{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:VQRHRP4N2VNTEW32G4QAHX32BG","short_pith_number":"pith:VQRHRP4N","schema_version":"1.0","canonical_sha256":"ac2278bf8dd55b325b7a372003df7a09ab21179fe1a93fe05e0cbdc3718f9c38","source":{"kind":"arxiv","id":"1204.1867","version":2},"attestation_state":"computed","paper":{"title":"Self-Similar Solutions to a Density-Dependent Reaction-Diffusion Model","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["q-bio.PE"],"primary_cat":"physics.bio-ph","authors_text":"Kannika Khompurngson, Waipot Ngamsaad","submitted_at":"2012-04-09T12:23:22Z","abstract_excerpt":"In this paper, we investigated a density-dependent reaction-diffusion equation, $u_t = (u^{m})_{xx} + u - u^{m}$. This equation is known as the extension of the Fisher or Kolmogoroff-Petrovsky-Piscounoff equation which is widely used in the population dynamics, combustion theory and plasma physics. By employing the suitable transformation, this equation was mapped to the anomalous diffusion equation where the nonlinear reaction term was eliminated. Due to its simpler form, some exact self-similar solutions with the compact support have been obtained. The solutions, evolving from an initial sta"},"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":"1204.1867","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"physics.bio-ph","submitted_at":"2012-04-09T12:23:22Z","cross_cats_sorted":["q-bio.PE"],"title_canon_sha256":"4c9bb0c49f95c783256c04a8e15d3db18190097c1037b4e7b80e8aa0833abbcf","abstract_canon_sha256":"65fde8adb583a3e80540d8da26b15cfdd4939b23096b78f72d018841486e0210"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:53:19.759127Z","signature_b64":"VbcuoxW/rPWkgIgPlIfjZIcYMNslglCyoP5FNMPs782ZGuJxwzPEsvRvT3XSwOCi1rvZyKDsAv8DeGsZX8/mAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ac2278bf8dd55b325b7a372003df7a09ab21179fe1a93fe05e0cbdc3718f9c38","last_reissued_at":"2026-05-18T03:53:19.758639Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:53:19.758639Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Self-Similar Solutions to a Density-Dependent Reaction-Diffusion Model","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["q-bio.PE"],"primary_cat":"physics.bio-ph","authors_text":"Kannika Khompurngson, Waipot Ngamsaad","submitted_at":"2012-04-09T12:23:22Z","abstract_excerpt":"In this paper, we investigated a density-dependent reaction-diffusion equation, $u_t = (u^{m})_{xx} + u - u^{m}$. This equation is known as the extension of the Fisher or Kolmogoroff-Petrovsky-Piscounoff equation which is widely used in the population dynamics, combustion theory and plasma physics. By employing the suitable transformation, this equation was mapped to the anomalous diffusion equation where the nonlinear reaction term was eliminated. Due to its simpler form, some exact self-similar solutions with the compact support have been obtained. The solutions, evolving from an initial sta"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.1867","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1204.1867","created_at":"2026-05-18T03:53:19.758709+00:00"},{"alias_kind":"arxiv_version","alias_value":"1204.1867v2","created_at":"2026-05-18T03:53:19.758709+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.1867","created_at":"2026-05-18T03:53:19.758709+00:00"},{"alias_kind":"pith_short_12","alias_value":"VQRHRP4N2VNT","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_16","alias_value":"VQRHRP4N2VNTEW32","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_8","alias_value":"VQRHRP4N","created_at":"2026-05-18T12:27:25.539911+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VQRHRP4N2VNTEW32G4QAHX32BG","json":"https://pith.science/pith/VQRHRP4N2VNTEW32G4QAHX32BG.json","graph_json":"https://pith.science/api/pith-number/VQRHRP4N2VNTEW32G4QAHX32BG/graph.json","events_json":"https://pith.science/api/pith-number/VQRHRP4N2VNTEW32G4QAHX32BG/events.json","paper":"https://pith.science/paper/VQRHRP4N"},"agent_actions":{"view_html":"https://pith.science/pith/VQRHRP4N2VNTEW32G4QAHX32BG","download_json":"https://pith.science/pith/VQRHRP4N2VNTEW32G4QAHX32BG.json","view_paper":"https://pith.science/paper/VQRHRP4N","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1204.1867&json=true","fetch_graph":"https://pith.science/api/pith-number/VQRHRP4N2VNTEW32G4QAHX32BG/graph.json","fetch_events":"https://pith.science/api/pith-number/VQRHRP4N2VNTEW32G4QAHX32BG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VQRHRP4N2VNTEW32G4QAHX32BG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VQRHRP4N2VNTEW32G4QAHX32BG/action/storage_attestation","attest_author":"https://pith.science/pith/VQRHRP4N2VNTEW32G4QAHX32BG/action/author_attestation","sign_citation":"https://pith.science/pith/VQRHRP4N2VNTEW32G4QAHX32BG/action/citation_signature","submit_replication":"https://pith.science/pith/VQRHRP4N2VNTEW32G4QAHX32BG/action/replication_record"}},"created_at":"2026-05-18T03:53:19.758709+00:00","updated_at":"2026-05-18T03:53:19.758709+00:00"}