{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:3VGKUCOA2HWKT4PYQCBKPJDO5M","short_pith_number":"pith:3VGKUCOA","canonical_record":{"source":{"id":"2605.13285","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-05-13T10:01:32Z","cross_cats_sorted":[],"title_canon_sha256":"20bd33e5f853fd0cf2f8884f80c19ac0f743c345eea972c7fd29464f06a022e1","abstract_canon_sha256":"5862f941ba77ad7e8588e10c5b30764268581e000e3826af7af7459588640afc"},"schema_version":"1.0"},"canonical_sha256":"dd4caa09c0d1eca9f1f88082a7a46eeb26bb1f93acd9834ab359e526f984a52c","source":{"kind":"arxiv","id":"2605.13285","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.13285","created_at":"2026-05-18T02:44:49Z"},{"alias_kind":"arxiv_version","alias_value":"2605.13285v1","created_at":"2026-05-18T02:44:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.13285","created_at":"2026-05-18T02:44:49Z"},{"alias_kind":"pith_short_12","alias_value":"3VGKUCOA2HWK","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"3VGKUCOA2HWKT4PY","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"3VGKUCOA","created_at":"2026-05-18T12:33:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:3VGKUCOA2HWKT4PYQCBKPJDO5M","target":"record","payload":{"canonical_record":{"source":{"id":"2605.13285","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-05-13T10:01:32Z","cross_cats_sorted":[],"title_canon_sha256":"20bd33e5f853fd0cf2f8884f80c19ac0f743c345eea972c7fd29464f06a022e1","abstract_canon_sha256":"5862f941ba77ad7e8588e10c5b30764268581e000e3826af7af7459588640afc"},"schema_version":"1.0"},"canonical_sha256":"dd4caa09c0d1eca9f1f88082a7a46eeb26bb1f93acd9834ab359e526f984a52c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:49.159997Z","signature_b64":"50A1DCf8DQcRjsrlSXcIfQclMeRwYsPfxYmC6uedtwpIgJFWpxK/ySXYqd1WiWcFBqOpynY5fjegHtK+YUKUDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dd4caa09c0d1eca9f1f88082a7a46eeb26bb1f93acd9834ab359e526f984a52c","last_reissued_at":"2026-05-18T02:44:49.159593Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:49.159593Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.13285","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":"2026-05-18T02:44:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"q69mYNibodcN7anYZT5ZfZtFFSA5UZeVZzhwhIKf1WkVnflIXNnspH3Blp6SDsjfxHBcr95oZuZ5GIHaOSMqBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-22T08:32:45.504665Z"},"content_sha256":"6cd5483381cd7c8cbbcb00e4088364796a0e08f0c23c7e92c3054e7743ba1a9d","schema_version":"1.0","event_id":"sha256:6cd5483381cd7c8cbbcb00e4088364796a0e08f0c23c7e92c3054e7743ba1a9d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:3VGKUCOA2HWKT4PYQCBKPJDO5M","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Forward and inverse problems for a time-fractional pseudo-parabolic equation with variable coefficients","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Global existence and uniqueness hold for the forward problem of a time-fractional pseudo-parabolic equation with time-dependent coefficient, and global existence holds for the associated inverse source problem.","cross_cats":[],"primary_cat":"math.AP","authors_text":"Elbek Husanov, Ravshan Ashurov","submitted_at":"2026-05-13T10:01:32Z","abstract_excerpt":"In this work, forward and inverse problems for a time-fractional pseudo-parabolic equation $D_t^{\\rho} [u(t) + \\mu Au(t)] + \\sigma(t) Au(t) = r(t)g$ are investigated in a Hilbert space, where $A$ is an unbounded, positive, self-adjoint operator. According to the known papers, the forward problem has been studied only in the case $\\sigma(t) = const$. The main novelty of the forward problem in this work is that the model is further generalized and investigated for a time-dependent coefficient $\\sigma(t)$. To determine the solution of the forward problem, the Fourier method is employed, and the g"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The global existence and uniqueness of the solution to the forward problem is proved for time-dependent σ(t), and the global existence of the solution to the inverse problem is proved by applying Schauder's fixed point theorem.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The functional F is assumed to be such that the associated mapping satisfies the hypotheses of Schauder's fixed point theorem, and the operator A possesses a discrete spectrum of positive eigenvalues with the requisite regularity.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Global existence and uniqueness are proved for the forward problem with variable σ(t) and for the inverse problem of recovering the source r(t) from a general overdetermination condition F[u(t)] = Φ(t).","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Global existence and uniqueness hold for the forward problem of a time-fractional pseudo-parabolic equation with time-dependent coefficient, and global existence holds for the associated inverse source problem.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"47ede6d7ec8a80200544e3893f3822a0978f6cd32238d27980047fa8aa843f51"},"source":{"id":"2605.13285","kind":"arxiv","version":1},"verdict":{"id":"d9e40b4b-0554-470a-bf2a-4b25eece73d9","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:21:40.928286Z","strongest_claim":"The global existence and uniqueness of the solution to the forward problem is proved for time-dependent σ(t), and the global existence of the solution to the inverse problem is proved by applying Schauder's fixed point theorem.","one_line_summary":"Global existence and uniqueness are proved for the forward problem with variable σ(t) and for the inverse problem of recovering the source r(t) from a general overdetermination condition F[u(t)] = Φ(t).","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The functional F is assumed to be such that the associated mapping satisfies the hypotheses of Schauder's fixed point theorem, and the operator A possesses a discrete spectrum of positive eigenvalues with the requisite regularity.","pith_extraction_headline":"Global existence and uniqueness hold for the forward problem of a time-fractional pseudo-parabolic equation with time-dependent coefficient, and global existence holds for the associated inverse source problem."},"references":{"count":17,"sample":[{"doi":"","year":2019,"title":"Lizama,Abstract linear fractional evolution equations, inHandbook of Fractional Calculus with Applications, Vol","work_id":"4f9aa771-3f20-468a-93a9-b711b2de5716","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2005,"title":"A. V. Pskhu,Fractional Differential Equations(Moscow, Russia, Nauka, 2005)","work_id":"3992c5b3-b0af-482d-8b97-61a7be341fe3","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1968,"title":"On a theory of heat conduction involving two temperatures,","work_id":"f2a98c24-ef28-4029-95c8-4dcdf1597793","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1960,"title":"Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks,","work_id":"c22a72a7-74bb-4bb3-9d47-e8bcf31ceafd","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1980,"title":"Determination of an unknown nonhomogeneous term in a linear partial differential equation from overspecified boundary data,","work_id":"3d62c7ec-5597-4c8e-bd81-045a2ae9c66b","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":17,"snapshot_sha256":"10898965441b4c42612e13b9af49b48aabd1e0161f1185b19374ed898f463562","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":"d9e40b4b-0554-470a-bf2a-4b25eece73d9"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:44:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RtPL1881otUJE6n+St1cWrafoSUisAyjelxyQU8TvuICDlfWjTmiX82x9X4RuveKQ486kkLipBxC2UII39QYAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-22T08:32:45.505741Z"},"content_sha256":"687cd1be86f3fc349cfeb7ffa3023ead64dd85dbb49da3d8ddeb7a7553822329","schema_version":"1.0","event_id":"sha256:687cd1be86f3fc349cfeb7ffa3023ead64dd85dbb49da3d8ddeb7a7553822329"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3VGKUCOA2HWKT4PYQCBKPJDO5M/bundle.json","state_url":"https://pith.science/pith/3VGKUCOA2HWKT4PYQCBKPJDO5M/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3VGKUCOA2HWKT4PYQCBKPJDO5M/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":"2026-05-22T08:32:45Z","links":{"resolver":"https://pith.science/pith/3VGKUCOA2HWKT4PYQCBKPJDO5M","bundle":"https://pith.science/pith/3VGKUCOA2HWKT4PYQCBKPJDO5M/bundle.json","state":"https://pith.science/pith/3VGKUCOA2HWKT4PYQCBKPJDO5M/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3VGKUCOA2HWKT4PYQCBKPJDO5M/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:3VGKUCOA2HWKT4PYQCBKPJDO5M","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":"5862f941ba77ad7e8588e10c5b30764268581e000e3826af7af7459588640afc","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-05-13T10:01:32Z","title_canon_sha256":"20bd33e5f853fd0cf2f8884f80c19ac0f743c345eea972c7fd29464f06a022e1"},"schema_version":"1.0","source":{"id":"2605.13285","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.13285","created_at":"2026-05-18T02:44:49Z"},{"alias_kind":"arxiv_version","alias_value":"2605.13285v1","created_at":"2026-05-18T02:44:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.13285","created_at":"2026-05-18T02:44:49Z"},{"alias_kind":"pith_short_12","alias_value":"3VGKUCOA2HWK","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"3VGKUCOA2HWKT4PY","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"3VGKUCOA","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:687cd1be86f3fc349cfeb7ffa3023ead64dd85dbb49da3d8ddeb7a7553822329","target":"graph","created_at":"2026-05-18T02:44:49Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"The global existence and uniqueness of the solution to the forward problem is proved for time-dependent σ(t), and the global existence of the solution to the inverse problem is proved by applying Schauder's fixed point theorem."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The functional F is assumed to be such that the associated mapping satisfies the hypotheses of Schauder's fixed point theorem, and the operator A possesses a discrete spectrum of positive eigenvalues with the requisite regularity."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Global existence and uniqueness are proved for the forward problem with variable σ(t) and for the inverse problem of recovering the source r(t) from a general overdetermination condition F[u(t)] = Φ(t)."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Global existence and uniqueness hold for the forward problem of a time-fractional pseudo-parabolic equation with time-dependent coefficient, and global existence holds for the associated inverse source problem."}],"snapshot_sha256":"47ede6d7ec8a80200544e3893f3822a0978f6cd32238d27980047fa8aa843f51"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this work, forward and inverse problems for a time-fractional pseudo-parabolic equation $D_t^{\\rho} [u(t) + \\mu Au(t)] + \\sigma(t) Au(t) = r(t)g$ are investigated in a Hilbert space, where $A$ is an unbounded, positive, self-adjoint operator. According to the known papers, the forward problem has been studied only in the case $\\sigma(t) = const$. The main novelty of the forward problem in this work is that the model is further generalized and investigated for a time-dependent coefficient $\\sigma(t)$. To determine the solution of the forward problem, the Fourier method is employed, and the g","authors_text":"Elbek Husanov, Ravshan Ashurov","cross_cats":[],"headline":"Global existence and uniqueness hold for the forward problem of a time-fractional pseudo-parabolic equation with time-dependent coefficient, and global existence holds for the associated inverse source problem.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-05-13T10:01:32Z","title":"Forward and inverse problems for a time-fractional pseudo-parabolic equation with variable coefficients"},"references":{"count":17,"internal_anchors":0,"resolved_work":17,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"Lizama,Abstract linear fractional evolution equations, inHandbook of Fractional Calculus with Applications, Vol","work_id":"4f9aa771-3f20-468a-93a9-b711b2de5716","year":2019},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"A. V. Pskhu,Fractional Differential Equations(Moscow, Russia, Nauka, 2005)","work_id":"3992c5b3-b0af-482d-8b97-61a7be341fe3","year":2005},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"On a theory of heat conduction involving two temperatures,","work_id":"f2a98c24-ef28-4029-95c8-4dcdf1597793","year":1968},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks,","work_id":"c22a72a7-74bb-4bb3-9d47-e8bcf31ceafd","year":1960},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"Determination of an unknown nonhomogeneous term in a linear partial differential equation from overspecified boundary data,","work_id":"3d62c7ec-5597-4c8e-bd81-045a2ae9c66b","year":1980}],"snapshot_sha256":"10898965441b4c42612e13b9af49b48aabd1e0161f1185b19374ed898f463562"},"source":{"id":"2605.13285","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-14T18:21:40.928286Z","id":"d9e40b4b-0554-470a-bf2a-4b25eece73d9","model_set":{"reader":"grok-4.3"},"one_line_summary":"Global existence and uniqueness are proved for the forward problem with variable σ(t) and for the inverse problem of recovering the source r(t) from a general overdetermination condition F[u(t)] = Φ(t).","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Global existence and uniqueness hold for the forward problem of a time-fractional pseudo-parabolic equation with time-dependent coefficient, and global existence holds for the associated inverse source problem.","strongest_claim":"The global existence and uniqueness of the solution to the forward problem is proved for time-dependent σ(t), and the global existence of the solution to the inverse problem is proved by applying Schauder's fixed point theorem.","weakest_assumption":"The functional F is assumed to be such that the associated mapping satisfies the hypotheses of Schauder's fixed point theorem, and the operator A possesses a discrete spectrum of positive eigenvalues with the requisite regularity."}},"verdict_id":"d9e40b4b-0554-470a-bf2a-4b25eece73d9"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:6cd5483381cd7c8cbbcb00e4088364796a0e08f0c23c7e92c3054e7743ba1a9d","target":"record","created_at":"2026-05-18T02:44:49Z","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":"5862f941ba77ad7e8588e10c5b30764268581e000e3826af7af7459588640afc","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-05-13T10:01:32Z","title_canon_sha256":"20bd33e5f853fd0cf2f8884f80c19ac0f743c345eea972c7fd29464f06a022e1"},"schema_version":"1.0","source":{"id":"2605.13285","kind":"arxiv","version":1}},"canonical_sha256":"dd4caa09c0d1eca9f1f88082a7a46eeb26bb1f93acd9834ab359e526f984a52c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dd4caa09c0d1eca9f1f88082a7a46eeb26bb1f93acd9834ab359e526f984a52c","first_computed_at":"2026-05-18T02:44:49.159593Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:44:49.159593Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"50A1DCf8DQcRjsrlSXcIfQclMeRwYsPfxYmC6uedtwpIgJFWpxK/ySXYqd1WiWcFBqOpynY5fjegHtK+YUKUDg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:44:49.159997Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.13285","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6cd5483381cd7c8cbbcb00e4088364796a0e08f0c23c7e92c3054e7743ba1a9d","sha256:687cd1be86f3fc349cfeb7ffa3023ead64dd85dbb49da3d8ddeb7a7553822329"],"state_sha256":"53039db696ad7abb313da5c532ad93ae2170595895767dd0731c28f386f9fa11"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eGnvMLaBo0insmALf9GB5gfmdgPHzTZ04xC9Z5tKxwwBDoGThmgzyFlglsNES70jDUMDXR2JWAK/MWm0RKhTAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-22T08:32:45.510299Z","bundle_sha256":"01053dedd642bb039c2033d64ad73ce69fbd3e8e6d5c1db84151c9c4fa662618"}}