Numerical Approaches of the Generalized Time-Fractional Burgers’ Equation with Time-Variable Coefficients

Nehad Ali Shah,Jae Dong Chung,Constantin Fetecau,Calogero Vetro,Dumitru Vieru

doi:10.1155/2021/8803182

Nehad Ali Shah, Jae Dong Chung + Show 3 more

Open Access

https://doi.org/10.1155/2021/8803182

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Abstract

The generalized time-fractional, one-dimensional, nonlinear Burgers equation with time-variable coefficients is numerically investigated. The classical Burgers equation is generalized by considering the generalized Atangana-Baleanu time-fractional derivative. The studied model contains as particular cases the Burgers equation with Atangana-Baleanu, Caputo-Fabrizio, and Caputo time-fractional derivatives. A numerical scheme, based on the finite-difference approximations and some integral representations of the two-parameter Mittag-Leffler functions, has been developed. Numerical solutions of a particular problem with initial and boundary values are determined by employing the proposed method. The numerical results are plotted to compare solutions corresponding to the problems with time-fractional derivatives with different kernels.

Highlights

The nonlinear convective–diffusive partial differential equations describe various mathematical models in important fields such as heat and mass transfer, fluid mechanics, and engineering
Saad et al [15] have extended the model of the Burgers equation to generalized models based on Liouville-Caputo, Caputo-Fabrizio, and Mittag-Leffler time-fractional derivatives
Baleanu and Shiri [16] numerically solved a system of fractional differential equations involving nonsingular MittagLeffler kernel using the collocation methods on discontinuous piecewise polynomial space

Summary

Introduction

The nonlinear convective–diffusive partial differential equations describe various mathematical models in important fields such as heat and mass transfer, fluid mechanics, and engineering. Saad et al [15] have extended the model of the Burgers equation to generalized models based on Liouville-Caputo, Caputo-Fabrizio, and Mittag-Leffler time-fractional derivatives. Baleanu and Shiri [16] numerically solved a system of fractional differential equations involving nonsingular MittagLeffler kernel using the collocation methods on discontinuous piecewise polynomial space. The generalization consists of considering the fractional differential Burgers’ equation with the generalized time-fractional Atangana-Baleanu fractional derivative with Mittag-Leffler kernel. A numerical scheme, based on the finite-difference approximations and some integral representations of the two-parameter Mittag-Leffler functions, has been developed along with the consistency, stability, and convergence of the proposed method. It is important to point out that the studied generalized model can be customized to generate solutions to the problems described by the time-fractional Atangana-Baleanu, Caputo-Fabrizio, and Caputo fractional derivatives. The numerical results are plotted to compare solutions corresponding to the problems with timefractional derivatives with different kernels

Preliminary Mathematics

Problem Formulation

ΔtβΓð2

Example

Conclusion