{"paper":{"title":"A Pedagogical Introduction to the Unified Transform Method: The Heat Equation on a Finite Interval","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Athanasios Paraskevopoulos","submitted_at":"2026-06-22T21:36:21Z","abstract_excerpt":"This paper presents a detailed application of the Unified Transform Method (Fokas method) to the one-dimensional heat equation on $[0,1]$ with Dirichlet boundary conditions. The analysis formulates the Initial-Boundary Value Problem and derives an integral representation of the solution via a generalised spatial Fourier transform with complex spectral parameter $\\lambda \\in \\mathbb{C}$, yielding the Global Relation -- an algebraic identity coupling the initial datum, prescribed boundary values, and unknown Neumann data. The unknowns are eliminated by exploiting the symmetry $\\lambda \\mapsto -\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.26148","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.26148/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}