Finite Element Numerical Methods Of The Coupled Burgers Equations With Space And Time Fractional Derivative

In this article,we study two numerical methods of the coupled Burgers equations with space fractional derivative and time fractional derivative respectively.Firstly,in order to seek the numerical solutions of the coupled Burgers equations with space fractional derivative,we use second-order backward difference scheme to ap-proximate in time direction,Galerkin finite element method approximation for spatial direction.Therefore,we establish finite element scheme of the coupled Burgers equations with space fractional derivative.We deal with the nonlinear term with space fractional derivative by calculating a series of expressions of fractional base functions with a nonlin-ear term,which base on citing the basic definition of fractional base function.In addition to this,we show the specific numerical calculation process.In numerical example,we can find the optimal error convergence order of the coupled Burgers equations'numerical solutions can reach(~2+?~2).Secondly,for the coupled Burgers equations with time fractional derivative,1-Galerkin finite element numerical calculation method is formulated by applying1 ap-proximation for the Caputo fractional-order derivative and Galerkin finite element method approximation for spatial direction.After choosing proper base functions,fully discrete equations transform into linear algebraic equations by calculating.From numerical re-sults,time convergence accuracy of the coupled Burgers equations with time fractional derivative can reach min(2-,2-),whereas spatial convergence accuracy can reach 2.