Exact Solutions And Conservation Laws Of Two Kinds Of Time Fractional Navier-Stokes Equations

In this paper,two kinds of time fractional Navier-Stokes equations are studied.Exact solutions and conservation laws of the(2+1)-dimensional NavierStokes equations containing Riemann-Liouville time derivative and the time fractional Navier-Stokes equation in sense of conformable derivative in cylindrical coordinates are constructed by means of Lie groups method,invariant subspace method and -homotopy analysis method and so on.The thesis is divided into five chapters as follows:In Chapter 1,we introduce the research background and significance of used methods and fractional calculus and Navier-Stokes equations.In Chapter 2,the definitions of fractional calculus,Bessel function,fractional Laplace transform and finite Hankel transform,and some lemmas,properties and two main methods are given.In Chapter 3,Lie symmetries and corresponding one-dimensional optimal system admitted by(2+1)-dimensional Navier-Stokes equations in sense of Riemann-Liouville are obtained.Then,based on the above results,we reduce this equations to lower dimensional equations with the help of Erd′elyi-Kober fractional differential operator and spatial compound transformation.Finally,we solve the reduced equations applying power series expansion method and invariant subspace method and obtain the group invariant solutions of(2+1)-dimensional Navier-Stokes equations and the conservation laws of this equations are derived using new Noether theorem.In Chapter 4,we study the analytical solutions of conformable time fractional Navier-Stokes equation in cylindrical coordinates applying Lie groups method,separation of variables method,-homotopy analysis method and the fractional Laplace and finite Hankel transforms,respectively.Furthermore,based on the obtained Lie algebra,the conservation laws of the equation are obtained using the new Noether theorem.In Chapter 5,we summarize the work of this paper and some topics to be researched in the future.