{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:GEW7PJ2LAS5VCWH3AWORAMFTWJ","short_pith_number":"pith:GEW7PJ2L","canonical_record":{"source":{"id":"1511.02017","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-11-06T09:55:54Z","cross_cats_sorted":["math.AP","math.NA"],"title_canon_sha256":"114f334465bdf11323ce2016dd7265e2658ab78822c395ad52bb1dd9733a26ae","abstract_canon_sha256":"3cd6785509b3477f5233c688a926ad674534877ac6fbe0e2cedcb7aa98e06fa1"},"schema_version":"1.0"},"canonical_sha256":"312df7a74b04bb5158fb059d1030b3b25457a5780f41d001582501e1637d5dc1","source":{"kind":"arxiv","id":"1511.02017","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.02017","created_at":"2026-05-18T01:25:11Z"},{"alias_kind":"arxiv_version","alias_value":"1511.02017v1","created_at":"2026-05-18T01:25:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.02017","created_at":"2026-05-18T01:25:11Z"},{"alias_kind":"pith_short_12","alias_value":"GEW7PJ2LAS5V","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"GEW7PJ2LAS5VCWH3","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"GEW7PJ2L","created_at":"2026-05-18T12:29:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:GEW7PJ2LAS5VCWH3AWORAMFTWJ","target":"record","payload":{"canonical_record":{"source":{"id":"1511.02017","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-11-06T09:55:54Z","cross_cats_sorted":["math.AP","math.NA"],"title_canon_sha256":"114f334465bdf11323ce2016dd7265e2658ab78822c395ad52bb1dd9733a26ae","abstract_canon_sha256":"3cd6785509b3477f5233c688a926ad674534877ac6fbe0e2cedcb7aa98e06fa1"},"schema_version":"1.0"},"canonical_sha256":"312df7a74b04bb5158fb059d1030b3b25457a5780f41d001582501e1637d5dc1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:11.143545Z","signature_b64":"/pYJk45YU/yfSRJSLkQGNbA/oZBbySKW8bcY0Mi6X8ouEjGdybns7hVJ0JN8sxVSfRX7oNx2DDk+LD+6QqrZDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"312df7a74b04bb5158fb059d1030b3b25457a5780f41d001582501e1637d5dc1","last_reissued_at":"2026-05-18T01:25:11.142988Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:11.142988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1511.02017","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":"2026-05-18T01:25:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Q2RvHtOCgs/AdehqLhoEaIXdHMpHQwbr402NliRhVkXKPMcCmtaqlyvDcuBJgKCnQqv3ggmtpLTZ/J/tYXgWAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T23:05:33.057783Z"},"content_sha256":"3d48d30cd8057c8730f738ea483321cec9e683ead2d781dd56ddcdff96a85d14","schema_version":"1.0","event_id":"sha256:3d48d30cd8057c8730f738ea483321cec9e683ead2d781dd56ddcdff96a85d14"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:GEW7PJ2LAS5VCWH3AWORAMFTWJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Caputo derivatives of fractional variable order: numerical approximations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.NA"],"primary_cat":"math.CA","authors_text":"Delfim F. M. Torres, Dina Tavares, Ricardo Almeida","submitted_at":"2015-11-06T09:55:54Z","abstract_excerpt":"We present a new numerical tool to solve partial differential equations involving Caputo derivatives of fractional variable order. Three Caputo-type fractional operators are considered, and for each one of them an approximation formula is obtained in terms of standard (integer-order) derivatives only. Estimations for the error of the approximations are also provided. We then compare the numerical approximation of some test function with its exact fractional derivative. We end with an exemplification of how the presented methods can be used to solve partial fractional differential equations of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.02017","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":"2026-05-18T01:25:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JzprNX5TBVD2AKKcsnsjC4Ojv9XPtKCekQgu4jtubQwFCAbu7thufkJWba++140Nbi9Y7Qp05Zb03OHf4OZ6Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T23:05:33.058129Z"},"content_sha256":"d3d0274a60cc13303928100de57f66025e47b8b67bed7c432a0da94219b492b3","schema_version":"1.0","event_id":"sha256:d3d0274a60cc13303928100de57f66025e47b8b67bed7c432a0da94219b492b3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GEW7PJ2LAS5VCWH3AWORAMFTWJ/bundle.json","state_url":"https://pith.science/pith/GEW7PJ2LAS5VCWH3AWORAMFTWJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GEW7PJ2LAS5VCWH3AWORAMFTWJ/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":"2026-06-02T23:05:33Z","links":{"resolver":"https://pith.science/pith/GEW7PJ2LAS5VCWH3AWORAMFTWJ","bundle":"https://pith.science/pith/GEW7PJ2LAS5VCWH3AWORAMFTWJ/bundle.json","state":"https://pith.science/pith/GEW7PJ2LAS5VCWH3AWORAMFTWJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GEW7PJ2LAS5VCWH3AWORAMFTWJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:GEW7PJ2LAS5VCWH3AWORAMFTWJ","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":"3cd6785509b3477f5233c688a926ad674534877ac6fbe0e2cedcb7aa98e06fa1","cross_cats_sorted":["math.AP","math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-11-06T09:55:54Z","title_canon_sha256":"114f334465bdf11323ce2016dd7265e2658ab78822c395ad52bb1dd9733a26ae"},"schema_version":"1.0","source":{"id":"1511.02017","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.02017","created_at":"2026-05-18T01:25:11Z"},{"alias_kind":"arxiv_version","alias_value":"1511.02017v1","created_at":"2026-05-18T01:25:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.02017","created_at":"2026-05-18T01:25:11Z"},{"alias_kind":"pith_short_12","alias_value":"GEW7PJ2LAS5V","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"GEW7PJ2LAS5VCWH3","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"GEW7PJ2L","created_at":"2026-05-18T12:29:22Z"}],"graph_snapshots":[{"event_id":"sha256:d3d0274a60cc13303928100de57f66025e47b8b67bed7c432a0da94219b492b3","target":"graph","created_at":"2026-05-18T01:25:11Z","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 new numerical tool to solve partial differential equations involving Caputo derivatives of fractional variable order. Three Caputo-type fractional operators are considered, and for each one of them an approximation formula is obtained in terms of standard (integer-order) derivatives only. Estimations for the error of the approximations are also provided. We then compare the numerical approximation of some test function with its exact fractional derivative. We end with an exemplification of how the presented methods can be used to solve partial fractional differential equations of ","authors_text":"Delfim F. M. Torres, Dina Tavares, Ricardo Almeida","cross_cats":["math.AP","math.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-11-06T09:55:54Z","title":"Caputo derivatives of fractional variable order: numerical approximations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.02017","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":"sha256:3d48d30cd8057c8730f738ea483321cec9e683ead2d781dd56ddcdff96a85d14","target":"record","created_at":"2026-05-18T01:25:11Z","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":"3cd6785509b3477f5233c688a926ad674534877ac6fbe0e2cedcb7aa98e06fa1","cross_cats_sorted":["math.AP","math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-11-06T09:55:54Z","title_canon_sha256":"114f334465bdf11323ce2016dd7265e2658ab78822c395ad52bb1dd9733a26ae"},"schema_version":"1.0","source":{"id":"1511.02017","kind":"arxiv","version":1}},"canonical_sha256":"312df7a74b04bb5158fb059d1030b3b25457a5780f41d001582501e1637d5dc1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"312df7a74b04bb5158fb059d1030b3b25457a5780f41d001582501e1637d5dc1","first_computed_at":"2026-05-18T01:25:11.142988Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:25:11.142988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/pYJk45YU/yfSRJSLkQGNbA/oZBbySKW8bcY0Mi6X8ouEjGdybns7hVJ0JN8sxVSfRX7oNx2DDk+LD+6QqrZDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:25:11.143545Z","signed_message":"canonical_sha256_bytes"},"source_id":"1511.02017","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3d48d30cd8057c8730f738ea483321cec9e683ead2d781dd56ddcdff96a85d14","sha256:d3d0274a60cc13303928100de57f66025e47b8b67bed7c432a0da94219b492b3"],"state_sha256":"f0386d6b6af0b175c9810bc072795bbc19d324a66be6519a70dcfbf3ec03103e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Klai0S2VtzmMotANGTpFkMwKOc89F5/YuOfx2qittTTz25cKj31pE6Se9J0CDQWqvrUpjfC+X4VaOV+svQZrDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T23:05:33.060177Z","bundle_sha256":"38fec3dabbf352843fb66c9448696817294f197e97bac624b655424e9aa260a2"}}