Efficient Numerical Solutions With Weighted Polynomials For Space-Fractional Partial Differential Equations

In this paper,three classical difference schemes and spectral collocation meth-ods are used to solve time-dependent space-fractional partial differential equations with non-smooth solutions.In the process of solving the fractional partial differential terms in space,the three-term recurrence relations of the space fractional differen-tial terms are obtained mainly by using weighted Jacobi polynomials(?1+x?^?1?1-x?^?2P_j^a,b?x?,a,b,?₁,?₂>-1).A weighted Jacobi polynomial spectral collocation method based on Jacobi-Gauss-Lobatto?JGL?points is proposed.For the time dimen-sion,the difference schemes are constructed by using backward Euler method,second-order backward difference method?BD2?and fourth-order Runge-Kutta method?RK4?to solve singular time-dependent space-fractional partial differential equations.Using the software of Matlab,the numerical examples are calculated and summarized.On the one hand,we can know how to select the number of spatial nodes and the value of?₁,?₂to minimize the numerical error about the spectral collocation method proposed in this paper.Moreover,the numerical examples show that the spectral collocation method proposed in this paper is better than the standard spectral collocation method??₁=?₂=0?in solving the time-dependent space-fractional partial differential equa-tions with non-smooth solutions.On the other hand,the linear and non-linear equations with non-smooth exact solution are solved.The methods and techniques of how to deal with the linear and non-linear terms in the difference schemes are presented,which can make the errors smaller.Numerical experiments on the time convergence rates show that the backward Euler difference scheme,the second-order backward differ-ence scheme and the fourth-order Runge-Kutta difference scheme used in this paper have approximate first-order,second-order and fourth-order time convergence rates,which are consistent with the theoretical results.We can know the correctness and effectiveness of difference schemes which used in this paper in solving time-dependent space-fractional partial differential equations with non-smooth solutions.