Invariant analysis, exact solutions and conservation laws of (2+1)-dimensional time fractional Navier–Stokes equations

Abstract

In this paper, we investigate the exact solutions and conservation laws of (2 + 1)-dimensional time fractional Navier–Stokes equations (TFNSE). Specifically, Lie symmetries and corresponding one-dimensional optimal system for TFNSE in Riemann–Liouville sense are obtained. Then, based on the admitted symmetries and optimal system, we reduce these equations to one-dimensional equations or (1 + 1)-dimensional fractional partial differential equations (PDEs) with the help of Erdélyi–Kober fractional differential operator and compound variable transformation. In addition, we solve the reduced PDEs applying power series expansion method and invariant subspace method, respectively. Furthermore, the conservation laws of TFNSE are derived using new Noether theorem.

Numerical conservation laws of time fractional diffusion PDEs

“…Conservation laws for time fractional diffusion problems have been object of several papers, e.g. [13,28,[33][34][35]. However, only papers [33,35] treat equation of type (2.1) defined on a 3D and a 1D space, respectively.…”

“…In the last decade, conservation laws for fractional differential problems have been derived by suitably extending some of these known methods for PDEs. In particular, techniques that rely on generalizations of Noether's theorem and variational Lie point symmetries have been applied to find conservation laws of fractional differential equations (FDE) with a fractional Lagrangian [13,34]. For nonlinearly self-adjoint FDEs that do not have a Lagrangian in the classical sense, a formal Lagrangian can be introduced and conservation laws are obtained by using modern techniques based on Lie group analysis of FDEs.…”

“…For nonlinearly self-adjoint FDEs that do not have a Lagrangian in the classical sense, a formal Lagrangian can be introduced and conservation laws are obtained by using modern techniques based on Lie group analysis of FDEs. This approach, proposed for the first time by Lukashchuk in 2015 [35], has been applied to time fractional PDEs [1,13,28,[32][33][34][35] and more recently to time and space fractional PDEs (see [46] and references therein).…”

Numerical conservation laws of time fractional diffusion PDEs

“…Conservation laws for time fractional diffusion problems have been object of several papers, e.g. [12,29,38,39,41]. However, only papers [38,41] treat equation of type (2.1) defined on a 3D and a 1D space, respectively.…”

“…In the last decade, conservation laws for fractional differential problems have been derived by suitably extending some of these known methods for PDEs. In particular, techniques that rely on generalizations of Noether's theorem and variational Lie point symmetries have been applied to find conservation laws of fractional differential equations (FDE) with a fractional Lagrangian [12,39]. For nonlinearly self-adjoint FDEs that do not have a Lagrangian in the classical sense, a formal Lagrangian can be introduced and conservation laws are obtained by using modern techniques based on Lie group analysis of FDEs.…”