Efficient Algorithm For Some Types Of Space Fractional Partial Differential Equations

Due to the nonlocal property of fractional derivative,fractional equation can well describes a lot of phenomena including fractal dispersion,genetic effect and long time memory process in science and engineering.In recent years,fractional equations are studied widely.However,it is difficult to obtain the analytical solutions of fractional partial differential models involving the special function,which is difficult to express.In this paper,we consider the efficient method for several types of space fractional partial equation,i.e.,nonlinear fractional Ginzburg-Landau equation,the conservative space Riemann-Liouville partial equation and conservative space Caputo equation with inhomogeneous boundary condition.The main contribution of this work is as follows:First of all,we proposed a split-step quasi-compact difference method to nonlinear fractional Ginzburg-Landau equation in one and two dimensions.In particular,an alternating direction implicit scheme is constructed for the two dimensional linear subproblem.The unconditional stability and convergence of the schemes are proved rigorously in the linear case.Numerical experiments are performed to confirm the high order and efficiency of the proposed method.Secondly,we consider two-sided Riemann-Liouville and Caputo fractional partial differential equation with the inhomogeneous boundary conditions.We establish a spectral penalty method(SPM)and the well-posedness of the corresponding weak problems and analyze sufficient conditions for the coercivity of the SPM for different types of fractional boundary value problems.This analysis allows us to estimate the proper values of the penalty parameters at boundary points.We present several numerical examples to verify the theory and demonstrate the high accuracy of SPM.Next,we mainly consider two-sided Riemann-Liouville fractional partial differential equation with inhomogeneous boundary condition.Due to the spectral relationship of two-sided fractional operator and general Jacobi basis(Jacobi polyfractonomials),we proposed a new fractional collocation method.The contribution of the paper is the collocation method is easy to express and we can get the explicit expression for the matrices of fractional derivatives.Furthermore,we construct the spectral collocation scheme for the fractional diffusion equation.Numerical tests confirm the efficiency of the scheme.Finally,we study two-sided Caputo fractional partial differential equation with inhomogeneous boundary condition.We construct the spectral element method with graded mesh,and we proposed the Hierarchical approximation for the spectral element method.We give the construction of Hierarchical matrix and precondition system,and obtain the fast solver which not only reduce the computational cost but also make a good approximation.Numerical tests also verify the efficiency of our method.In summary,we consider the efficient numerical schemes for nonlinear fractional Ginzburg-Landau equation and two-sided fractional with general boundary condition.These algorithms can be developed for the fractional diffusion equation,even or nonlinear fractional partial differential equations.