{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:RB6QGJXW7LKYXJZFEQSTRPLNMA","short_pith_number":"pith:RB6QGJXW","canonical_record":{"source":{"id":"1507.08957","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2015-07-31T17:40:41Z","cross_cats_sorted":[],"title_canon_sha256":"1aaa4ae8e0259741d5587b8503a65501500aeca38a114165373c4b38b6eaeac6","abstract_canon_sha256":"42e4fe3cc46198f8e3889887515e36615157480bc6439821a32a7251e049ed28"},"schema_version":"1.0"},"canonical_sha256":"887d0326f6fad58ba725242538bd6d602234e5fe1914f58549f12293a1db2807","source":{"kind":"arxiv","id":"1507.08957","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.08957","created_at":"2026-05-18T01:36:02Z"},{"alias_kind":"arxiv_version","alias_value":"1507.08957v1","created_at":"2026-05-18T01:36:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.08957","created_at":"2026-05-18T01:36:02Z"},{"alias_kind":"pith_short_12","alias_value":"RB6QGJXW7LKY","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_16","alias_value":"RB6QGJXW7LKYXJZF","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_8","alias_value":"RB6QGJXW","created_at":"2026-05-18T12:29:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:RB6QGJXW7LKYXJZFEQSTRPLNMA","target":"record","payload":{"canonical_record":{"source":{"id":"1507.08957","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2015-07-31T17:40:41Z","cross_cats_sorted":[],"title_canon_sha256":"1aaa4ae8e0259741d5587b8503a65501500aeca38a114165373c4b38b6eaeac6","abstract_canon_sha256":"42e4fe3cc46198f8e3889887515e36615157480bc6439821a32a7251e049ed28"},"schema_version":"1.0"},"canonical_sha256":"887d0326f6fad58ba725242538bd6d602234e5fe1914f58549f12293a1db2807","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:02.502914Z","signature_b64":"UnZnjaFK1hGkg7v6j3lFOrM/MU7qHdWSAjzpDS8/7W5qpTmq1gn2Ogcg5VeQfGFdj+wosNGH6DNzI7slHpYDBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"887d0326f6fad58ba725242538bd6d602234e5fe1914f58549f12293a1db2807","last_reissued_at":"2026-05-18T01:36:02.502304Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:02.502304Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.08957","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:36:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9DIlyGS4cr5BqTgmO13HP8VTG4BZyL2DUh1kHn2ZwTbzsz8HQNF7bDoqIX+9jQUUurAiBveQKEbbOvJe7wpuAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T17:04:11.461597Z"},"content_sha256":"3ea8fcf7579dc638ec986676385e5351422f8d081869df2b3413d7557f136752","schema_version":"1.0","event_id":"sha256:3ea8fcf7579dc638ec986676385e5351422f8d081869df2b3413d7557f136752"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:RB6QGJXW7LKYXJZFEQSTRPLNMA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"High order parameter-robust numerical method for a system of (M>=2) coupled singularly perturbed parabolic reaction-diffusion problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Mukesh Kumar, S. Chandra Sekhara Rao","submitted_at":"2015-07-31T17:40:41Z","abstract_excerpt":"We present a high order parameter-robust numerical method for a system of (M>=2) coupled singularly perturbed parabolic reaction-diffusion problems. A small perturbation parameter {\\epsilon} is multiplied with the second order spatial derivatives in all the equations. The parabolic boundary layer appears in the solution of the problem when the perturbation parameter {\\epsilon} tends to zero. To obtain a high order approximation to the solution of this problem, we propose a numerical method that employs the Crank-Nicolson method on an uniform mesh in time direction, together with a hybrid finit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08957","kind":"arxiv","version":1},"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":"2026-05-18T01:36:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"J8F5drJdPJFsoa0LfDO3Ntc3On5BC0xQ8Mm4tH8yVFLi2hwPEKC0ecyCkeJf7YLwWf4tHEW6oliH9i9UHJtvBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T17:04:11.461950Z"},"content_sha256":"3c87b419f44b02e0b31d1b92938da53748f194d6541c3a23dead1102f74ab08b","schema_version":"1.0","event_id":"sha256:3c87b419f44b02e0b31d1b92938da53748f194d6541c3a23dead1102f74ab08b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RB6QGJXW7LKYXJZFEQSTRPLNMA/bundle.json","state_url":"https://pith.science/pith/RB6QGJXW7LKYXJZFEQSTRPLNMA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RB6QGJXW7LKYXJZFEQSTRPLNMA/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":"2026-07-02T17:04:11Z","links":{"resolver":"https://pith.science/pith/RB6QGJXW7LKYXJZFEQSTRPLNMA","bundle":"https://pith.science/pith/RB6QGJXW7LKYXJZFEQSTRPLNMA/bundle.json","state":"https://pith.science/pith/RB6QGJXW7LKYXJZFEQSTRPLNMA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RB6QGJXW7LKYXJZFEQSTRPLNMA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:RB6QGJXW7LKYXJZFEQSTRPLNMA","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":"42e4fe3cc46198f8e3889887515e36615157480bc6439821a32a7251e049ed28","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2015-07-31T17:40:41Z","title_canon_sha256":"1aaa4ae8e0259741d5587b8503a65501500aeca38a114165373c4b38b6eaeac6"},"schema_version":"1.0","source":{"id":"1507.08957","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.08957","created_at":"2026-05-18T01:36:02Z"},{"alias_kind":"arxiv_version","alias_value":"1507.08957v1","created_at":"2026-05-18T01:36:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.08957","created_at":"2026-05-18T01:36:02Z"},{"alias_kind":"pith_short_12","alias_value":"RB6QGJXW7LKY","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_16","alias_value":"RB6QGJXW7LKYXJZF","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_8","alias_value":"RB6QGJXW","created_at":"2026-05-18T12:29:39Z"}],"graph_snapshots":[{"event_id":"sha256:3c87b419f44b02e0b31d1b92938da53748f194d6541c3a23dead1102f74ab08b","target":"graph","created_at":"2026-05-18T01:36:02Z","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":"We present a high order parameter-robust numerical method for a system of (M>=2) coupled singularly perturbed parabolic reaction-diffusion problems. A small perturbation parameter {\\epsilon} is multiplied with the second order spatial derivatives in all the equations. The parabolic boundary layer appears in the solution of the problem when the perturbation parameter {\\epsilon} tends to zero. To obtain a high order approximation to the solution of this problem, we propose a numerical method that employs the Crank-Nicolson method on an uniform mesh in time direction, together with a hybrid finit","authors_text":"Mukesh Kumar, S. Chandra Sekhara Rao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2015-07-31T17:40:41Z","title":"High order parameter-robust numerical method for a system of (M>=2) coupled singularly perturbed parabolic reaction-diffusion problems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08957","kind":"arxiv","version":1},"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":"sha256:3ea8fcf7579dc638ec986676385e5351422f8d081869df2b3413d7557f136752","target":"record","created_at":"2026-05-18T01:36:02Z","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":"42e4fe3cc46198f8e3889887515e36615157480bc6439821a32a7251e049ed28","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2015-07-31T17:40:41Z","title_canon_sha256":"1aaa4ae8e0259741d5587b8503a65501500aeca38a114165373c4b38b6eaeac6"},"schema_version":"1.0","source":{"id":"1507.08957","kind":"arxiv","version":1}},"canonical_sha256":"887d0326f6fad58ba725242538bd6d602234e5fe1914f58549f12293a1db2807","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"887d0326f6fad58ba725242538bd6d602234e5fe1914f58549f12293a1db2807","first_computed_at":"2026-05-18T01:36:02.502304Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:36:02.502304Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UnZnjaFK1hGkg7v6j3lFOrM/MU7qHdWSAjzpDS8/7W5qpTmq1gn2Ogcg5VeQfGFdj+wosNGH6DNzI7slHpYDBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:36:02.502914Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.08957","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3ea8fcf7579dc638ec986676385e5351422f8d081869df2b3413d7557f136752","sha256:3c87b419f44b02e0b31d1b92938da53748f194d6541c3a23dead1102f74ab08b"],"state_sha256":"0bcb07f7ca626305fe14a552d6f6cc9544365eedfca1b50acd063179b6c63a2c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eU5+SEG7rwjGGCZix5pLaD54NfZm8DhE7X3RrBf/QLPPvekAipCT8/yKONq4OG4AUIdszTOjFz6NmOcAy3XQDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-02T17:04:11.463968Z","bundle_sha256":"0a6949cb6f1ea7aeeb6b067d9c86a8832362e520f01b440f35ca1d24f5b7a342"}}