The New Semianalytical Technique for the Solution of Fractional-Order Navier-Stokes Equation

Abstract

In this paper, we introduce a modified method which is constructed by mixing the residual power series method and the Elzaki transformation. Precisely, we provide the details of implementing the suggested technique to investigate the fractional-order nonlinear models. Second, we test the efficiency and the validity of the technique on the fractional-order Navier-Stokes models. Then, we apply this new method to analyze the fractional-order nonlinear system of Navier-Stokes models. Finally, we provide 3-D graphical plots to support the impact of the fractional derivative acting on the behavior of the obtained profile solutions to the suggested models.

A Posteriori Error Estimates for a Nonconforming Finite Element Discretization of the Stokes–Biot System

“…ere exists a unique solution (U h , P h ) ∈ H h × M h to a discrete problem (24) and if the solution (U, P) ∈ H × M of the continuous problem (19) is smooth enough, then we have…”

“…e model features two different kinds of coupling across the interface: Stokes-Darcy coupling [2][3][4][5][6][7][8][9][10] and fluid-structure interaction (FSI) [11][12][13][14][15]. e well-posedness of the mathematical model based on the Stokes-Biot system for the coupling between a fluid and a poroelastic structure is studied in [16][17][18][19]. A numerical study of the problem, using Navier-Stokes equations for the fluid, is presented in [11,20], utilizing a variational multiscale approach to stabilize the finite element spaces.…”

“…e approach utilizes a Lagrange multiplier method to impose weakly the interface conditions. To our best knowledge, there is no a posteriori error estimation for the strongly coupled mixed formulation (19) ( [32], Section 3) of the coupled Stokes-Biot problem where a nonconforming finite element method is used. Here, we develop such a posteriori error analysis.…”

“…is is the natural discretization of the weak formulation (19) only with the penalizing term J(U h , V h ) added, where…”

“…(9) Normal Jump G E,n E in Ω p : for each edge/face E ∈ E h (Ω p ), we considerw E � K 1 ∪ K 2 . As G E,n E ∈ [P 0 (E)] d , we set w E ≔ − G E,n E b E ∈ H 1First the weak formulation(19) with V � (0, w E ) ∈ H × X p yields e previous bounds of r 4,K , R 3 , and the obvious estimateh E ≤ h K imply that + Ψ K 1 + Ψ K 2 .Interface elements on Γ fp (Υ 5,K and Υ 6,K ): to estimate Υ 5,K and Υ 6,K , we fix an edge E included in Γ fp and, for a constant r E fixed later on and a unit vector i, we considerw E ≔ r E b E i, (132)that clearly belongs to H. We take W � (w E , 0) and the weak formulation (19) yields where K f (resp. K p ) is the unique triangle/tetrahedron included in Ω f (resp.…”