An Improved Algorithm Based on Finite Difference Schemes for Fractional Boundary Value Problems with Nonsmooth Solution

In this paper, an efficient algorithm is presented by the extrapolation technique to improve the accuracy of finite difference schemes for solving the fractional boundary value problems with nonsmooth solution. Two popular finite difference schemes, the weighted shifted Grünwald difference (WSGD) scheme and the fractional centered difference (FCD) scheme, are revisited and stability of the schemes is shown in maximum norm. Based on the analysis of leading singularity of exact solution for the underlying problem, it is demonstrated that, with the use of the proposed algorithm, the improved WSGD and FCD schemes can achieve higher accuracy than the original ones for nonsmooth solution. To further improve the accuracy for solving problems with small fractional order, an extended algorithm dealing with two-term singularities correction is also developed. Several numerical examples are given to validate our theoretical prediction. It is shown that both accuracy and convergence rate of numerical solutions can be significantly improved by using the proposed algorithms.