A Review on Harmonic Wavelets and Their Fractional Extension

Abstract

<jats:p>In this paper a review on harmonic wavelets and their fractional generalization, within the local fractional calculus, will be discussed. The main properties of harmonic wavelets and fractional harmonic wavelets will be given, by taking into account of their characteristic features in the Fourier domain. It will be shown that the local fractional derivatives of fractional wavelets have a very simple expression thus opening new frontiers in the solution of fractional differential problems.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited.</jats:p>

Least Squares Differential Quadrature Method for the Generalized Bagley–Torvik Fractional Differential Equation

“…We define the error function in the following way: e(x) � y(x) − y(x), where y(x) is the exact solution for problems (1) and (2) and we obtain (using (18) and (19)) the differential equation for the error function:…”

“…Bagley-Torvik-type fractional differential equations have been studied both numerically and analytically in numerous articles. Among the methods used to solve this equation, we mention the following: numerical methods [5], Adomian decomposition method [6], discrete spline methods [7], Haar wavelet method [8,9], homotopy perturbation method [10], sinc collocation method [11,12], cubic spline method [13], quadratic spline solution [14], B-spline collocation method [15,16], Chebyshev collocation method [17], hybrid functions approximation [18], harmonic wavelets [19], predictor-corrector method of Adams type [20], spectral methods [21], fractional natural decomposition method [22], and finite element method [23]. e motivation of this paper is to introduce a new method for obtaining analytical approximate solutions for fractional differential equations.…”

New numerical simulation for fractional Benney–Lin equation arising in falling film problems using two novel techniques

“…Particularly, with the revolution in computer technology and symbolic programming, many new algorithms are proposed by many mathematicians, engineers and physicists. With the help of these techniques, many researchers analyze various classes of nonlinear systems and present some simulating results [12–56]. For instance, authors in [34, 35] studied logistic models within the frame of FC and illustrated some interesting consequences.…”

New Caputo-Fabrizio fractional order SEIASqEqHR model for COVID-19 epidemic transmission with genetic algorithm based control strategy

“…In [21] , Biao Tang et al constructed the paradigm in order to predict the dynamics of the novel coronavirus transmission via the ordinary differential equations. Several problems in the real world can be formulated utilizing fractional-order mathematical paradigms, for example see [22] , [23] , [24] , [25] , [26] , [27] , [28] , [29] , [30] , [31] , [32] , [33] , [34] , [35] , [36] .…”