{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:PZYIFQBB2XJGS7VVZJHCS5SQFN","short_pith_number":"pith:PZYIFQBB","canonical_record":{"source":{"id":"1504.08178","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2015-04-30T11:52:50Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"2d3027d396d530d8301c23232185604373aedae00753e9a34fc57d2e76c84d06","abstract_canon_sha256":"fcc5f6e9a49147f74b2c4bb9b7ef36ed59e24ad1d75111dfd9d1144f5f83b3ed"},"schema_version":"1.0"},"canonical_sha256":"7e7082c021d5d2697eb5ca4e2976502b68a13f6d630e9007e63c157709f22b5d","source":{"kind":"arxiv","id":"1504.08178","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.08178","created_at":"2026-05-18T02:17:23Z"},{"alias_kind":"arxiv_version","alias_value":"1504.08178v1","created_at":"2026-05-18T02:17:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.08178","created_at":"2026-05-18T02:17:23Z"},{"alias_kind":"pith_short_12","alias_value":"PZYIFQBB2XJG","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"PZYIFQBB2XJGS7VV","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"PZYIFQBB","created_at":"2026-05-18T12:29:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:PZYIFQBB2XJGS7VVZJHCS5SQFN","target":"record","payload":{"canonical_record":{"source":{"id":"1504.08178","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2015-04-30T11:52:50Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"2d3027d396d530d8301c23232185604373aedae00753e9a34fc57d2e76c84d06","abstract_canon_sha256":"fcc5f6e9a49147f74b2c4bb9b7ef36ed59e24ad1d75111dfd9d1144f5f83b3ed"},"schema_version":"1.0"},"canonical_sha256":"7e7082c021d5d2697eb5ca4e2976502b68a13f6d630e9007e63c157709f22b5d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:17:23.121356Z","signature_b64":"Ic1zM0W4y6ywBfi7gz4GLWzW8iBtQlrK5D/tGGaVMdxDlb+AzZBiCl2h8mVOxKIzHuOGcrv6Pehbp4r4eXKJDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7e7082c021d5d2697eb5ca4e2976502b68a13f6d630e9007e63c157709f22b5d","last_reissued_at":"2026-05-18T02:17:23.120702Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:17:23.120702Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1504.08178","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":"2026-05-18T02:17:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/nPAIrfzwVaCbWObfnPxRf/dyNeGWEabFvscvsHYzP6cPTB79Gdugn0ga64h9TS+7+dBI+JI2JDz9BQHuFLTAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T20:57:16.053113Z"},"content_sha256":"6639b37e204d3546a63509541631582842106f3ea2adf3d39ede76d801e3c944","schema_version":"1.0","event_id":"sha256:6639b37e204d3546a63509541631582842106f3ea2adf3d39ede76d801e3c944"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:PZYIFQBB2XJGS7VVZJHCS5SQFN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Numerical method based on Galerkin approximation for the fractional advection-dispersion equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.NA","authors_text":"Harendra Singh, Manas Ranjan Sahoo, Om Prakash Singh","submitted_at":"2015-04-30T11:52:50Z","abstract_excerpt":"We use a concept of weak asymptotic solution for homogeneous as well as non-homogeneous fractional advection dispersion type equations. Using Legendre scaling functions as basis, a numerical method based on Galerkin approximation is proposed. This leads to a system of fractional ordinary differential equations whose solutions in turn give approximate solution for the advection-dispersion equations of fractional order. Under certain assumptions on the approximate solutions, it is shown that this sequence of approximate solutions forms a weak asymptotic solution. Numerical examples are given to "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.08178","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":"2026-05-18T02:17:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"czsCnhl8zEh1/TODU/MtZcGgpIgbGQR50EfDRsBaep5Lb5RNW1zGdi3f8s2gwGjb32LjpkwsWFwX80itCsVLCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T20:57:16.053473Z"},"content_sha256":"0ee48918633877f920f3c794c49bc408f91be7d82568a4bebed1ae4ff0acc434","schema_version":"1.0","event_id":"sha256:0ee48918633877f920f3c794c49bc408f91be7d82568a4bebed1ae4ff0acc434"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PZYIFQBB2XJGS7VVZJHCS5SQFN/bundle.json","state_url":"https://pith.science/pith/PZYIFQBB2XJGS7VVZJHCS5SQFN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PZYIFQBB2XJGS7VVZJHCS5SQFN/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":"2026-06-22T20:57:16Z","links":{"resolver":"https://pith.science/pith/PZYIFQBB2XJGS7VVZJHCS5SQFN","bundle":"https://pith.science/pith/PZYIFQBB2XJGS7VVZJHCS5SQFN/bundle.json","state":"https://pith.science/pith/PZYIFQBB2XJGS7VVZJHCS5SQFN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PZYIFQBB2XJGS7VVZJHCS5SQFN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:PZYIFQBB2XJGS7VVZJHCS5SQFN","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":"fcc5f6e9a49147f74b2c4bb9b7ef36ed59e24ad1d75111dfd9d1144f5f83b3ed","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2015-04-30T11:52:50Z","title_canon_sha256":"2d3027d396d530d8301c23232185604373aedae00753e9a34fc57d2e76c84d06"},"schema_version":"1.0","source":{"id":"1504.08178","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.08178","created_at":"2026-05-18T02:17:23Z"},{"alias_kind":"arxiv_version","alias_value":"1504.08178v1","created_at":"2026-05-18T02:17:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.08178","created_at":"2026-05-18T02:17:23Z"},{"alias_kind":"pith_short_12","alias_value":"PZYIFQBB2XJG","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"PZYIFQBB2XJGS7VV","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"PZYIFQBB","created_at":"2026-05-18T12:29:37Z"}],"graph_snapshots":[{"event_id":"sha256:0ee48918633877f920f3c794c49bc408f91be7d82568a4bebed1ae4ff0acc434","target":"graph","created_at":"2026-05-18T02:17:23Z","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 use a concept of weak asymptotic solution for homogeneous as well as non-homogeneous fractional advection dispersion type equations. Using Legendre scaling functions as basis, a numerical method based on Galerkin approximation is proposed. This leads to a system of fractional ordinary differential equations whose solutions in turn give approximate solution for the advection-dispersion equations of fractional order. Under certain assumptions on the approximate solutions, it is shown that this sequence of approximate solutions forms a weak asymptotic solution. Numerical examples are given to ","authors_text":"Harendra Singh, Manas Ranjan Sahoo, Om Prakash Singh","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2015-04-30T11:52:50Z","title":"Numerical method based on Galerkin approximation for the fractional advection-dispersion equation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.08178","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":"sha256:6639b37e204d3546a63509541631582842106f3ea2adf3d39ede76d801e3c944","target":"record","created_at":"2026-05-18T02:17:23Z","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":"fcc5f6e9a49147f74b2c4bb9b7ef36ed59e24ad1d75111dfd9d1144f5f83b3ed","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2015-04-30T11:52:50Z","title_canon_sha256":"2d3027d396d530d8301c23232185604373aedae00753e9a34fc57d2e76c84d06"},"schema_version":"1.0","source":{"id":"1504.08178","kind":"arxiv","version":1}},"canonical_sha256":"7e7082c021d5d2697eb5ca4e2976502b68a13f6d630e9007e63c157709f22b5d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7e7082c021d5d2697eb5ca4e2976502b68a13f6d630e9007e63c157709f22b5d","first_computed_at":"2026-05-18T02:17:23.120702Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:17:23.120702Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ic1zM0W4y6ywBfi7gz4GLWzW8iBtQlrK5D/tGGaVMdxDlb+AzZBiCl2h8mVOxKIzHuOGcrv6Pehbp4r4eXKJDg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:17:23.121356Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.08178","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6639b37e204d3546a63509541631582842106f3ea2adf3d39ede76d801e3c944","sha256:0ee48918633877f920f3c794c49bc408f91be7d82568a4bebed1ae4ff0acc434"],"state_sha256":"034843fd97290a4dc256096c5f0ae9c18e7bc97399e17a4512da9a4f3e9997ef"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"C5x8yNHNFZ4pf5f9UsPLQn3cmX6juR9lyCl655JgZnLXCoCnwE3Ikyg/ksNejUJffTzOBPtgTlb8v87UJRyfDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T20:57:16.055463Z","bundle_sha256":"5e1c00e79d579aefc4cdb89736d30917268ae93b90003d085509b818aa45bb69"}}