Spectral Collocation Methods for Fractional Integro-Differential Equations with Weakly Singular Kernels

In this paper, we propose and analyze a spectral approximation for the numerical solutions of fractional integro-differential equations with weakly kernels. First, the original equations are transformed into an equivalent weakly singular Volterra integral equation, which possesses nonsmooth solutions. To eliminate the singularity of the solution, we introduce some suitable smoothing transformations, and then use Jacobi spectral collocation method to approximate the resulting equation. Later, the spectral accuracy of the proposed method is investigated in the infinity norm and weighted L2 norm. Finally, some numerical examples are considered to verify the obtained theoretical results.

Automated summary

In this paper, a spectral approximation method is proposed and analyzed for the numerical solutions of fractional integro-differential equations with weakly kernels. The method transforms the original equations into an equivalent weakly singular Volterra integral equation and introduces smoothing transformations to eliminate singularity. The proposed method is investigated for spectral accuracy and validated with numerical examples.