New fractional pseudospectral methods with accurate convergence rates for fractional differential equations
| Autor: | Esmail Babolian, Shahnam Javadi, Shervan Erfani |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Matrix (mathematics)
Recurrence relation ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION Convergence (routing) MathematicsofComputing_NUMERICALANALYSIS Applied mathematics Inverse Barycentric coordinate system Birkhoff interpolation Analysis Mathematics Quadrature (mathematics) Interpolation |
| Zdroj: | ETNA - Electronic Transactions on Numerical Analysis. 54:150-175 |
| ISSN: | 1068-9613 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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