{"paper":{"title":"Fractional calculus via variable-transform-based spectral approximations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Variable transforms applied to Chebyshev polynomials yield stable spectral approximations for fractional integral operators.","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Kuan Xu, Xiaolin Liu","submitted_at":"2026-04-28T09:27:11Z","abstract_excerpt":"We present a novel and unifying framework for constructing spectral approximations to fractional integral operators. These spectral approximations are based on transplanted Chebyshev polynomials, which are obtained by composing Chebyshev polynomials with a variable transform. When an algebraic transform is used, the framework produces spectral approximations based on Jacobi fractional polynomials. When an exponential transform is used, it yields a versatile spectral approximation that is applicable to a much broader class of fractional calculus problems. The construction of such spectral appro"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"These spectral approximations lead to stable and fast spectral methods for fractional calculus. The spectral approximation based on the double-exponential transform is demonstrated through extensive numerical examples that are intractable for existing spectral methods.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The variable transforms are assumed to preserve spectral accuracy and numerical stability for the fractional integral operator without introducing hidden instabilities or requiring problem-specific tuning of the transform parameters.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Variable-transform-based transplanted Chebyshev polynomials provide stable, optimal-complexity spectral approximations to fractional integrals, including Jacobi fractional polynomials and double-exponential variants applicable to broader problems.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Variable transforms applied to Chebyshev polynomials yield stable spectral approximations for fractional integral operators.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"aa22cfa0ef2a50bae51ebe07600d4b7307369b1602a0d2fc1c3b0c6e26ba0f7a"},"source":{"id":"2604.25417","kind":"arxiv","version":3},"verdict":{"id":"352f0da8-ddf2-4465-be06-6d444dea04ee","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T15:45:34.807645Z","strongest_claim":"These spectral approximations lead to stable and fast spectral methods for fractional calculus. The spectral approximation based on the double-exponential transform is demonstrated through extensive numerical examples that are intractable for existing spectral methods.","one_line_summary":"Variable-transform-based transplanted Chebyshev polynomials provide stable, optimal-complexity spectral approximations to fractional integrals, including Jacobi fractional polynomials and double-exponential variants applicable to broader problems.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The variable transforms are assumed to preserve spectral accuracy and numerical stability for the fractional integral operator without introducing hidden instabilities or requiring problem-specific tuning of the transform parameters.","pith_extraction_headline":"Variable transforms applied to Chebyshev polynomials yield stable spectral approximations for fractional integral operators."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.25417/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T04:40:23.149418Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T21:09:31.154413Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"757fcdc6206790dd91777aa63149d25302eb55a49f8124e51241fe70e2a0cd87"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}