{"paper":{"title":"Forward and inverse problems for a time-fractional pseudo-parabolic equation with variable coefficients","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Global existence and uniqueness hold for the forward problem of a time-fractional pseudo-parabolic equation with time-dependent coefficient, and global existence holds for the associated inverse source problem.","cross_cats":[],"primary_cat":"math.AP","authors_text":"Elbek Husanov, Ravshan Ashurov","submitted_at":"2026-05-13T10:01:32Z","abstract_excerpt":"In this work, forward and inverse problems for a time-fractional pseudo-parabolic equation $D_t^{\\rho} [u(t) + \\mu Au(t)] + \\sigma(t) Au(t) = r(t)g$ are investigated in a Hilbert space, where $A$ is an unbounded, positive, self-adjoint operator. According to the known papers, the forward problem has been studied only in the case $\\sigma(t) = const$. The main novelty of the forward problem in this work is that the model is further generalized and investigated for a time-dependent coefficient $\\sigma(t)$. To determine the solution of the forward problem, the Fourier method is employed, and the g"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The global existence and uniqueness of the solution to the forward problem is proved for time-dependent σ(t), and the global existence of the solution to the inverse problem is proved by applying Schauder's fixed point theorem.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The functional F is assumed to be such that the associated mapping satisfies the hypotheses of Schauder's fixed point theorem, and the operator A possesses a discrete spectrum of positive eigenvalues with the requisite regularity.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Global existence and uniqueness are proved for the forward problem with variable σ(t) and for the inverse problem of recovering the source r(t) from a general overdetermination condition F[u(t)] = Φ(t).","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Global existence and uniqueness hold for the forward problem of a time-fractional pseudo-parabolic equation with time-dependent coefficient, and global existence holds for the associated inverse source problem.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"47ede6d7ec8a80200544e3893f3822a0978f6cd32238d27980047fa8aa843f51"},"source":{"id":"2605.13285","kind":"arxiv","version":1},"verdict":{"id":"d9e40b4b-0554-470a-bf2a-4b25eece73d9","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:21:40.928286Z","strongest_claim":"The global existence and uniqueness of the solution to the forward problem is proved for time-dependent σ(t), and the global existence of the solution to the inverse problem is proved by applying Schauder's fixed point theorem.","one_line_summary":"Global existence and uniqueness are proved for the forward problem with variable σ(t) and for the inverse problem of recovering the source r(t) from a general overdetermination condition F[u(t)] = Φ(t).","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The functional F is assumed to be such that the associated mapping satisfies the hypotheses of Schauder's fixed point theorem, and the operator A possesses a discrete spectrum of positive eigenvalues with the requisite regularity.","pith_extraction_headline":"Global existence and uniqueness hold for the forward problem of a time-fractional pseudo-parabolic equation with time-dependent coefficient, and global existence holds for the associated inverse source problem."},"references":{"count":17,"sample":[{"doi":"","year":2019,"title":"Lizama,Abstract linear fractional evolution equations, inHandbook of Fractional Calculus with Applications, Vol","work_id":"4f9aa771-3f20-468a-93a9-b711b2de5716","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2005,"title":"A. V. Pskhu,Fractional Differential Equations(Moscow, Russia, Nauka, 2005)","work_id":"3992c5b3-b0af-482d-8b97-61a7be341fe3","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1968,"title":"On a theory of heat conduction involving two temperatures,","work_id":"f2a98c24-ef28-4029-95c8-4dcdf1597793","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1960,"title":"Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks,","work_id":"c22a72a7-74bb-4bb3-9d47-e8bcf31ceafd","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1980,"title":"Determination of an unknown nonhomogeneous term in a linear partial differential equation from overspecified boundary data,","work_id":"3d62c7ec-5597-4c8e-bd81-045a2ae9c66b","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":17,"snapshot_sha256":"10898965441b4c42612e13b9af49b48aabd1e0161f1185b19374ed898f463562","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"}