Abstract
In this paper, we present a numerical scheme used to solve the nonlinear time fractional Navier–Stokes equations in two dimensions. We first employ the meshless local Petrov–Galerkin (MLPG) method based on a local weak formulation to form the system of discretized equations and then we will approximate the time fractional derivative interpreted in the sense of Caputo by a simple quadrature formula. The moving Kriging interpolation which possesses the Kronecker delta property is applied to construct shape functions. This research aims to extend and develop further the applicability of the truly MLPG method to the generalized incompressible Navier–Stokes equations. Two numerical examples are provided to illustrate the accuracy and efficiency of the proposed algorithm. Very good agreement between the numerically and analytically computed solutions can be observed in the verification. The present MLPG method has proved its efficiency and reliability for solving the two-dimensional time fractional Navier–Stokes equations arising in fluid dynamics as well as several other problems in science and engineering.
Highlights
In recent years, substantially contributed works associated with the fractional differential equations (FDE), sometimes called as extraordinary differential equations, have been published both theoretical and numerical aspects on account of its applications in almost all branches of sciences and engineering
Research methodology we introduce the governing time fractional Navier–Stokes equations (NSE) in Cartesian coordinate system and the meshless local Petrov–Galerkin (MLPG) formulation and numerical implementation including local weak form and discretization techniques are described in detail
In this paper, we have presented a numerical scheme used to obtain the approximate solution of the time fractional Navier–Stokes equations
Summary
Substantially contributed works associated with the fractional differential equations (FDE), sometimes called as extraordinary differential equations, have been published both theoretical and numerical aspects on account of its applications in almost all branches of sciences and engineering. Various physical phenomena in fluid mechanics, viscoelasticity, control theory of dynamical systems, chemical physics, biology, stochastic processes, finance, and other sciences can be successfully described by fractional models. Even though there have been a lot of achievements on the theoretical analysis, exact solutions of most FDE cannot be derived explicitly. Approximate analytical and numerical solutions can be obtained using procedures such as linearization, perturbation, or discretization. As a result, proposing a new algorithm to find the numerical solution of FDE is of practical significance.
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