{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:RWPWDLWEXXGLCEYA22COXZOGQ6","short_pith_number":"pith:RWPWDLWE","canonical_record":{"source":{"id":"1810.09544","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-10-19T03:01:09Z","cross_cats_sorted":["math-ph","math.CA","math.MP"],"title_canon_sha256":"915cc0912075dfbf212aa60586421a4f13aceda6607b41084e06b97ca89687f7","abstract_canon_sha256":"ce939a2997a611f2d5616c20d61d3ad822fbdb0a77729afe82babe23f44335b4"},"schema_version":"1.0"},"canonical_sha256":"8d9f61aec4bdccb11300d684ebe5c6878e9ae9ed927eca048a851a08a798a30b","source":{"kind":"arxiv","id":"1810.09544","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.09544","created_at":"2026-05-18T00:02:39Z"},{"alias_kind":"arxiv_version","alias_value":"1810.09544v1","created_at":"2026-05-18T00:02:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.09544","created_at":"2026-05-18T00:02:39Z"},{"alias_kind":"pith_short_12","alias_value":"RWPWDLWEXXGL","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_16","alias_value":"RWPWDLWEXXGLCEYA","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_8","alias_value":"RWPWDLWE","created_at":"2026-05-18T12:32:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:RWPWDLWEXXGLCEYA22COXZOGQ6","target":"record","payload":{"canonical_record":{"source":{"id":"1810.09544","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-10-19T03:01:09Z","cross_cats_sorted":["math-ph","math.CA","math.MP"],"title_canon_sha256":"915cc0912075dfbf212aa60586421a4f13aceda6607b41084e06b97ca89687f7","abstract_canon_sha256":"ce939a2997a611f2d5616c20d61d3ad822fbdb0a77729afe82babe23f44335b4"},"schema_version":"1.0"},"canonical_sha256":"8d9f61aec4bdccb11300d684ebe5c6878e9ae9ed927eca048a851a08a798a30b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:02:39.286215Z","signature_b64":"zj34TQdRnHduF0jd+xcS1II26bR7uukaqPj+ApqpQDoEPS86itNyJrj+E16EBBUgjeU0E7zL8Q5XI6zyJ/UUDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8d9f61aec4bdccb11300d684ebe5c6878e9ae9ed927eca048a851a08a798a30b","last_reissued_at":"2026-05-18T00:02:39.285565Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:02:39.285565Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1810.09544","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":"2026-05-18T00:02:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3dVS2urNn/dz3rV3LsUZ6XgiQtGuKdEKaoJZIV8RvvV5cEz32jOAYtktul7H87LtMjjNxRZrp6fRFPM353WvCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T19:43:08.228550Z"},"content_sha256":"b2d7994f05ff08465fe895e5e508540fc107e3178ad6224e61b132583fce16d3","schema_version":"1.0","event_id":"sha256:b2d7994f05ff08465fe895e5e508540fc107e3178ad6224e61b132583fce16d3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:RWPWDLWEXXGLCEYA22COXZOGQ6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Solving the nonlinear biharmonic equation by the Laplace-Adomian and Adomian Decomposition Methods","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CA","math.MP"],"primary_cat":"math.AP","authors_text":"Chun Sing Leung, Man Kwong Mak, Tiberiu Harko","submitted_at":"2018-10-19T03:01:09Z","abstract_excerpt":"The biharmonic equation, as well as its nonlinear and inhomogeneous generalizations, plays an important role in engineering and physics. In particular the focusing biharmonic nonlinear Schr\\\"{o}dinger equation, and its standing wave solutions, have been intensively investigated. In the present paper we consider the applications of the Laplace-Adomian and Adomian Decomposition Methods for obtaining semi-analytical solutions of the generalized biharmonic equations of the type $\\Delta ^{2}y+\\alpha \\Delta y+\\omega y+b^{2}+g\\left( y\\right) =f$, where $\\alpha $, $\\omega $ and $b$ are constants, and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.09544","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":"2026-05-18T00:02:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1qp7+tWTn+C1mzy/7zVMdHE52ARSJlFjczA12aA/ARIl7/bOJ2uiqoSfqug4SbIu/FUVk+gkqBmM/PRua2qrCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T19:43:08.229259Z"},"content_sha256":"555bbd996e8b57fd8822bc975b67f42fa1711d2ec892f41db58bb6e00d7481e6","schema_version":"1.0","event_id":"sha256:555bbd996e8b57fd8822bc975b67f42fa1711d2ec892f41db58bb6e00d7481e6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RWPWDLWEXXGLCEYA22COXZOGQ6/bundle.json","state_url":"https://pith.science/pith/RWPWDLWEXXGLCEYA22COXZOGQ6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RWPWDLWEXXGLCEYA22COXZOGQ6/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":"2026-05-31T19:43:08Z","links":{"resolver":"https://pith.science/pith/RWPWDLWEXXGLCEYA22COXZOGQ6","bundle":"https://pith.science/pith/RWPWDLWEXXGLCEYA22COXZOGQ6/bundle.json","state":"https://pith.science/pith/RWPWDLWEXXGLCEYA22COXZOGQ6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RWPWDLWEXXGLCEYA22COXZOGQ6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:RWPWDLWEXXGLCEYA22COXZOGQ6","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":"ce939a2997a611f2d5616c20d61d3ad822fbdb0a77729afe82babe23f44335b4","cross_cats_sorted":["math-ph","math.CA","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-10-19T03:01:09Z","title_canon_sha256":"915cc0912075dfbf212aa60586421a4f13aceda6607b41084e06b97ca89687f7"},"schema_version":"1.0","source":{"id":"1810.09544","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.09544","created_at":"2026-05-18T00:02:39Z"},{"alias_kind":"arxiv_version","alias_value":"1810.09544v1","created_at":"2026-05-18T00:02:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.09544","created_at":"2026-05-18T00:02:39Z"},{"alias_kind":"pith_short_12","alias_value":"RWPWDLWEXXGL","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_16","alias_value":"RWPWDLWEXXGLCEYA","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_8","alias_value":"RWPWDLWE","created_at":"2026-05-18T12:32:50Z"}],"graph_snapshots":[{"event_id":"sha256:555bbd996e8b57fd8822bc975b67f42fa1711d2ec892f41db58bb6e00d7481e6","target":"graph","created_at":"2026-05-18T00:02:39Z","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":"The biharmonic equation, as well as its nonlinear and inhomogeneous generalizations, plays an important role in engineering and physics. In particular the focusing biharmonic nonlinear Schr\\\"{o}dinger equation, and its standing wave solutions, have been intensively investigated. In the present paper we consider the applications of the Laplace-Adomian and Adomian Decomposition Methods for obtaining semi-analytical solutions of the generalized biharmonic equations of the type $\\Delta ^{2}y+\\alpha \\Delta y+\\omega y+b^{2}+g\\left( y\\right) =f$, where $\\alpha $, $\\omega $ and $b$ are constants, and ","authors_text":"Chun Sing Leung, Man Kwong Mak, Tiberiu Harko","cross_cats":["math-ph","math.CA","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-10-19T03:01:09Z","title":"Solving the nonlinear biharmonic equation by the Laplace-Adomian and Adomian Decomposition Methods"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.09544","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:b2d7994f05ff08465fe895e5e508540fc107e3178ad6224e61b132583fce16d3","target":"record","created_at":"2026-05-18T00:02:39Z","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":"ce939a2997a611f2d5616c20d61d3ad822fbdb0a77729afe82babe23f44335b4","cross_cats_sorted":["math-ph","math.CA","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-10-19T03:01:09Z","title_canon_sha256":"915cc0912075dfbf212aa60586421a4f13aceda6607b41084e06b97ca89687f7"},"schema_version":"1.0","source":{"id":"1810.09544","kind":"arxiv","version":1}},"canonical_sha256":"8d9f61aec4bdccb11300d684ebe5c6878e9ae9ed927eca048a851a08a798a30b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8d9f61aec4bdccb11300d684ebe5c6878e9ae9ed927eca048a851a08a798a30b","first_computed_at":"2026-05-18T00:02:39.285565Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:02:39.285565Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zj34TQdRnHduF0jd+xcS1II26bR7uukaqPj+ApqpQDoEPS86itNyJrj+E16EBBUgjeU0E7zL8Q5XI6zyJ/UUDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:02:39.286215Z","signed_message":"canonical_sha256_bytes"},"source_id":"1810.09544","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b2d7994f05ff08465fe895e5e508540fc107e3178ad6224e61b132583fce16d3","sha256:555bbd996e8b57fd8822bc975b67f42fa1711d2ec892f41db58bb6e00d7481e6"],"state_sha256":"b013fd3bd1cf0b237b920d9fb7cb21b448fcd43a9bcc664660c2fc02786b4834"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3jI0Hx9PqUG1lRkSeq/tFymlU6rEjk/o+KQ/191Km3gt/eKWtmmeJQxRvd9cbDRoZ1/+pc+MXNCwsRZBsKvECg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T19:43:08.233278Z","bundle_sha256":"fde5425cb9c64e9e0db5afd0c497d0c32a533e5765d2422343c3f77c2af6e75e"}}