Numerical study of a class of variable order nonlinear fractional differential equation in terms of Bernstein polynomials

Numerical study of a class of variable order nonlinear fractional differential equation in terms of Bernstein polynomials

In this paper, we use Bernstein polynomials to seek the numerical solution of a class of nonlinear variable order fractional differential equation. The fractional derivative is described in the Caputo sense. Three different kinds of operational matrixes with Bernstein polynomials are derived and are...

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Main Authors: Yi-ming Chen, Li-qing Liu, Dayan Liu, Driss Boutat
Format: Article
Language:English
Published: Elsevier 2018-12-01
Series:Ain Shams Engineering Journal
Online Access:http://www.sciencedirect.com/science/article/pii/S2090447916301046
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spelling doaj-98fca5c41d334c87bb2a653b614ef2562021-06-02T10:33:21ZengElsevierAin Shams Engineering Journal2090-44792018-12-019412351241Numerical study of a class of variable order nonlinear fractional differential equation in terms of Bernstein polynomialsYi-ming Chen0Li-qing Liu1Dayan Liu2Driss Boutat3College of Sciences, Yanshan University, Qinhuangdao 066004, Hebei, China; Loire Valley Institute for Advanced Studies, PRISME (INSA-Institut National des sciences appliquees/University of Oreleans)-88, Boulevard, Lahitolle, 18000 Bourges, France; Corresponding author at: College of Sciences, Yanshan University, Qinhuangdao 066004, Hebei, China, Loire Valley Institute for Advanced Studies, PRISME (INSA-Institut National des sciences appliquees/University of Oreleans)-88, Boulevard, Lahitolle, 18000 Bourges, France.College of Sciences, Yanshan University, Qinhuangdao 066004, Hebei, ChinaINSA Centre Val de Loire, Université d’Orléans, PRISME EA 4229, 18022 Bourges, FranceINSA Centre Val de Loire, Université d’Orléans, PRISME EA 4229, 18022 Bourges, FranceIn this paper, we use Bernstein polynomials to seek the numerical solution of a class of nonlinear variable order fractional differential equation. The fractional derivative is described in the Caputo sense. Three different kinds of operational matrixes with Bernstein polynomials are derived and are utilized to transform the initial equation into the products of several dependent matrixes which can also be regarded as the system of nonlinear equations after dispersing the variable. By solving the system of equations, the numerical solutions are acquired. Numerical examples are provided to show that the method is computationally efficient and accurate. Keywords: Bernstein polynomials, Variable order fractional nonlinear differential equation, Operational matrix, Numerical solution, Convergence analysis, The absolute errorhttp://www.sciencedirect.com/science/article/pii/S2090447916301046
collection DOAJ
language English
format Article
sources DOAJ
author Yi-ming Chen
Li-qing Liu
Dayan Liu
Driss Boutat
spellingShingle Yi-ming Chen
Li-qing Liu
Dayan Liu
Driss Boutat
Numerical study of a class of variable order nonlinear fractional differential equation in terms of Bernstein polynomials
Ain Shams Engineering Journal
author_facet Yi-ming Chen
Li-qing Liu
Dayan Liu
Driss Boutat
author_sort Yi-ming Chen
title Numerical study of a class of variable order nonlinear fractional differential equation in terms of Bernstein polynomials
title_short Numerical study of a class of variable order nonlinear fractional differential equation in terms of Bernstein polynomials
title_full Numerical study of a class of variable order nonlinear fractional differential equation in terms of Bernstein polynomials
title_fullStr Numerical study of a class of variable order nonlinear fractional differential equation in terms of Bernstein polynomials
title_full_unstemmed Numerical study of a class of variable order nonlinear fractional differential equation in terms of Bernstein polynomials
title_sort numerical study of a class of variable order nonlinear fractional differential equation in terms of bernstein polynomials
publisher Elsevier
series Ain Shams Engineering Journal
issn 2090-4479
publishDate 2018-12-01
description In this paper, we use Bernstein polynomials to seek the numerical solution of a class of nonlinear variable order fractional differential equation. The fractional derivative is described in the Caputo sense. Three different kinds of operational matrixes with Bernstein polynomials are derived and are utilized to transform the initial equation into the products of several dependent matrixes which can also be regarded as the system of nonlinear equations after dispersing the variable. By solving the system of equations, the numerical solutions are acquired. Numerical examples are provided to show that the method is computationally efficient and accurate. Keywords: Bernstein polynomials, Variable order fractional nonlinear differential equation, Operational matrix, Numerical solution, Convergence analysis, The absolute error
url http://www.sciencedirect.com/science/article/pii/S2090447916301046
work_keys_str_mv AT yimingchen numericalstudyofaclassofvariableordernonlinearfractionaldifferentialequationintermsofbernsteinpolynomials
AT liqingliu numericalstudyofaclassofvariableordernonlinearfractionaldifferentialequationintermsofbernsteinpolynomials
AT dayanliu numericalstudyofaclassofvariableordernonlinearfractionaldifferentialequationintermsofbernsteinpolynomials
AT drissboutat numericalstudyofaclassofvariableordernonlinearfractionaldifferentialequationintermsofbernsteinpolynomials
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