{"paper":{"title":"Approximations with Non-Symmetric Green's Kernels and their Application to Fractional Differential Equations","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"Non-symmetric Green's kernels produce optimal-order spline interpolants for fractional differential equations in reproducing kernel Banach spaces.","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Nick Fisher","submitted_at":"2026-05-12T20:09:42Z","abstract_excerpt":"Several kernel-based methods for the numerical solution of fractional differential equations have been developed in the recent past; however, these techniques exclusively relied on the use of radial basis function approximations. In the present work, we consider the non-symmetric Green's kernel perspective on fractional order spline interpolation and its application to a kernel Galerkin method for the numerical solution of certain fractional order differential equation. Unfortunately, the reliance on a non-symmetric kernel requires that our theoretical analysis of the kernel interpolants must "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we are able to prove that the proposed kernel interpolants obtain optimal order convergence rates in a reproducing kernel Banach space.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The fractional differential operator admits a well-defined non-symmetric Green's kernel that can be used to construct the spline interpolant outside the reproducing kernel Hilbert space setting.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Non-symmetric Green's kernels yield optimal-order convergent approximations for fractional differential equations in reproducing kernel Banach spaces.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Non-symmetric Green's kernels produce optimal-order spline interpolants for fractional differential equations in reproducing kernel Banach spaces.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"4dee2898d6ee9249d37c11dbfb75d5e46b4540d939276ad3393d6be0050ddc3d"},"source":{"id":"2605.12707","kind":"arxiv","version":1},"verdict":{"id":"939a8d9f-a4b7-490b-ba30-dff91b980b31","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T19:58:06.215803Z","strongest_claim":"we are able to prove that the proposed kernel interpolants obtain optimal order convergence rates in a reproducing kernel Banach space.","one_line_summary":"Non-symmetric Green's kernels yield optimal-order convergent approximations for fractional differential equations in reproducing kernel Banach spaces.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The fractional differential operator admits a well-defined non-symmetric Green's kernel that can be used to construct the spline interpolant outside the reproducing kernel Hilbert space setting.","pith_extraction_headline":"Non-symmetric Green's kernels produce optimal-order spline interpolants for fractional differential equations in reproducing kernel Banach spaces."},"references":{"count":44,"sample":[{"doi":"","year":null,"title":"An introduction to the Hilbert-Schmidt SVD using iterated Brownian bridge kernels , author=. Numer. Algor. , volume=","work_id":"18c90fc8-034f-44ef-b703-b6513f557bf4","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Variational formulation for the stationary fractional advection dispersion equation , author=. Num. Meth. for PDE's , volume=","work_id":"b2685823-3557-4148-bfd5-63839184822a","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"M. Esmaeilbeigi and O. Chatrabgoun and M. Cheraghi , journal=. The role of","work_id":"4d8d8eaa-eac7-4b33-b9fc-dc3dfdd3a3f6","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Total Positivity , author=","work_id":"39d09f6e-54ba-4f5a-b03a-b9af0e755477","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Numerical simulation for solute transport in fractal porous media , author=. ANZIAM J. , volume=","work_id":"3e635943-dd5f-4bd5-a7ad-3ab90f2b221a","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":44,"snapshot_sha256":"0da15bc5154371d9d606616431c214aeb1818eadd81499643bcec6a6ca53e33e","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"1d846d2e0638c11e8f317068c2de4244c4172cb55018ed080706379caedb8def"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}