism responsible for photothermal influence on the third-order nonlinear optical effects in the nanosystems. A 532 nm wavelength emitted by the second harmonic of a nanosecond Nd:YAG laser system was employed as an optical source in our measurements. Based on the importance of the vectorial nature of light for dictating the amplitude of the beams and splitting properties, within this work is highlighted the attractive energy transfer functions and phase modulation that can be envisioned by photothermally-controlled interactions.

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“…In our opinion, this demonstration is less convincing, because by absence of an integrating factor, the high frequencies are not suppressed and because the plots are not given as log-log plots. One should compare with our approach in [11] (Section 6) where we give a log-log plot of the modulus of the frequency response.…”

Section: The Frequency Response For the Hahn Polynomialsmentioning

confidence: 99%

The Fractional Orthogonal Derivative

Diekema¹

2015

Mathematics

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This paper builds on the notion of the so-called orthogonal derivative, where an n-th order derivative is approximated by an integral involving an orthogonal polynomial of degree n. This notion was reviewed in great detail in a paper by the author and Koornwinder in 2012. Here, an approximation of the Weyl or Riemann-Liouville fractional derivative is considered by replacing the n-th derivative by its approximation in the formula for the fractional derivative. In the case of, for instance, Jacobi polynomials, an explicit formula for the kernel of this approximate fractional derivative can be given. Next, we consider the fractional derivative as a filter and compute the frequency response in the continuous case for the Jacobi polynomials and in the discrete case for the Hahn polynomials. The frequency response in this case is a confluent hypergeometric function. A different approach is discussed, which starts with this explicit frequency response and then obtains the approximate fractional derivative by taking the inverse Fourier transform.

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The Fractional Orthogonal Difference with Applications

Cited by 4 publications

References 13 publications

Higher Accuracy Order in Differentiation-by-Integration

Higher Accuracy Order in Differentiation-by-Integration

Optical phase-change in plasmonic nanoparticles by a two-wave mixing

The Fractional Orthogonal Derivative

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