Multiparameter Fractional Differentiation with non singular kernel

AbstractIn this work, a nonlinear singular variable-order fractional Emden–Fowler equation involved with derivative with non-singular kernel (in the Atangana–Baleanu–Caputo type) is introduced and a computational method is proposed for its numerical solution. The desired method is established upon the shifted Jacobi polynomials and their operational matrix of variable-order fractional differentiation (which is extracted in the present study) together with the spectral collocation method. The presented method transforms obtaining the solution of the main problem into obtaining the solution of an algebraic system of equations. Several numerical examples are examined to show the validity and the high accuracy of the established method.

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APPLICATION OF THE CAPUTO-FABRIZIO FRACTIONAL DERIVATIVE WITHOUT SINGULAR KERNEL TO KORTEWEG-DE VRIES-BURGERS EQUATION∗

Mathematical Modelling and Analysis ◽

10.3846/13926292.2016.1145607 ◽

2016 ◽

Vol 21 (2) ◽

pp. 188-198 ◽

Cited By ~ 82

Author(s):

Emile Franc Doungmo Goufo

Keyword(s):

Fractional Derivative ◽

Fractional Order ◽

Existence And Uniqueness ◽

Burgers Equation ◽

Continuous Solution ◽

Singular Kernel ◽

Fractional Differentiation ◽

De Vries ◽

Perturbation Parameters ◽

Derivative Order

In order to bring a broader outlook on some unusual irregularities observed in wave motions and liquids’ movements, we explore the possibility of extending the analysis of Korteweg–de Vries–Burgers equation with two perturbation’s levels to the concepts of fractional differentiation with no singularity. We make use of the newly developed Caputo-Fabrizio fractional derivative with no singular kernel to establish the model. For existence and uniqueness of the continuous solution to the model, conditions on the perturbation parameters ν, µ and the derivative order α are provided. Numerical approximations are performed for some values of the perturbation parameters. This shows similar behaviors of the solution for close values of the fractional order α.

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Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

Nonlinear Analysis Modelling and Control ◽

10.15388/na.2003.8.1.15178 ◽

2003 ◽

Vol 8 (1) ◽

pp. 61-75

Author(s):

V. Litovchenko

Keyword(s):

Functional Analysis ◽

Cauchy Problem ◽

Fundamental Solution ◽

Initial Function ◽

Well Posedness ◽

Fractional Differentiation ◽

Basic Tool ◽

Conjugate Space ◽

Analysis Technique ◽

The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

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MIXED FRACTIONAL DIFFERENTIATION OPERATORS IN HÖLDER SPACES DEFINED BY USUAL HÖLDER CONDITION

Chronos Journal ◽

10.31618/2658-7556-2019-38-11-1 ◽

2019 ◽

Author(s):

T. Mamatov ◽

R. Sabirova ◽

D. Barakaev

Keyword(s):

Fractional Derivative ◽

Fractional Differentiation ◽

Hölder Condition ◽

Hölder Spaces ◽

Main Interest ◽

Hölder Class ◽

Holder Class ◽

Function Of Two Variables ◽

Holder Spaces

We study mixed fractional derivative in Marchaud form of function of two variables in Hölder spaces of different orders in each variables. The main interest being in the evaluation of the latter for the mixed fractional derivative in the cases Hölder class defined by usual Hölder condition

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