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

Shervan Erfani, Esmail Babolian, and Shahnam Javadi

### Abstract

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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