{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:Y76R3Y57Z4L4TORXXUQIJJC7C5","short_pith_number":"pith:Y76R3Y57","canonical_record":{"source":{"id":"1811.04016","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2018-11-09T17:10:41Z","cross_cats_sorted":[],"title_canon_sha256":"5d06b6f80bdd19286b0fa833632a009d2b95fffcdec0f1fa9d4eef59fe9b4456","abstract_canon_sha256":"463f59ec6864c891c8942599330129c31c23597a27fa8974f1878152ffcd9725"},"schema_version":"1.0"},"canonical_sha256":"c7fd1de3bfcf17c9ba37bd2084a45f176f38d58bac6a35462ab9d0a71fccfc01","source":{"kind":"arxiv","id":"1811.04016","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.04016","created_at":"2026-05-17T23:53:41Z"},{"alias_kind":"arxiv_version","alias_value":"1811.04016v2","created_at":"2026-05-17T23:53:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.04016","created_at":"2026-05-17T23:53:41Z"},{"alias_kind":"pith_short_12","alias_value":"Y76R3Y57Z4L4","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_16","alias_value":"Y76R3Y57Z4L4TORX","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_8","alias_value":"Y76R3Y57","created_at":"2026-05-18T12:33:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:Y76R3Y57Z4L4TORXXUQIJJC7C5","target":"record","payload":{"canonical_record":{"source":{"id":"1811.04016","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2018-11-09T17:10:41Z","cross_cats_sorted":[],"title_canon_sha256":"5d06b6f80bdd19286b0fa833632a009d2b95fffcdec0f1fa9d4eef59fe9b4456","abstract_canon_sha256":"463f59ec6864c891c8942599330129c31c23597a27fa8974f1878152ffcd9725"},"schema_version":"1.0"},"canonical_sha256":"c7fd1de3bfcf17c9ba37bd2084a45f176f38d58bac6a35462ab9d0a71fccfc01","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:53:41.597078Z","signature_b64":"7sWPx+fRuqJr4q+IQx84ozpSYN1Y7IGNjZmo/nZ1K3H1dkGNsVnb7rljJuxrCkef3u6d205Px/CCknRY/QNdCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c7fd1de3bfcf17c9ba37bd2084a45f176f38d58bac6a35462ab9d0a71fccfc01","last_reissued_at":"2026-05-17T23:53:41.596458Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:53:41.596458Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1811.04016","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:53:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jRrcLCmFqw8yrfSGKFIneBciafTs1RYkeNjadPCUCLdLL1F1aBUkNTq2pNL82R/rbH2Ta90Epy7sGdCkUH/pAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T08:16:35.547017Z"},"content_sha256":"3fe0c92e26ae0650a79fabdcd47ad9fdaa95ed42575b8ad32d1061dd591172f3","schema_version":"1.0","event_id":"sha256:3fe0c92e26ae0650a79fabdcd47ad9fdaa95ed42575b8ad32d1061dd591172f3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:Y76R3Y57Z4L4TORXXUQIJJC7C5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Singularly perturbed reaction-diffusion problems with discontinuities in the initial and/or the boundary data","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Eugene O'Riordan, Jose Luis Gracia","submitted_at":"2018-11-09T17:10:41Z","abstract_excerpt":"Numerical approximations to the solutions of three different problem classes of singularly perturbed parabolic reaction-diffusion problems, each with a discontinuity in the bound\\-ary-initial data, are generated. For each problem class, an analytical function associated with the discontinuity in the data, is identified. Parameter-uniform numerical approximations to the difference between the analytical function and the solution of the singularly perturbed problem are generated using piecewise-uniform Shishkin meshes. Numerical results are given to illustrate all the theoretical error bounds es"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.04016","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:53:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"M4gf/ve0GpC3alvoAjE92f7wNWOm1Pg2hjdC6dLcQ/9obRv/sED98LOZXTW+8cgIlNKgCIi+1lfi1NTkRzYjAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T08:16:35.547386Z"},"content_sha256":"024ca796dbfe8d7e9716404d49e56ea1f23142eff1f932de8a285d802d27eb65","schema_version":"1.0","event_id":"sha256:024ca796dbfe8d7e9716404d49e56ea1f23142eff1f932de8a285d802d27eb65"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Y76R3Y57Z4L4TORXXUQIJJC7C5/bundle.json","state_url":"https://pith.science/pith/Y76R3Y57Z4L4TORXXUQIJJC7C5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Y76R3Y57Z4L4TORXXUQIJJC7C5/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-07-02T08:16:35Z","links":{"resolver":"https://pith.science/pith/Y76R3Y57Z4L4TORXXUQIJJC7C5","bundle":"https://pith.science/pith/Y76R3Y57Z4L4TORXXUQIJJC7C5/bundle.json","state":"https://pith.science/pith/Y76R3Y57Z4L4TORXXUQIJJC7C5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Y76R3Y57Z4L4TORXXUQIJJC7C5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:Y76R3Y57Z4L4TORXXUQIJJC7C5","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"463f59ec6864c891c8942599330129c31c23597a27fa8974f1878152ffcd9725","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2018-11-09T17:10:41Z","title_canon_sha256":"5d06b6f80bdd19286b0fa833632a009d2b95fffcdec0f1fa9d4eef59fe9b4456"},"schema_version":"1.0","source":{"id":"1811.04016","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.04016","created_at":"2026-05-17T23:53:41Z"},{"alias_kind":"arxiv_version","alias_value":"1811.04016v2","created_at":"2026-05-17T23:53:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.04016","created_at":"2026-05-17T23:53:41Z"},{"alias_kind":"pith_short_12","alias_value":"Y76R3Y57Z4L4","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_16","alias_value":"Y76R3Y57Z4L4TORX","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_8","alias_value":"Y76R3Y57","created_at":"2026-05-18T12:33:04Z"}],"graph_snapshots":[{"event_id":"sha256:024ca796dbfe8d7e9716404d49e56ea1f23142eff1f932de8a285d802d27eb65","target":"graph","created_at":"2026-05-17T23:53:41Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Numerical approximations to the solutions of three different problem classes of singularly perturbed parabolic reaction-diffusion problems, each with a discontinuity in the bound\\-ary-initial data, are generated. For each problem class, an analytical function associated with the discontinuity in the data, is identified. Parameter-uniform numerical approximations to the difference between the analytical function and the solution of the singularly perturbed problem are generated using piecewise-uniform Shishkin meshes. Numerical results are given to illustrate all the theoretical error bounds es","authors_text":"Eugene O'Riordan, Jose Luis Gracia","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2018-11-09T17:10:41Z","title":"Singularly perturbed reaction-diffusion problems with discontinuities in the initial and/or the boundary data"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.04016","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:3fe0c92e26ae0650a79fabdcd47ad9fdaa95ed42575b8ad32d1061dd591172f3","target":"record","created_at":"2026-05-17T23:53:41Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"463f59ec6864c891c8942599330129c31c23597a27fa8974f1878152ffcd9725","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2018-11-09T17:10:41Z","title_canon_sha256":"5d06b6f80bdd19286b0fa833632a009d2b95fffcdec0f1fa9d4eef59fe9b4456"},"schema_version":"1.0","source":{"id":"1811.04016","kind":"arxiv","version":2}},"canonical_sha256":"c7fd1de3bfcf17c9ba37bd2084a45f176f38d58bac6a35462ab9d0a71fccfc01","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c7fd1de3bfcf17c9ba37bd2084a45f176f38d58bac6a35462ab9d0a71fccfc01","first_computed_at":"2026-05-17T23:53:41.596458Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:53:41.596458Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7sWPx+fRuqJr4q+IQx84ozpSYN1Y7IGNjZmo/nZ1K3H1dkGNsVnb7rljJuxrCkef3u6d205Px/CCknRY/QNdCQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:53:41.597078Z","signed_message":"canonical_sha256_bytes"},"source_id":"1811.04016","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3fe0c92e26ae0650a79fabdcd47ad9fdaa95ed42575b8ad32d1061dd591172f3","sha256:024ca796dbfe8d7e9716404d49e56ea1f23142eff1f932de8a285d802d27eb65"],"state_sha256":"81cb85e3b8c9c2569940d9c8f8d5ad6eefd2dcbf1ffce81f36ffe1ad2262094b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iDYRO/SzxFnfk7uyXoiOskgvM7QVPai3vQ3mltnuSsGGo0Hjow7TaMbZVamd/Y7z8TM3+GtYvXB/T/4LdrbeDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-02T08:16:35.549376Z","bundle_sha256":"1ae292ed444afa1d7e90ff18a57012fabef5c9bd501efe713a69ecaff9723d51"}}