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Contour-Based Fast Fractional Integration in a Jacobi Fractional Polynomial Basis

  • Xiaolin Liu
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Abstract

We develop a contour-based method for applying Riemann--Liouville fractional integrals in a Jacobi fractional polynomial basis with an endpoint weight. A Gamma-ratio representation reduces the action to contour quadrature and banded ultraspherical solves. For a fixed number of contour nodes, the cost and storage are linear in the polynomial dimension. For fractional equations, we construct selected diagonals of a short rational approximation directly and use their banded factorizations as reusable right preconditioners for Krylov iterations. Numerical experiments show accurate integral actions and substantially reduced solution costs across Abel, multi-order, BBO, deferred-correction, eigenvalue, and boundary-value problems.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026contourbased,
  title = {{Contour-Based Fast Fractional Integration in a Jacobi Fractional Polynomial Basis}},
  author = {Xiaolin Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.39388},
  eprint = {2610.39388},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.39388},
}
RIS
TY  - UNPB
TI  - Contour-Based Fast Fractional Integration in a Jacobi Fractional Polynomial Basis
AU  - Xiaolin Liu
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.39388
UR  - https://arxiv.org/abs/2610.39388
AB  - We develop a contour-based method for applying Riemann--Liouville fractional integrals in a Jacobi fractional polynomial basis with an endpoint weight. A Gamma-ratio representation reduces the action to contour quadrature and banded ultraspherical solves. For a fixed number of contour nodes, the cost and storage are linear in the polynomial dimension. For fractional equations, we construct selected diagonals of a short rational approximation directly and use their banded factorizations as reusable right preconditioners for Krylov iterations. Numerical experiments show accurate integral actions and substantially reduced solution costs across Abel, multi-order, BBO, deferred-correction, eigenvalue, and boundary-value problems.
ER  -