A High Order Approximation And Estimation Of Derivative Order For Time Fractional Diffusion Equation

An efficient,stable and high order numerical algorithm based on a finite difference scheme in time and Legendre spectral collocation method in space is proposed for time fractional diffusion equation,and the derivative order of the time fractional diffusion equation is estimated by an iterative method in this paper.For the time fractional diffusion equation.Firstly,we discrete equation in time by using the four degree Lagrange interpolation,and obtain the finite difference scheme and the weak form.Secondly,we rigorously prove the stability and make error estimation of the weak form.Then,we use Legendre spectral collocation method to approximate the semi discrete solution in space,estimate the error and prove that the convergence order of the fully discrete solution is O(?t5-?+N-m),where ?t,N and m are the time step,polynomial degree and regularity of the exact solution respectively.Finally,we give some numerical examples,the numerical results indicate that algorithm is very efficient.For the derivative order estimation problem of the time fractional diffusion equation.Firstly,we derive the equation satisfied by the derivative du/d? of the solution u with respect to ? of the time fractional diffusion equation,and construct an iterative algorithm for estimating the fractional derivative ? when the source term f is known.Secondly,we use linear Lagrange interpolating polynomial to discrete the equation in time and construct the finite difference scheme and the weak form.Then the semi discrete solution in the weak form is approximated by Legendre polynomial,the fully discrete scheme is obtained,and the corresponding stability analysis and error estimation are given.Finally,some numerical examples are given the numerical results indicate that the derived equation is correct,the numerical algorithm is effective,and the iterative algorithm converges with good accuracy.