Study On The Finite Element Method For Fractional Differential Equations With Non-homogeneous Boundary Conditions

Abstract:First, the development of fractional differential equations is narrated in this paper, the basic concept and properties of the fractional calculus and fractional Sobolev spaces are reviewed. After elaborating the finite element method based on variational principles, general steps for the finite element approximation to the fractional boundary value problems are introduced. Bilinear basis functions are substituted into the variational equation and numerical results are given by using MATLAB software. Second, the steady state fractional advection dispersion equation with non-homogeneous boundary condition is solved by using finite element method. The main idea is as follows:The homogeneous transformation for the non-homogeneous boundary condition is derived, then variational formulation of the equation could be got. The existence and uniqueness results are proved in the case of0<β<1/2and1/2<β<1where β is the range of the fractional integral term. Experimental results based on piecewise linear basis functions are presented to confirm the conclusions. Last, after analyzing the element method for the fractional differential equation with Riemann-Liouville derivative where the left and right boundary value conditions both are non-homogeneous, the difficulties of the general homogeneous transformation are pointed out. That is, the integral of new source term on the domain isn’t convergent, which cause the solution of the finite element equation can not be got. Under this circumstance, a new variational formulation is given, numerical results show that the corresponding finite element equation has solution. The well-posedness of variational formulation for this equation is proved where the left boundary value is zero and the right is not.5Pictures,6Tables,63References.