Volume 54, pp. 150-175, 2021.

New fractional pseudospectral methods with accurate convergence rates for fractional differential equations

Shervan Erfani, Esmail Babolian, and Shahnam Javadi


The main purpose of this paper is to introduce generalized fractional pseudospectral integration and differentiation matrices using a family of fractional interpolants, called fractional Lagrange interpolants. We develop novel approaches to the numerical solution of fractional differential equations with a singular behavior at an end-point. To achieve this goal, we present efficient and stable methods based on three-term recurrence relations, generalized barycentric representations, and Jacobi-Gauss quadrature rules to evaluate the corresponding matrices. In a special case, we prove the equivalence of the proposed fractional pseudospectral methods using a suitable fractional Birkhoff interpolation problem. In fact, the fractional integration matrix yields the stable inverse of the fractional differentiation matrix, and the resulting system is well-conditioned. We develop efficient implementation procedures for providing optimal error estimates with accurate convergence rates for the interpolation operators and the proposed schemes in the $ L^{2}$-norm. Some numerical results are given to illustrate the accuracy and performance of the algorithms and the convergence rates.

Full Text (PDF) [1.3 MB], BibTeX

Key words

convergence rate, fractional differential equations, fractional Birkhoff interpolation, fractional pseudospectral methods, fractional Lagrange interpolants, singularity

AMS subject classifications

26A33, 41A05, 65M06, 65M12, 65L60

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