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Huygens Principle of Secondary Wavelets

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In the next section we will see that this computation can be avoided by using the connection coefficients. Theorem 9. Let us notice that the value of the scalar product and then of the integral depends on the non-vanishing values of the characteristic function. In fact the non-vanishing values of the characteristic functions are.

These derivatives are given by the following theorems. Theorem The sinc-fractional derivative 3. So that the computation of the sinc-fractional derivative of a function, that can be expressed as wavelet series, is reduced to the computation of the scalar product:. From these inequalities, by taking into account 6. Equations 6. Its sinc-fractional derivative can be computed according to. The sinc-fractional derivative of the wavelet representation 4. Proof : Let us start from Equation 3.

From here, by using 6. A good approximation of this function, in terms of Shannon wavelet expansion 4. Sinc function is playing a fundamental role in mathematics and physics. Due to the many properties of this function it deserves a special role in applications. In recent years some Authors have proposed [ 20 ] a fractional derivative based on this function. Moreover a wavelet theory based on the sinc function has been settled thus extending the many features of the Sinc.

In this paper the sinc-fractional derivative has been extended to the Shannon wavelet space, in order to give the explicit analytical form of the fractional derivatives of functions belonging to the wavelet space. It has been shown that the sinc-fractional derivative is the most natural and suitable choice of fractional operator when dealing with functions that can be represented as Shannon wavelet series. The author confirms being the sole contributor of this work and has approved it for publication.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Dalir M, Bashour M. Applications of fractional calculus. Appl Math Sci. Google Scholar. Tarasov VE. Fractional vector calculus and fractional Maxwells equations. Ann Phys.

Electromagnetic fields on fractals. Mod Phys Lett A — Local fractional Z transforms with applications to signals on Cantor sets. Abstr Appl Anal. Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives. Phys Lett A — Local fractional series expansion method for solving wave and diffusion equations on Cantor sets. Maxwells equations on Cantor sets: a local fractional approach. Adv High Energy Phys.

Space-time fractional Rosenou-Haynam equation: lie symmetry analysis, explicit solutions and conservation laws. Adv Diff Equat. Lie symmetry analysis, exact solutions and conservation laws for the time fractional Caudrey Dodd Gibbon Sawada Kotera equation. Commun Nonlinear Sci Numer Simul.

Lie symmetry analysis and conservation laws for the time fractional simplified modified Kawahara equation. Open Phys. Optimal system, nonlinear self-adjointness and conservation laws for generalized shallow water wave equation. Lie symmetry analysis and explicit solutions for the time fractional generalized Burgers-Huxley equation. Opt Quant Electron.

Wavelets in Physics. Access. Edited by J. C. van den Berg, Agricultural University , Wageningen, The Netherlands. Publisher: Cambridge University Press. A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases, and then Many areas of physics have seen this paradigm shift, including molecular dynamics, chaos theory, ab initio calculations, astrophysics, gravitational.

Lie symmetry analysis, exact solutions and conservation laws for the time fractional modified Zakharov Kuznetsov equation. Nonlinear Anal Model Control — Physica A — Time fractional third-order variant Boussinesq system: symmetry analysis, explicit solutions, conservation laws and numerical approximations. Eur Phys J Plus Wang X. Fractional geometric calculus: toward a unified mathematical language for physics and engineering. Fractional derivatives in complex planes. Nonlinear Anal Theor Methods Appl. Ortigueira MD. A coherent approach to non-integer order derivatives.

Signal Process. CrossRef Full Text. Cattani C, Guariglia E. J King Saud Univ.

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