SM-Algorithms for Approximating the Variable-Order Fractional Derivative of High Order

2017 ◽  
Vol 151 (1-4) ◽  
pp. 293-311 ◽  
Author(s):  
B. P. Moghaddam ◽  
J. A. T. Machado
Keyword(s):  
Fractional Derivative ◽  
High Order ◽  
Variable Order ◽  
Variable Order Fractional Derivative

scholarly journals On the sub–diffusion fractional initial value problem with time variable order

2021 ◽  
Vol 10 (1) ◽  
pp. 1301-1315
Author(s):  
Eduardo Cuesta ◽  
Mokhtar Kirane ◽  
Ahmed Alsaedi ◽  
Bashir Ahmad
Keyword(s):  
Asymptotic Behavior ◽  
Fractional Derivative ◽  
Initial Value Problem ◽  
Regularity Result ◽  
Time Variable ◽  
Variable Order ◽  
Initial Value ◽  
Integration By Parts ◽  
Integration By Parts Formula ◽  
Variable Order Fractional Derivative

Abstract We consider a fractional derivative with order varying in time. Then, we derive for it a Leibniz' inequality and an integration by parts formula. We also study an initial value problem with our time variable order fractional derivative and present a regularity result for it, and a study on the asymptotic behavior.

scholarly journals A Numerical Scheme Based on the Chebyshev Functions to Find Approximate Solutions of the Coupled Nonlinear Sine-Gordon Equations with Fractional Variable Orders

2021 ◽  
Vol 2021 ◽  
pp. 1-20
Author(s):  
MohammadHossein Derakhshan
Keyword(s):  
Fractional Derivative ◽  
Numerical Method ◽  
Numerical Scheme ◽  
Numerical Approximation ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Variable Order ◽  
Algebraic Equations ◽  
Variable Order Fractional Derivative ◽  
Chebyshev Functions

In this article, a numerical method based on the shifted Chebyshev functions for the numerical approximation of the coupled nonlinear variable-order fractional sine-Gordon equations is shown. The variable-order fractional derivative is considered in the sense of Caputo-Prabhakar. To solve the problem, first, we obtain the operational matrix of the Caputo-Prabhakar fractional derivative of shifted Chebyshev polynomials. Then, this matrix and collocation method are used to reduce the solution of the nonlinear coupled variable-order fractional sine-Gordon equations to a system of algebraic equations which is technically simpler for handling. Convergence and error analysis are examined. Finally, some examples are given to test the proposed numerical method to illustrate the accuracy and efficiency of the proposed method.

Jacobi Spectral Collocation Method for the Time Variable-Order Fractional Mobile-Immobile Advection-Dispersion Solute Transport Model

2016 ◽  
Vol 6 (3) ◽  
pp. 337-352 ◽  
Author(s):  
Heping Ma ◽  
Yubo Yang
Keyword(s):  
Fractional Derivative ◽  
Solute Transport ◽  
Collocation Method ◽  
Dispersion Model ◽  
Time Variable ◽  
Spectral Collocation Method ◽  
Variable Order ◽  
Spectral Collocation ◽  
Advection Dispersion ◽  
Variable Order Fractional Derivative

AbstractAn efficient high order numerical method is presented to solve the mobile-immobile advection-dispersion model with the Coimbra time variable-order fractional derivative, which is used to simulate solute transport in watershed catchments and rivers. On establishing an efficient recursive algorithm based on the properties of Jacobi polynomials to approximate the Coimbra variable-order fractional derivative operator, we use spectral collocation method with both temporal and spatial discretisation to solve the time variable-order fractional mobile-immobile advection-dispersion model. Numerical examples then illustrate the effectiveness and high order convergence of our approach.

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