On exact traveling-wave solutions for local fractional Korteweg-de Vries equation

Yang, Xiao‐Jun; Machado, J. A. Tenreiro; Băleanu, Dumitru; Cattani, Carlo

doi:10.1063/1.4960543

Cited by 211 publications

(117 citation statements)

References 18 publications

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“…[34][35][36][37] There is an alternative operator (called local FC) to model the local FODEs in fractal electric circuits, 38 free damped vibrations, 39 shallow water surfaces 40 and populations. [41][42][43] The fractal partial differential equations (FPDEs) in mathematical physics were also discussed in Refs.…”

Section: Introductionmentioning

confidence: 99%

Non-Differentiable Exact Solutions for the Nonlinear Odes Defined on Fractal Sets

2017

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In the present paper, a family of the special functions via the celebrated Mittag-Leffler function defined on the Cantor sets is investigated. The nonlinear local fractional ODEs (NLFODEs) are presented by following the rules of local fractional derivative (LFD). The exact solutions for these problems are also discussed with the aid of the non-differentiable charts on Cantor sets. The obtained results are important for describing the characteristics of the fractal special functions.

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Section: Introductionmentioning

confidence: 99%

Non-Differentiable Exact Solutions for the Nonlinear Odes Defined on Fractal Sets

2017

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“…Therefore, many researchers have tried their best to use different techniques to find the analytical and numerical solutions of these problems, for example, Adomian decomposition method (ADM), 5 spline collocation method (SCM), 6 fractional transform method (FTM), 7 homotopy perturbation method (HPM), 8 operational tau method (OTM), 9 shifted Chebyshev polynomial method (SCPM), 10 rationalized Haar function method (RHFM), 11 exp-function method, 12 traveling wave transformation method, 13 and Cole-Hopf transformation method, 14 and also see the work in Yang et al, 15 Sayevand and Pichaghchi, 16 and Wang and Liu. 17 Recently, Yang et al 18 did a comprehensive study of the methods which have been used for the solutions of the problems containing fractional derivatives and integral operators.…”

Section: Introductionmentioning

confidence: 99%

Chebyshev wavelet method to nonlinear fractional Volterra–Fredholm integro-differential equations with mixed boundary conditions

Mohyud‐Din

Khan

Arif

et al. 2017

Advances in Mechanical Engineering

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This research work addresses the numerical solutions of nonlinear fractional integro-differential equations with mixed boundary conditions, using Chebyshev wavelet method. The basic idea of this work started from the Caputo definition of fractional differential operator. The fractional derivatives are replaced by Caputo operator, and the solution is approximated by wavelet family of functions. The numerical scheme by Chebyshev wavelet method is constructed through a very simple and straightforward way. The numerical results of the current method are compared with the exact solutions of the problems, which show that the proposed method has a strong agreement with the exact solutions of the problems. The numerical solutions of the present method are also compared with steepest decent method and Adomian decomposition method. The comparison with other methods reveals that this method has the highest degree of accuracy than those methods.

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“…Therefore, for the study of numerical solutions of FDEs, there are variety of analytical methods, which were found in literature. Among them, most useful and common methods are presented in [1,4,5,8,9,11,14,15,[17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Residual power series method for time-fractional Schrödinger equations

Yu¹,

Kumar²,

Kumar³

et al. 2016

J. Nonlinear Sci. Appl.

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In this paper, the residual power series method (RPSM) is effectively applied to find the exact solutions of fractional-order time dependent Schrödinger equations. The competency of the method is examined by applying it to the several numerical examples. Mainly, we find that our solutions obtained by the proposed method are completely compatible with the solutions available in the literature. The obtained results interpret that the proposed method is very effective and simple for handling different types of fractional differential equations (FDEs).

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On exact traveling-wave solutions for local fractional Korteweg-de Vries equation

Cited by 211 publications

References 18 publications

Non-Differentiable Exact Solutions for the Nonlinear Odes Defined on Fractal Sets

Non-Differentiable Exact Solutions for the Nonlinear Odes Defined on Fractal Sets

Chebyshev wavelet method to nonlinear fractional Volterra–Fredholm integro-differential equations with mixed boundary conditions

Residual power series method for time-fractional Schrödinger equations

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