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Moreover, for Fredholm weakly singular integral equations of the second kind, i.e., $\\lambda u(x)- I(a,b,x) =f(x)$ with $ \\lambda \\neq 0$, also have global convergence rate $\\mathcal{O}\\left(h^2\\right)$ and $\\mathcal{O}\\l"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1909.10756","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-09-24T08:17:43Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"2a06ea565755db3286f0262dc6400a527ad048480354ccc2538317b42a868afe","abstract_canon_sha256":"4945537430dd80b8388259b8ae29664fbd9a16387eaab8cb0e81015324183308"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:28:48.742388Z","signature_b64":"CDLy0xO2q4z8HjkWkIOtyKNHbBEm7P/wtOkoF8Jjxns+UguP9dZKF0AvcBA5YIrIi4zfg2G+MbRDRTy/wNQZCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"05af7f3830d82c1d5f37e54e246cd8ba36e0b3ad8bdde65eb9b62a3f404d39b3","last_reissued_at":"2026-07-05T01:28:48.741793Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:28:48.741793Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A sharp error estimate of piecewise polynomial collocation for nonlocal problems with weakly singular kernels","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Jiankang Shi, Jiming Wu, Minghua Chen, Wenya Qi","submitted_at":"2019-09-24T08:17:43Z","abstract_excerpt":"As is well known, using piecewise linear polynomial collocation (PLC) and piecewise quadratic polynomial collocation (PQC), respectively, to approximate the weakly singular integral $$I(a,b,x) =\\int^b_a \\frac{u(y)}{|x-y|^\\gamma}dy, \\quad x \\in (a,b) ,\\quad 0< \\gamma <1,$$ have the local truncation error $\\mathcal{O}\\left(h^2\\right)$ and $\\mathcal{O}\\left(h^{4-\\gamma}\\right)$. 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