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Mathematically, the bibliography on initial-boundary value problems for ultraparabolic equations is not extensive although the problems have many applications related to option pricing, multi parameter Br"},"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":"1408.1351","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-08-06T16:38:25Z","cross_cats_sorted":["math.NA","math.SP"],"title_canon_sha256":"4c368d7103bd90886b2fff31bec93dc97986669bec281459e9cef6af77018695","abstract_canon_sha256":"5104abf349f85b6e364cdf8a4d32fdd9c7114d3a2edc17f7db273c2c6d48d547"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:45:39.265149Z","signature_b64":"+h88ScoPeNF40d5Iyos/RpekE40uNrwZFl36wR6YaJSVed02Rs0Koav7Nxv+W4xNh1K0xLtmqXPjHUnp4BycCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5e8447b7a5fde4e2d250530977f481a1e01955c1d09737f42c90fca68a083e54","last_reissued_at":"2026-05-18T02:45:39.264568Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:45:39.264568Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A numerical approach to approximation for an ultraparabolic equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA","math.SP"],"primary_cat":"math.AP","authors_text":"Le Trong Lan, Nguyen Huy Tuan, Nguyen Thi Yen Ngoc, Vo Anh Khoa","submitted_at":"2014-08-06T16:38:25Z","abstract_excerpt":"We study the following ultraparabolic equation\n  \\[ \\frac{\\partial}{\\partial t}u\\left(t,s\\right)+\\frac{\\partial}{\\partial s}u\\left(t,s\\right)+\\mathcal{L}u\\left(t,s\\right)=f\\left(u\\left(t,s\\right),t,s\\right),\\quad\\left(t,s\\right)\\in\\left(0,T\\right)\\times\\left(0,T\\right), \\]\n  where $\\mathcal{L}$ is a positive-definite, self-adjoint operator with compact inverse and $f$ is a nonlinear function. 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