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

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.

Evaluating the Stability of Numerical Schemes for Fluid Solvers in Game Technology

“…Recent investigations have analysed the timefractional Navier-Stokes equation [39] and the SWEs [40] for bespoke systems. Many numerical schemes have been proposed for solving FDEs [41,42] as well as for those applicable to fluid simulation, e.g., the fractional Burgers equation [43], the fractional diffusion equation [44] (relevant to the incompressible Navier-Stokes equation), and the fractional parabolic differential equations [45] (relevant to vorticitystream function formulation of fluids).…”