The Crank‐Nicolson/interpolating stabilized element‐free Galerkin method to investigate the fractional Galilei invariant advection‐diffusion equation

Abstract

Recently, finding a stable and convergent numerical procedure to simulate the fractional partial differential equations (PDEs) is one of the interesting topics. Meanwhile, the fractional advection‐diffusion equation is a challenge model numerically and analytically. This paper develops a new meshless numerical procedure to simulate the fractional Galilei invariant advection‐diffusion equation. The fractional derivative is the Riemann‐Liouville fractional derivative sense. At the first stage, a difference scheme with the second‐order accuracy has been employed to get a semi‐discrete plan. After this procedure, the unconditional stability has been investigated, analytically. At the second stage, a meshless weak form based upon the interpolating stabilized element‐free Galerkin (ISEFG) method has been used to achieve a full‐discrete scheme. As for the full‐discrete scheme, the order of convergence is scriptOfalse(τ2+rm+1false). Two examples are studied, and simulation results are reported to verify the theoretical results.

Approximation of Caputo‐Prabhakar derivative with application in solving time fractional advection‐diffusion equation

“…44 It also describes transport of pollutants in the atmosphere in long-range, 45 forced cooling in turbo generators, 46 thermal pollution in river systems, 47 and flow in porous media. 48 There are various numerical approaches available in the literature for solving time fractional advection-diffusion equation numerically, [49][50][51][52][53][54][55][56][57][58] and so forth however, papers available in the literature are quite limited which motivates us to study the effective numerical method for the solution of time-fractional advection-diffusion equation where the fractional derivative is taken in the Caputo-Prabhakar sense.…”

Robust ∞ model reduction for the continuous fractional‐order two‐dimensional Roesser system: The 0 < ε ≤ 1 case

“…This manuscript focuses on the $$$ {H}_{\infty } $$$ model reduction problem for continuous FO 2D Roesser system with the FO $$$ 0&lt;\varepsilon \le 1 $$$ and the robust $$$ {H}_{\infty } $$$ model reduction problem for continuous FO 2D Roesser system with the FO $$$ 0&lt;\varepsilon \le 1 $$$ with polytopic uncertainties. Fractional calculus extends integer calculus, one of the most influential mathematical tools worldwide [18–21]. Due to their memory properties, FO systems can better explain specific actual system models [22–25], such as control processing [26], circuit systems [27], electrical noises [28], and semi‐crystalline polymers [29], than integer‐order systems.…”

Integrated radial basis functions to simulate modified anomalous sub‐diffusion equation

“…A transport equation for confined structures has been used to calculate the ionic currents through various transmembrane proteins in Khodadadian and Heitzinger [24]. Also for more details see [25][26][27][37][38][39][40].…”