{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:MMIAUE6UJFBBJJPM45F3KGWQ6L","short_pith_number":"pith:MMIAUE6U","schema_version":"1.0","canonical_sha256":"63100a13d4494214a5ece74bb51ad0f2d2e13e180ca0cce5860b6a490617eb65","source":{"kind":"arxiv","id":"1609.04775","version":1},"attestation_state":"computed","paper":{"title":"A Caputo fractional derivative of a function with respect to another function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ricardo Almeida","submitted_at":"2016-09-12T15:52:40Z","abstract_excerpt":"In this paper we consider a Caputo type fractional derivative with respect to another function. Some properties, like the semigroup law, a relationship between the fractional derivative and the fractional integral, Taylor's Theorem, Fermat's Theorem, etc, are studied. Also, a numerical method to deal with such operators, consisting in approximating the fractional derivative by a sum that depends on the first-order derivative, is presented. Relying on examples, we show the efficiency and applicability of the method. Finally, an application of the fractional derivative, by considering a Populati"},"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":"1609.04775","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-09-12T15:52:40Z","cross_cats_sorted":[],"title_canon_sha256":"26ed6ca2797a4c820dff8ed1adb28546308f351dd6413be04a3d7f1ce4d5561c","abstract_canon_sha256":"2c04fa56c645a5dc3583c8ad9133af1ef62eb265057662e9d3ce2fefe4b69477"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:02:36.633794Z","signature_b64":"BAYwZJ6yfdIt6ZEdGEOlPP2RaGw/UYB132TpXW5/QkvMXyEhld2Byry05bGrmO4mn1kArqGt2zuDolFNqhFjDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"63100a13d4494214a5ece74bb51ad0f2d2e13e180ca0cce5860b6a490617eb65","last_reissued_at":"2026-05-18T01:02:36.633123Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:02:36.633123Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Caputo fractional derivative of a function with respect to another function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ricardo Almeida","submitted_at":"2016-09-12T15:52:40Z","abstract_excerpt":"In this paper we consider a Caputo type fractional derivative with respect to another function. Some properties, like the semigroup law, a relationship between the fractional derivative and the fractional integral, Taylor's Theorem, Fermat's Theorem, etc, are studied. Also, a numerical method to deal with such operators, consisting in approximating the fractional derivative by a sum that depends on the first-order derivative, is presented. Relying on examples, we show the efficiency and applicability of the method. Finally, an application of the fractional derivative, by considering a Populati"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.04775","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1609.04775","created_at":"2026-05-18T01:02:36.633201+00:00"},{"alias_kind":"arxiv_version","alias_value":"1609.04775v1","created_at":"2026-05-18T01:02:36.633201+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.04775","created_at":"2026-05-18T01:02:36.633201+00:00"},{"alias_kind":"pith_short_12","alias_value":"MMIAUE6UJFBB","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_16","alias_value":"MMIAUE6UJFBBJJPM","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_8","alias_value":"MMIAUE6U","created_at":"2026-05-18T12:30:32.724797+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/MMIAUE6UJFBBJJPM45F3KGWQ6L","json":"https://pith.science/pith/MMIAUE6UJFBBJJPM45F3KGWQ6L.json","graph_json":"https://pith.science/api/pith-number/MMIAUE6UJFBBJJPM45F3KGWQ6L/graph.json","events_json":"https://pith.science/api/pith-number/MMIAUE6UJFBBJJPM45F3KGWQ6L/events.json","paper":"https://pith.science/paper/MMIAUE6U"},"agent_actions":{"view_html":"https://pith.science/pith/MMIAUE6UJFBBJJPM45F3KGWQ6L","download_json":"https://pith.science/pith/MMIAUE6UJFBBJJPM45F3KGWQ6L.json","view_paper":"https://pith.science/paper/MMIAUE6U","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1609.04775&json=true","fetch_graph":"https://pith.science/api/pith-number/MMIAUE6UJFBBJJPM45F3KGWQ6L/graph.json","fetch_events":"https://pith.science/api/pith-number/MMIAUE6UJFBBJJPM45F3KGWQ6L/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MMIAUE6UJFBBJJPM45F3KGWQ6L/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MMIAUE6UJFBBJJPM45F3KGWQ6L/action/storage_attestation","attest_author":"https://pith.science/pith/MMIAUE6UJFBBJJPM45F3KGWQ6L/action/author_attestation","sign_citation":"https://pith.science/pith/MMIAUE6UJFBBJJPM45F3KGWQ6L/action/citation_signature","submit_replication":"https://pith.science/pith/MMIAUE6UJFBBJJPM45F3KGWQ6L/action/replication_record"}},"created_at":"2026-05-18T01:02:36.633201+00:00","updated_at":"2026-05-18T01:02:36.633201+00:00"}