Fractional PDEs and Neutrosophic Structures in MR-Metric Spaces: Theory and Applications

This paper develops a unified framework that integrates fractional partial differential equations (PDEs) and neutrosophic logic within the novel setting of MR-metric spaces. Theoretically, we establish three main contributions. First, we define the fractional MR-Laplacian via spectral theory on compact MR-metric measure spaces, under explicit hypotheses on the underlying operator Δ_M (an additional datum, not determined by the MR-metric), providing a precise characterization of its domain and functional setting. Second, we prove a fractional Poincaré inequality that relates the L²-norm of a zero-mean function to its Gagliardo-type seminorm defined through the MR-metric kernel M(x,y,y)^(−2α). We give an elementary proof with an explicit constant and, under explicit hypotheses (two-sided sub-Gaussian heat kernel bounds and comparability of M(x,y,y) with the underlying metric), we prove an equivalence between the spectral fractional energy and a Gagliardo-type energy with kernel M(x,y,y)^(−(Q+αβ)), where Q and β are the volume and walk exponents. Third, we prove a general existence and uniqueness theorem for weak solutions of a fractional heat equation on compact MR-metric spaces using the Galerkin method, with energy estimates that depend only on T and on the data. In applications, we demonstrate the versatility of the framework through four concrete case studies: (1) an explicit construction of a neutrosophic MR-metric space on a three-element set, validating the abstract axioms; (2) an illustration of the fractional Poincaré inequality on a weighted circular graph, where the relevant constants are computed numerically; (3) a model of anomalous diffusion on the Sierpinski gasket, where the on-diagonal heat kernel of the fractional operator satisfies two-sided bounds of order t^(−d_s/(2α)) as t→0⁺; and (4) a formulation of semi-supervised graph learning as a fractional heat equation with the labelled nodes clamped, showing how the fractional parameter α controls the non-locality of label propagation. This work bridges fractional calculus, analysis on non-smooth metric spaces, and neutrosophic uncertainty modeling, offering a tool for both theoretical investigation and practical applications in data science and physics. The neutrosophic structure enters through the definition and the example of Section 3.1; it is not used in the analytic results of Section 2.